The locations medium height over sea level is 8 meters.
The over all population of Ekmul is 2009 persons, 996 of them are male and 1013 of them are female.
The population of this place consists of 678 minors and 1331 adults, with 173 of them being 60 years and older.
1602 inhabitants of the location Ekmul in Mexico live in indigenous households. 704 of the citizens 5 years and older, speak an indigenous language. The count of people who only speak an indigenous language without having knowledge of the Spanish language is 1 people, while 687 as well speak Spanish.
A legal claim on health care and social insurance benefits have 1142 citizens of Ekmul.
In Ekmul exist about 472 households.
Of these households 453 are common houses or apartments, 8 are without floor and about 114 consist of one room only.
188 of the normal households have sanitary installations, 381 are connected to the public water supply, 431 have access to electricity.
The economic situation allows 27 households to own a computer, 250 own a washing machine and 415 households are equipped with one ore more televisions.
Besides the 163 analphabets aged 15 or older, about 4 minors between 6 and 14 are not visiting a school.
131 inhabitants of the population of 15 years and older did not visit a school and 676 persons did not finish the school. 333 visited only the 6 years lasting primary school, 302 visited and finished the college or similar scholar institutions.
A total of 151 aged 15 to 24 years visited a school, the medium time school is visited through the whole population is 7 years.
Do You know the places close to Ekmul ?


The kitchen is not only the location for much of our household consumption, it's also the source of much of our household waste production, including one of the worst greenhouse-gas-producing waste products — food waste.
According to a National Resources Defense Council report, getting food from farm to table uses 10% of our nation's energy budget. This morning on MPR, Jonathan Foley, director of the Institute on the Environment at the University of Minnesota, said that we use about 40 percent of the (non-ice-covered) land on the planet to grow food, and "70% of all the water we consume is used to irrigate crops." Agriculture is one of the biggest contributors to climate change.
That's an extremely costly food chain and yet approximately 40% of all food produced in the world is never eaten. According to the EPA, 21% of our municipal waste is food waste. More food goes to landfills and incinerators than any other type of material.
Not only are we losing an estimated $165 billion in food that could be used to feed hungry people, but food in landfills produces methane as it decays. Methane is 21 times more potent as a greenhouse gas than carbon dioxide so this is a very big deal. Landfills are the source of 20% of all methane emissions.
With the kitchen remodel we wanted to see if we could reduce food waste — and encourage recycling — with better design. There were three locations in our kitchen were food waste occurred: the refrigerator, the fresh food bowls on the counter, the longer-term storage space.
Refrigerator: We looked at a variety of refrigerators. If money were no object, we probably would have chosen refrigerator drawers because it is easier to see all the food in the drawer. But as I noted in an earlier post, drawers are expensive and the ones we could afford were less energy efficient than standard refrigerators.
A larger countertop fruit bowl with air circulation should keep the fruit visible so it gets eaten.
We decided on a 17 cubic foot countertop-depth refrigerator. Not only did it fit our tiny space well, but because it is shallower than a standard fridge, we can see and reach to the back of each shelf. Food can't hide.
Fresh Food Bowls: We had numerous fresh fruit/veg bowls on the counter and hanging by the window. These were replaced with one multi-level countertop bowl made of wire that promotes air circulation.
Longer Storage: It's happened one too many times that we've walked into the kitchen and wondered where that awful smell was coming from, only to discover rotting onions or well-sprouted potatoes or liquid apples inside a cabinet. To prevent the problem of "out of sight, out of mind," we decided to use wire baskets on open shelves for storage of these kinds of foods.
With a smaller fridge, we need to buy less and shop more often. This means our food will be fresher and since we can ride our bike to the grocery store, we might get in more exercise, too!
When we grow our own food, we can harvest as needed and not refrigerate food at all.
Harder for me to learn is to cook the proper amount of food so we don't have a fridge overflowing with leftovers. Although it's been years since the kids left home, I've only managed to recalculate my cooking down to 4 people, not the 2 who actually comprise our household.
Not that leftovers are bad. We love them — when we remember to eat them. The key to leftovers is creating a system so you know when the food went into the fridge and then eating it before it goes bad. That's a design issue I'll tackle another time.
Our household food waste gets composted either in our worm bins, in our home compost bins, or by the City of Minneapolis organics composting program.
We wanted to ensure our design of the kitchen allowed for the various kinds of composting we do but I didn't want to fill the counter with compost containers. In the best of all possible kitchen design worlds, we would have had holes cut in the back of the counter where compost could have been pushed down into the right container for its use. But we don't live in that perfect world.
Our solution is one countertop bin that holds a combination of worm and compost-bin scraps and one taller, narrow bin under the sink for city compost. "Special" scraps, like egg shells that I'll be drying and grinding to put in my tomato bed, are put into small plastic bins under the sink until they are used.
Narrow bin holds city compost and two kitty litter bins were just the right size for garbage and recycling.
There are a lot of miscellaneous items that could be recycled … but where to put them so the kitchen doesn't become a dumping ground. We decided on a large sink cabinet (36″). With such a large space, we no longer have to look at garbage cans in the room and there is plenty of space for a compost bin, a recycling bin, and a garbage bin.
These cubbies hold items leaving the house to be returned for recycling or reuse.
Once again, design can only goes so far. We need to continue to develop better habits. In particular, we need to continually challenge ourselves to buy food with little or no packaging.
* I am not paid to advertise any products or services, nor am I promoting a product, service or company.
This entry was posted in Food/gardening, mini challenges, Reduce/reuse/recycle and tagged food waste, recycling. Bookmark the permalink.
Thanks for all you do for our neighborhood and our world. I always enjoy your posts. I had a chance to tour a composting site – the Mulch Store – and a recycling plant last week as part of the Master Recycler class run by Hennepin County. So much to learn – so little time….
Great thinking overall, but make sure you consider which fruits and veg you are storing together, especially ones that produce ethylene gas and accelerate ripening (and spoiling) of their neighbors: http://www.thekitchn.com/food-science-ethylene-gas-130275, and think about things that should be stored in the dark (like potatoes).
That is a really good point! Thanks for the link.


text:
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Outside » Outside Features
Missing Out: Why Dikembe Mutombo may have helped Boise State win last Saturday
I carry within me a deep, nagging and mostly illogical need to watch pivotal sporting events.
by Mike Bookey
I carry within me a deep, nagging and mostly illogical need to watch pivotal sporting events. If I miss a big bowl game or an NCAA tournament upset because one of my wife's college friends is getting married (this happens about every four months) or I get stuck at work or the cable goes out, I find myself feeling absent from the cultural fabric of the present tense. It's as if the sporting world has left me on the side of the road and continued on without me. Sure, I can catch the next bus and watch the 1:00 a.m. replay, but that's not the same. It's almost cheating.
This is what happened on Saturday. I missed the Oregon State vs. Boise State game. All of it. Didn't even catch the highlights. I was at the Bend Roots Revival all day Saturday (totally worth it) and finally asked someone wearing an OSU hat what happened and he gave me the score. It was both defeating and relieving all at once. On one hand, I hadn't missed what would have been one of the biggest upsets of the young season, but conversely, I live in Oregon and I enjoy when things from Oregon do well, be they beers, pregnant men or football teams.
Now here's where it gets weird. When I miss games like this and the team I'm pulling for loses, I feel just a tiny, little bit, kinda responsible. This is wholly irrational thinking; I realize that. But I have a name for it: The Mutombo Effect.
I'm, of course, referring to longtime NBA center, Dikembe Mutombo, the man who in 1994 laid on his back on the floor of Key Arena, clutching the ball above his head as he wept celebratory tears, his Denver Nuggets having pulled off a game-five upset victory to knock the Sonics out of the first round of the playoffs. I was 11 years old and a devout Sonics fan, but was also a boy whose good friend was having a birthday party at an indoor swimming pool equipped with a high dive. Figuring that Gary Payton and Shawn Kemp could overcome the jitters that had hindered their squad throughout the series and polish off the Nuggets, I opted for the party. I had to - my mom had already RSVPed and, again, they had a high dive.
I was standing on the side of the pool when my friend's dad approached. I asked the score. He shook his head and I had to hold back tears. In downtown Seattle I figured Shawn Kemp was sitting at his locker wondering why one of his biggest fans had let him down in favor of indoor swimming (and probably also how to care for his 17 children, but mostly about why I'd deserted him).
From then on, the Mutombo Effect became the only streak of superstition to, however irrationally, sneak into my life. And it haunts me at times - partially because actually believing one's television viewing schedule could affect the outcome of major sporting events is borderline psychotic behavior, but also because there will always be big games I'll miss.
Damn you, Dikembe Mutombo.
Speaking of Left Field
The Year Football Broke: The past season was at once tragic, intriguing and exciting. Thank God it's over.
Goodbye, Bend: On five years of writing about sports and other things you might have cared about
The Best We Can Do?: The anticlimactic ending to an otherwise dynamite bowl season
Latest in Outside Features
The Journey Back Om
Youth Sports, Upgraded
Monitoring Glaciers
More by Mike Bookey
A Strange Week: The weirdness of the NCAA Tournament's first two rounds
An overview of this past week in the NCAA.
Two Days in March: Or, the least productive work week of the year
Ways to respond to your boss when he/she questions you watching the game during your work hours.
Looking back on an interesting football season.
© 2020 The Source Weekly - Bend

redpajama_set_name: RedPajamaCommonCrawl


The 2017 water polo season was a tough season to say the least. Finishing up with an overall record of 11-13 and going winless in league, along with a no-show in CIF, the season was somewhat disappointing. The team faced a lot of obstacles but nonetheless fought through and played with their best effort in every game.
The 11-13 record along with no wins in league was not completely shocking: "Sure, the outcome is not what we hoped for, but there were many factors to having that record," said Varsity head coach Jeff Powers. The team had injuries and a lack of experience that ultimately factored in the outcome of the team's record.
Senior Gabe Chaine says that he has no regrets and enjoyed the time he got to play during the season. "I had fun this season…a little sadness does come to me knowing that this is my last year (at Bosco) and we didn't make a big CIF run." His last year was not what he hoped for but he is satisfied with what he left at Bosco, especially knowing that the team next is due to have a solid season.
The 2018 season is looking more promising than this year, with everybody returning but two players. The team will have the needed experience that was sorely missed this year. Coach Powers explains, "This was definitely a learning season for next year for the returning players which is basically our whole team." He goes on and adds, "The players next year will now know what is needed to compete in this league." The Trinity league is one of the best leagues in the nation and Coach Powers feels confident about his player's abilities to compete in this league.
This means that the current juniors and sophomores will have more important roles and make a bigger impact than they did this year. The 2017 season was not a complete disappointment, because this season really prepared the returning players. The team can now reflect and learn from the mistakes they made this year and correct them for the better. Therefore the team is optimistic about next year and there chances to win league and ultimately make a big CIF run in the 2018 season.


Globalization is discussed often as a relatively new phenomenon, arising sometime after the spread of the internet. But new research from Utah State University and Arizona State University published in the Proceedings of the National Academy of Sciences shows that the roots of globalization stretch back 10,000 years.
Question: How did the idea for this research come about?
Answer: Archaeologists around the world have been working to gather radiocarbon data from very old human trash for quite a long time. Recent efforts to pull together this radiocarbon data analyzed in recent decades have opened up interesting research opportunities, and that's where our work started.
The availability of this long-term data has allowed us to look at how human societies use energy over time. The question is: Is human energy use synchronized across space and time in any way? That's a proxy for trying to understand the evolution of human societies, because energy use is a proxy for the size of human societies, and the complexity of human societies.
Q: What did early human energy consumption looked like?
A: In small-scale societies — for example, hunter-gatherer groups — people used natural resources to fuel their societies and economies. If you look at early human societies around the globe, you might expect that they follow natural resource availability driven by solar energy. And as societies developed, they transitioned from solar to fossil fuels.
Q: What were your findings, and what makes those findings significant?
A: The findings are significant because they help us understand general patterns of the evolution of human societies. They help us understand that the globalization we talk about today is nothing new. It has happened in the past. The main finding is that if you look at energy consumption in societies, in different places around the world, their correlation decreases with distance. This suggests that societies' energy consumption may be linked by trade networks — networks of exchange, migration and conflict. So, energy consumption within societies is more correlated by processes within those societies rather than an outside driver, the availability of solar energy, for example.
In modern times, we're facing this issue of human societies becoming highly correlated around the world due to exactly these processes: economic exchange and migration. The entire globe is now linked, and that is a modern consequence of modern technology. Globalization has been going on for a very long time, and will continue. The scale of globalization is the only thing that is new today.


text:
Karnataka HC Sends Notice To Law Ministry for a PIL Filed on Constitutional Validity of Contempt of Court Act
By Adhiti Sharma
Adhiti Sharma
The Karnataka High Court sent a notice to the Ministry of law and Justice on Tuesday regarding a PIL that challenged the constitutional validity of Section 2(c) (1) of the Contempt of Courts Act 1971. It was challenged on the grounds that it violates articles 14 and 19 of the constitution which is vague and arbitrary.
The petition was collectively filed by journalists N Ram and Krishna Prasad, Arun Shourie who was the former Union minister, and Prashant Bhushan a Senior Advocate. The petition is likely to be heard on 22nd February 2020.
What is Section (2)(1) of Contempt of Court Act?
Section (2)(1) defines criminal contempt as the publication or performing of an act that "scandalizes or tends to scandalize, or lowers or tends to lower the authority of any court". The division bench ordered notice to the Attorney general of India as central laws like these were facing challenges based on their constitutional validity.
What did the petition say?
The Supreme Court allowed Ram and Prashant Bhushan in August, to withdraw the challenge on the clause giving them the liberty to approach any other judicial forum regarding the same but not the top court.
As several petitions were already pending on the issue the petitioners had decided to withdraw their plea earlier. It was stated in the plea before the High Court that "the petitioners are concerned about the Section (2)(1) and the chilling effect it has on the freedom of speech".
Freedom of speech can be restricted only when there are reasonable grounds for it, for example, the emergency period, etc. According to the petitioners, the provision does not amount to any reasonable restriction under Article 19(2) of the constitution. They further reason it out that, it does not pass the overbreadth test which means the section is vague and also hinders a citizen's freedom of speech. Even though proximate harm is absent.
They argued that the issue of "Scandalizing the court" is rooted in Colonial times where no-one was given the right to go against the court. According to the petitioners, such obsolete laws do not have any place in a democratic society like ours.
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text:
\section*{Introduction}

Category theory has, from the early beginning, played an important role in quantum groups. In the operator algebraic approach to quantum groups the key result connecting the two areas is due to Woronowicz~\cite{WorTK}. Generalizing the classical Tannaka-Krein duality he showed that by associating to a compact quantum group its representation category together with the canonical fiber functor, we get a duality between compact quantum groups on one side, and C$^*$-tensor categories with conjugates and a unitary fiber functor, on the other. Therefore, in principle, all properties of a compact quantum group $G$ can be formulated entirely in terms of its representation category $\Rep G$ and canonical fiber functor. Remarkably, a lot of properties depend on $\Rep G$ alone. A systematic study of such properties was made possible by Bichon, De Rijdt and Vaes~~\cite{BDRV}, who showed how, given two monoidally equivalent (that is, having equivalent representation categories) compact quantum groups, to construct a linking C$^*$-algebra connecting the two. The linking algebra is equipped with ergodic actions of both quantum groups, and by fixing one quantum group $G$ and varying the other, or in other words, by considering all possible unitary fiber functors on $\Rep G$, we get all ergodic actions of $G$ of full quantum multiplicity. This categorical point of view on actions, together with the construction of linking algebra, has been extremely successful. It has been applied to a variety of seemingly unrelated problems, from a study of random walks on discrete quantum groups~\cite{DRV} to $K$-theoretic computations~\cite{Voigt}.

The results of Bichon, De Rijdt and Vaes were later generalized by Pinzari and Ro\-berts~\cite{PR}, who showed how to describe all ergodic actions of $G$ in terms of $\Rep G$. Namely, for every ergodic action they constructed a ``spectral functor'' from $\Rep G$ into the category of Hilbert spaces, and then gave an abstract characterization of such functors. Their result implies, in particular, that isomorphism classes of ergodic actions of monoidally equivalent compact quantum groups are in canonical correspondence with each other. Soon afterwards De Rijdt and Vander Vennet~\cite{DRV} showed that the same is true even for nonergodic actions. Their argument bypasses category theory altogether and is based on induction using the linking algebra. A natural problem completing this circle of ideas is nevertheless to find a description of actions of $G$ entirely in terms of~$\Rep G$. Our goal in this paper is to do exactly that. By modifying the definition of a spectral functor and the axioms of Pinzari and Roberts, we show that actions of a compact quantum group~$G$ correspond to a class of functors, which we call weak unitary tensor functors, from $\Rep G$ into categories $\Hilb A$ of C$^*$-correspondences over C$^*$-algebras $A$. It should become apparent from our results, and is not difficult to show directly, that in the case $A={\mathbb C}$ our definitions/axioms are equivalent to the ones given by Pinzari and Roberts. Overall, the construction of a C$^*$-algebra from a functor $\Rep G\to\Hilb A$ follows familiar lines going back to Woronowicz~\cite{WorTK}. Since some of the maps involved are not adjointable, we just have to be more careful not to overuse various dualities.

A different solution to the same problem is suggested by recent work of De Commer and Yamashita~\cite{DCY}. Complementing the results of Pinzari and Roberts, they showed that ergodic actions of~$G$ can also be described in terms of semisimple $(\Rep G)$-module C$^*$-categories with a fixed simple generating object. In fact, a significant part of their arguments does not involve ergodicity/semisimplicity in any way, and we show that indeed by discarding these assumptions we get a characterization of general actions in terms of module categories. The relation between two categorical pictures can be described as follows. Given a right $(\Rep G)$-module C$^*$-category and an object $M$ in it, the functor $\Mor(M,M\otimes\cdot)$ has a canonical structure of a weak unitary tensor functor. Therefore, using the analogy with representation theory, we can say that the relation between two pictures is as between a cyclic representation and its matrix coefficient defined by the cyclic vector. From this point of view weak unitary tensor functors are categorifications of positive definite functions. It is interesting that the module category approach, being less economical than the approach via weak tensor functors, seems, nevertheless, more suitable for classification of actions, at least for representation categories described by simple universal properties~\cite{DCY2}.

In the last section of this paper we discuss some examples and applications of our general results. The construction of a C$^*$-algebra from a weak unitary tensor functor is reminiscent of various crossed product type constructions. To make the connection more explicit, we reformulate this construction in a category-free way. This will make it clear that for duals of discrete groups it generalizes such constructions as cross-sectional algebras of Fell bundles or crossed products by Hilbert bimodules. We also show that categorical point of view on actions naturally leads to a construction of deformation of C$^*$-algebras by  $2$-cocycles on discrete quantum groups.

\bigskip

\section{Spectral functors} \label{s1}

Let us first fix the notation; overall, we follow the same conventions as in \cite{NTbook}. Consider a compact quantum group $G$. The Hopf $*$-algebra of matrix coefficients of finite dimensional representations of $G$ is denoted by $({\mathbb C}[G],\Delta)$. A finite dimensional representation $U$ of $G$ is an invertible element of $B(H_U)\otimes C(G)$ such that $(\iota\otimes\Delta)(U)=U_{12}U_{13}$. The tensor product of two representations $U$ and $V$ is denoted by $U\times V$ and is defined by $U\times V=U_{13}V_{23}$.

The contragredient representation to a finite dimensional representation $U$ is defined by
$$
U^c=(j\otimes\iota)(U^{-1})\in B(H_U^*)\otimes C(G),
$$
where $j$ is the canonical anti-isomorphism $B(H_U)\cong B(H_U^*)$. When $H_U$ is a Hilbert space, we identify the dual space $H_U^*$ with the complex conjugate Hilbert space $\bar H_U$.

We denote the Woronowicz character $f_1\in{\mathbb C}[G]^*$ by $\rho$. For every finite dimensional representation~$U$ of $G$ we have a representation $\pi_U$ of the algebra ${\mathbb C}[G]^*$ on $H_U$ defined by $\pi_U(\omega)=(\iota\otimes\omega)(U)$. Given a finite dimensional unitary representation $U$ of $G$, the conjugate representation is defined by
$$
\bar U=(j(\pi_U(\rho)^{1/2})\otimes1)U^c(j(\pi_U(\rho)^{-1/2})\otimes1)\in B(\bar H_U)\otimes C(G).
$$
This is a unitary representation equivalent to $U^c$, and $\pi_{\bar U}(\rho)=j(\pi_U(\rho)^{-1})$. Unitarity of $\bar U$ essentially characterizes $\rho$: if $U$ is an irreducible unitary representation, then $\pi_U(\rho)$ is the unique strictly positive operator in $B(H_U)$ such that the above definition of $\bar U$ gives a unitary element and such that $\Tr(\pi_U(\rho))=\Tr(\pi_U(\rho)^{-1})$. We will usually suppress $\pi_U$ and simply write $\rho\xi$ for $\xi\in H_U$ instead of~$\pi_U(\rho)\xi$.

\medskip

Denote by $\Rep G$ the C$^*$-tensor category of finite dimensional unitary representations of $G$. In this category $\bar U$ is conjugate to $U$, in the sense that there exist morphisms $R_U\colon\un\to \bar U\times U$ and $\bar R_U\colon\un\to U\times \bar U$, where $\un$ is the trivial representation of $G$ on the one-dimensional space ${\mathbb C}$, such that the compositions
$$
U \xrightarrow{\iota \otimes R_U} U\otimes \bar U \otimes U \xrightarrow{\bar R^*_U \otimes \iota} U \quad \mbox{and} \quad \bar U \xrightarrow{\iota \otimes \bar R_U} \bar U \otimes U \otimes \bar U \xrightarrow{R^*_U \otimes \iota}\bar U
$$
are the identity morphisms. Using the Woronowicz character $\rho$ we can define such morphisms by
\begin{equation}\label{econj}
R_U(1)=\sum_i\bar\xi_i\otimes\rho^{-1/2}\xi_i,\ \ \bar R_U(1)=\sum_i\rho^{1/2}\xi_i\otimes\bar\xi_i,
\end{equation}
where $\{\xi_i\}_i$ is an orthonormal basis in $H_U$. Note that the above expressions do not depend on the choice of an orthonormal basis, and
$$
\|R_U\|=\|\bar R_U\|=(\dim_q U)^{1/2},
$$
where $\dim_qU=\Tr\pi_U(\rho)$ is the quantum dimension of $U$.

\medskip

Consider now a continuous left action $\theta$ of $G$ on a C$^*$-algebra $B$, so $\theta\colon B\to C(G)\otimes B$ is an injective $*$-homomorphism such that $(\Delta\otimes\iota)\theta=(\iota\otimes\theta)\theta$ and $(C(G)\otimes1)\theta(B)$ is dense in $C(G)\otimes B$. Consider the $*$-subalgebra ${\mathcal B}\subset B$ consisting of elements $x\in B$ such that $\theta(x)$ lies in the algebraic tensor product ${\mathbb C}[G]\otimes B$. Equivalently, ${\mathcal B}$ is the linear span of elements of the form $(h\otimes\iota)((a\otimes1)\theta(x))$, where $a\in {\mathbb C}[G]$, $x\in B$ and $h$ is the Haar state on $G$. We call ${\mathcal B}$ the algebra of regular elements in~$B$. It is a dense $*$-subalgebra of $B$, and $\theta$ defines a left  coaction of the Hopf $*$-algebra $({\mathbb C}[G],\Delta)$ on it. As follows from \cite[Lemma~2.5]{DCY0}, the positive map $E=(h\otimes\iota)\theta\colon B\to B^G$ is faithful on ${\mathcal B}$, in the sense that $E(x^*x)\ne0$ for every nonzero $x\in{\mathcal B}$.

Conversely, assume we have a left  coaction $\theta$ of the Hopf $*$-algebra $({\mathbb C}[G],\Delta)$ on a $*$-algebra ${\mathcal B}$. By slightly extending the definition in \cite{DCY} we say that $\theta$ is an algebraic action of $G$ if the following conditions are satisfied:
\begin{itemize}
\item[(i)] the fixed point algebra $A={\mathcal B}^G=\{x\in{\mathcal B}\mid\theta(x)=1\otimes x\}$ is a C$^*$-algebra;
\item[(ii)] the projection $E=(h\otimes\iota)\theta\colon{\mathcal B}\to A$ is positive and faithful, so $E(x^*x)\ge0$ and $E(x^*x)\ne0$ for $x\ne0$;
\item[(iii)] $E(x^*a^*ax)\le\|a\|^2E(x^*x)$ for all $a\in A$ and $x\in{\mathcal B}$.
\end{itemize}
Note that condition (iii) follows from (i) and (ii) if ${\mathcal B}$ is unital with unit $1\in A$. Note also that conditions (ii) and (iii) can be formulated by saying that ${\mathcal B}$ is a right pre-Hilbert $A$-module with inner product $\langle x,y\rangle=E(x^*y)$, and the operators of multiplication on the left by elements of $A$ are bounded.

Under the above conditions (i)-(iii) it is not difficult to show that the $*$-algebra ${\mathcal B}$ admits a unique C$^*$-completion $B$ such that $\theta$ extends to a continuous left action of the reduced form of $G$ on $B$, see \cite[Proposition~4.4]{DCY}. Namely, ${\mathcal B}$ is faithfully represented by operators of multiplication on the left on the right pre-Hilbert $A$-module~${\mathcal B}$ with inner product $\langle x,y\rangle=E(x^*y)$, and this defines a norm on~${\mathcal B}$. Note that in general the subalgebra of regular elements in the completion $B$ of ${\mathcal B}$ can be strictly larger than~${\mathcal B}$.

\medskip

Given a finite dimensional unitary representation $U$ of $G$, we can consider $H_U$ as a left comodule over $({\mathbb C}[G],\Delta)$ by defining
$$
\delta_U\colon H_U\to{\mathbb C}[G]\otimes H_U\ \ \text{by}\ \ \delta_U(\xi)=U^*_{21}(1\otimes\xi).
$$
Then, if $\theta$ is a continuous left action of $G$ on a C$^*$-algebra $B$, we can consider comodule maps $H_U\to B$. The linear span of images of all such maps is denoted by $B(U)$ and is called the spectral subspace of $B$ corresponding to $U$. Choosing representatives $U_\alpha$ of isomorphism classes of irreducible unitary representations of $G$, for the subalgebra ${\mathcal B}\subset B$ of regular elements we get
$$
{\mathcal B}=\bigoplus_\alpha B(U_\alpha).
$$
Consider the tensor product comodule $H_U\otimes{\mathcal B}$.
We denote by $(H_U\otimes B)^G$ the subcomodule of invariant vectors, so
$$
(H_U\otimes B)^G=\{X\in H_U\otimes B\mid U_{12}X_{13}=(\iota\otimes\theta)(X)\}.
$$
In other words, if $\{\xi_i\}_i$ is an orthonormal basis in $H_U$ and $U=(u_{ij})_{i,j}$ is written in the matrix form with respect to this basis, then
$$
(H_U\otimes B)^G=\{X=\sum_i\xi_i\otimes x_i\mid\theta(x_i)=\sum_ju_{ij}\otimes x_j\ \ \text{for all}\ \ i\}.
$$
Note that using Frobenius reciprocity we can identify $(H_U\otimes B)^G$ with $\Hom_G(H_{\bar U},{\mathcal B})$, but we are not going to do this. The spectral subspaces can be recovered from $(H_U\otimes B)^G$ using the canonical surjective maps
$$
\bar H_U\otimes (H_U\otimes B)^G\to B(\bar U),\ \ \bar\xi\otimes X\mapsto(\bar\xi\otimes\iota)(X),
$$
which are isomorphisms for irreducible $U$.

The spaces $(H_U\otimes B)^G$ is our main object of interest. Clearly, if $A=B^G$, then these spaces are $A$-bimodules. Furthermore, if $X=\sum_i\xi_i\otimes x_i$ and $Y=\sum_i\xi_i\otimes y_i$ are vectors in $(H_U\otimes B)^G$, then the element $\sum_i x_i^*y_i$ is $G$-invariant. Hence $(H_U\otimes B)^G$ is a right Hilbert $A$-module with inner product $\langle X,Y\rangle=\sum_i x_i^*y_i$. This inner product is independent of the choice of an orthonormal basis, and by slightly abusing notation it can be written as $\langle X,Y\rangle=X^*Y$.

Given two finite dimensional unitary representation $U$ and $V$ of $G$, we have a map
$$
(H_U\otimes B)^G\otimes (H_V\otimes B)^G\to (H_{U\times V}\otimes B)^G, \ \ X\otimes Y\mapsto X_{13}Y_{23}.
$$
In other words, if we fix orthonormal bases $\{\xi_i\}_i$ in $H_U$ and $\{\zeta_j\}_j$ in $H_V$, then for $X=\sum_i\xi_i\otimes x_i$ and $Y=\sum_j\zeta_j\otimes y_j$ we have
$$
X\otimes Y\mapsto\sum_{i,j}\xi_j\otimes\zeta_j\otimes x_iy_j.
$$
It is obvious that this map defines an isometric map $(H_U\otimes B)^G\otimes_A (H_V\otimes B)^G\to (H_{U\times V}\otimes B)^G$.

\medskip

We are now ready to define spectral functors.

\begin{definition}
Given a continuous left action of a compact quantum group $G$ on a C$^*$-algebra $B$ with fixed point algebra $A$, the associated spectral functor is the unitary functor $F$ from $\Rep G$ into the C$^*$-tensor category $\Hilb A$ of C$^*$-correspondences over $A$ defined by
$$
F(U)=(H_U\otimes B)^G\ \ \text{with inner product}\ \ \langle X,Y\rangle=X^*Y
$$
for representations $U$, and $F(T)=T\otimes\iota$ for morphisms, together with the $A$-bilinear isometries
$$
F_{2,U,V}\colon F(U)\otimes_A F(V)\to F(U\times V),\ \ X\otimes Y\mapsto X_{13}Y_{23}.
$$
\end{definition}

A few comments are in order. By a C$^*$-correspondence over $A$ we mean a right Hilbert $A$-module together with a {\em nondegenerate} left action of $A$ on it. We have to explain why the left action on $(H_U\otimes B)^G$ is nondegenerate in the nonunital case. This is a consequence of the following simple lemma.

\begin{lemma}
If $\theta$ is a continuous left action of a compact quantum group $G$ on a C$^*$-algebra~$B$, then the fixed point algebra $A=B^G$ is a nondegenerate C$^*$-subalgebra of $B$.
\end{lemma}

\bp Let $\{e_s\}_s$ be an approximate unit in $A$. Define an $A$-valued inner product on ${\mathcal B}$ by $\langle x,y\rangle=E(x^*y)$. Then $xe_s\to x$ in the norm defined by this inner product for every $x\in{\mathcal B}$. By \cite[Lemma~2.5]{DCY0}, on every spectral subspace $B(U)\subset{\mathcal B}$ the norm defined by the inner product is equivalent to the C$^*$-norm. Therefore $xe_s\to x$ in the C$^*$-norm for every $x\in{\mathcal B}$, hence for every $x\in B$.
\ep

C$^*$-correspondences over $A$ form a C$^*$-tensor category $\Hilb A$ with adjointable $A$-bilinear maps as morphisms. We emphasize that the isometries $F_{2,U,V}$ in the definition of the spectral functor are not claimed to be adjointable, and therefore formally they are not morphisms in $\Hilb A$.

Finally, recall that two natural notions of isometry between Hilbert modules coincide: if~$M$ and~$N$ are right Hilbert $A$-modules, and $T\colon M\to N$ is an $A$-linear map such that $\|TX\|=\|X\|$ for all~$X\in M$, then $\langle TX,TY\rangle=\langle X,Y\rangle$ for all $X,Y\in M$, see e.g.~\cite[Theorem~3.5]{Lance}.

\bigskip

\section{Weak tensor functors}\label{s2}

Our goal is to give an abstract characterization of spectral functors. Here is the main definition.

\begin{definition}\label{dwutf}
Given a C$^*$-algebra $A$ and a strict C$^*$-tensor category ${\mathcal C}$ with unit object $\un$, by a weak unitary tensor functor ${\mathcal C}\to\Hilb A$ we mean a linear functor $F\colon{\mathcal C}\to\Hilb A$ together with natural $A$-bilinear isometries $F_2=F_{2,U,V}\colon F(U)\otimes_A F(V)\to F(U\otimes V)$ such that the following conditions are satisfied:
\begin{itemize}
\item[(i)] $F(\un)=A$;
\item[(ii)] $F(T)^*=F(T^*)$ for any morphism $T$ in ${\mathcal C}$;
\item[(iii)] $F_2\colon A\otimes_AF(U)\to F(\un\otimes U)=F(U)$ maps $a\otimes X$ into $aX$, and similarly $F_2\colon F(U)\otimes_A A\to F(U)$ maps $X\otimes a$ into $Xa$;
\item[(iv)] the diagrams
\begin{equation*}
\xymatrix{ F(U)\otimes_A F(V)\otimes_A F(W) \ar[d]_{\iota\otimes F_2}
\ar[r]^{\ \ F_2\otimes\iota}
 & F(U\otimes V)\otimes_A F(W)\ar[d]^{F_2}\\
F(U)\otimes_A F(V\otimes W) \ar[r]^{\ \ \ F_2}
& F(U\otimes V\otimes W)}
\end{equation*}
commute;
\item[(v)] for all objects $U$ and $V$ in ${\mathcal C}$ and every vector $X\in F(U)$, the right $A$-linear map $S_X=S_{X,V}\colon F(V)\to F(U\otimes V)$ mapping $Y\in F(V)$ into $F_2(X\otimes Y)$ is adjointable, and the diagrams
\begin{equation*}
\xymatrix{ F(U\otimes V)\otimes_A F(W) \ar[d]_{S^*_X\otimes\iota}
\ar[r]^{\ \ \ F_2}
 & F(U\otimes V \otimes W)\ar[d]^{S^*_X}\\
F(V)\otimes_A F(W) \ar[r]^{\ \ \ F_2}
& F(V\otimes W)}
\end{equation*}
commute.
\end{itemize}
\end{definition}

Note that any unitary tensor functor ${\mathcal C}\to\Hilb A$ defines a weak unitary tensor functor. In other words, if conditions (i)-(iv) are satisfied and the maps $F_2$ are surjective, then condition (v) is also satisfied. Indeed, the map $S_X\colon F(V)\to F(U\otimes V)$ is adjointable, because by assumption $F_2$ is unitary and the map $Y\mapsto X\otimes Y$ is adjointable, with adjoint given by $X'\otimes Y'\mapsto \langle X,X'\rangle Y'$. Since
$$
S_XF_2=F_2(S_X\otimes\iota)\colon F(V)\otimes_A F(W)\to F(U\otimes V\otimes W),
$$
by taking the adjoints we get $F_2^*S^*_X=(S^*_X\otimes\iota)F_2^*$. This is equivalent to commutativity of the diagram in (v) by unitarity of $F_2$.

Note also that if we consider $F$ simply as a functor into the category of vector spaces, then $S_X$ is a natural transformation from $F$ to $F(U\otimes\cdot)$, and so $S_X^*$ is a natural transformation from $F(U\otimes\cdot)$ to~$F$, or in other words,
\begin{equation} \label{enats}
S_X^*F(\iota\otimes T)=F(T)S^*_X
\end{equation}
for morphisms $T$ in ${\mathcal C}$.

\medskip

Given a continuous left action of a compact quantum group $G$ on a C$^*$-algebra $B$ with fixed point algebra $A$, the associated spectral functor $\Rep G\to\Hilb A$ is a weak unitary tensor functor. Indeed, properties (i)-(iv) are immediate, while (v) follows by observing that the adjoint of the map
$$
S_X\colon (H_V\otimes B)^G\to (H_{U\times V}\otimes B)^G, \ \ Y\mapsto X_{13}Y_{23},
$$
is given by $S^*_XZ=X^*_{13}Z$. In other words, if $X=\sum_i\xi_i\otimes x_i$ and $Z=\sum_{i,j}\xi_i\otimes\zeta_j\otimes z_{ij}$ for orthonormal bases $\{\xi_i\}_i$ in $H_U$ and $\{\zeta_j\}_j$ in $H_V$, then
\begin{equation}\label{esxstar}
S^*_XZ=\sum_{i,j}\zeta_j\otimes x_i^*z_{ij}\in (H_V\otimes B)^G.
\end{equation}

The following is our main result.

\begin{theorem}\label{tmain}
Assume $G$ is a reduced compact quantum group and $A$ is a C$^*$-algebra. Then by associating to an action of $G$ on a C$^*$-algebra its spectral functor we get a bijection between isomorphism classes of triples $(B,\theta,\psi)$, where $\theta$ is a continuous left action of $G$ on a C$^*$-algebra $B$ and $\psi\colon A\to B$ is an embedding such that $B^G=\psi(A)$, and natural unitary monoidal isomorphism classes of weak unitary tensor functors $\Rep G\to\Hilb A$.
\end{theorem}

In the proof we will identify $A$ with $\psi(A)$ and simply talk about actions with fixed point algebra~$A$.

\medskip

The main part of the proof is, of course, a construction of an action from a weak unitary tensor functor $F\colon\Rep G\to\Hilb A$. We will define this action in a series of lemmas.

\medskip

Choose representatives $U_\alpha$ of isomorphism classes of irreducible unitary representations of $G$, and write~$H_\alpha$ instead of $H_{U_\alpha}$ for the underlying Hilbert spaces. We assume that there exists an index $e$ such that $U_e=\un$. Consider the space
$$
{\mathcal B}_F=\bigoplus_\alpha\bar H_\alpha\otimes F(U_\alpha).
$$
It will also be convenient to consider a much larger space. Choose a small C$^*$-tensor subcategory ${\mathcal C}\subset\Rep G$ containing the objects $U_\alpha$, and then put
$$
\tilde{\mathcal B}_F=\bigoplus_U\bar H_U\otimes F(U),
$$
where the summation is over all objects in ${\mathcal C}$. We have a canonical linear map $\pi\colon\tilde{\mathcal B}_F\to{\mathcal B}_F$ defined as follows. For a finite dimensional unitary representation $U$ of $G$, choose isometries $w_i\in\Mor(U_{\alpha_i},U)$ such that $\sum_iw_iw_i^*=\iota$. Then put
$$
\pi(\bar\xi\otimes X)=\sum_i\bar w^*_i\bar\xi\otimes F(w_i^*)X,
$$
where $\bar w_i\bar\zeta=\overline{w_i\zeta}$, so $\bar w^*_i\bar\xi=\overline{w_i^*\xi}$. This definition is independent of the choice of isometries $w_i$, since for any other choice $v_j$ there exists a unitary matrix $(u_{ij})_{i,j}$ such that $w_i=\sum_ju_{ij}v_j$. One property of~$\pi$ that we will regularly use, is that if $\bar\xi\otimes X\in\bar H_U\otimes F(U)$ and $w\in\Mor(U,V)$ is an isometry, then
\begin{equation} \label{epi}
\pi\big(\overline{w\xi}\otimes F(w)X\big)=\pi(\bar\xi\otimes X).
\end{equation}

Define a product on $\tilde{\mathcal B}_F$ by
$$
(\bar\xi\otimes X)\cdot(\bar\zeta\otimes Y)=\overline{(\xi\otimes\zeta)}\otimes F_2(X\otimes Y).
$$
It is immediate that this product is associative. Considering ${\mathcal B}_F$ as a subspace of $\tilde{\mathcal B}_F$, we define a product on ${\mathcal B}_F$ by
$$
xy=\pi(x\cdot y)\ \ \text{for}\ \ x,y\in{\mathcal B}_F.
$$

\begin{lemma}
The map $\pi\colon\tilde{\mathcal B}_F\to{\mathcal B}_F$ is a homomorphism, hence the product on ${\mathcal B}_F$ is associative.
\end{lemma}

\bp We have to check that $\pi(\pi(x)\cdot\pi(y))=\pi(x\cdot y)$ for all $x,y\in\tilde{\mathcal B}_F$. Take $x=\bar\xi\otimes X\in\bar H_U\otimes F(U)$, $y=\bar\zeta\otimes Y\in\bar H_V\otimes F(V)$ and choose isometries $u_i\in\Mor(U_{\alpha_i},U)$, $v_j\in\Mor(U_{\alpha_j},V)$ and $w_{ijk}\in\Mor(U_{\alpha_k},U_{\alpha_i}\times U_{\alpha_j})$ defining decompositions of $U$, $V$ and $U_{\alpha_i}\times U_{\alpha_j}$ into irreducibles. Then
$$
\pi(\pi(x)\cdot\pi(y))=\sum_{i,j,k}\overline{w_{ijk}^*(u_i^*\xi\otimes v_j^*\zeta)}\otimes F(w^*_{ijk})F_2(F(u_i^*)X\otimes F(v_j^*)Y),
$$
while
$$
\pi(x\cdot y)=\sum_{i,j,k}\overline{w_{ijk}^*(u_i^*\xi\otimes v_j^*\zeta)}\otimes F(w^*_{ijk}(u^*_i\otimes v_j^*))F_2(X\otimes Y).
$$
By naturality of $F_2$ these expressions are equal.
\ep

We can identify the space $\bar H_{e}\otimes F(U_e)=\bar{\mathbb C}\otimes A\subset{\mathcal B}_F$ with $A$. Under this identification, the space $A$, with its original product, becomes a subalgebra of ${\mathcal B}_F$. Furthermore, the left and right multiplications on $\bar H_\alpha\otimes F(U_\alpha)$ by elements of $A\subset {\mathcal B}_F$ are defined by the $A$-bimodule structure on~$F(U_\alpha)$, that is, the product on ${\mathcal B}_F$ has the property
$$
a(\bar\xi\otimes X)=\bar\xi\otimes aX,\ \ (\bar\xi\otimes X)a=\bar\xi\otimes Xa.
$$

Our next goal is to define an involution on ${\mathcal B}_F$. For a finite dimensional unitary representation $U$ of $G$ consider the standard solution $(R_U,\bar R_U)$ of the conjugate equations for $U$ defined by~\eqref{econj}.

\begin{lemma}\label{lbullet}
For every $X\in F(U)$ there exists a unique element $X^\bullet\in F(\bar U)$ such that
$$
\langle X^\bullet,Y\rangle=F(\bar R_U^*)F_2(X\otimes Y)\ \ \text{for all}\ \ Y\in F(\bar U).
$$
If the C$^*$-algebra $A$ is unital, then $X^\bullet=S^*_XF(\bar R_U)(1)$. We also have
$$
\langle X,Y\rangle=F(R_U^*)F_2(X^\bullet\otimes Y)\ \ \text{for all}\ \ Y\in F(U).
$$
\end{lemma}

\bp The uniqueness is clear. In order to prove the existence assume first that $A$ is unital. We then have
$$
\langle S^*_XF(\bar R_U)(1),Y\rangle=\langle F(\bar R_U)(1),F_2(X\otimes Y)\rangle
=F(\bar R_U^*)F_2(X\otimes Y),
$$
so $X^\bullet=S^*_XF(\bar R_U)(1)$. If $A$ is nonunital, then a similar computation shows that for any $X\in F(U)$ and $a\in A$ the element $(aX)^\bullet$ exists and $(aX)^\bullet=S^*_XF(\bar R_U)(a^*)$. But this is enough, since by Cohen's factorization theorem any element of $F(U)$ has the form $aX$.

To prove the last statement in the formulation, assume once again that $A$ is unital, the nonunital case requires only a minor modification. For $Y\in F(U)$ we compute:
\begin{align*}
F(R^*_U)F_2(X^\bullet\otimes Y)
&=F(R^*_U)F_2(S^*_XF(\bar R_U)(1)\otimes Y)\\
&=F(R^*_U)S^*_XF_2(F(\bar R_U)(1)\otimes Y)\\
&=S^*_XF(\iota\otimes R^*_U)F(\bar R_U\otimes\iota)Y&\text{by \eqref{enats}}\\
&=S^*_XY.
\end{align*}
Here $S_X$ is the map $A\to F(U)$, $a\mapsto F_2(a\otimes X)=aX$, so $S^*_XY=\langle X,Y\rangle$.
\ep

This lemma implies that the correspondences $F(U)$ and $F(\bar U)$ are, in some sense, dual to each other. In general, this is not the duality in the C$^*$-categorical sense. Already the simplest examples, such as the spectral functor associated with the action of $\T$ by rotations on the unit disk, show that the objects $F(U)$ do not necessarily have conjugates in $\Hilb A$.

Similarly, for every vector $\xi\in H_U$ define a vector $\xi^\bullet\in H_{\bar U}$ by
$$
\xi^\bullet=(\iota\otimes\bar\xi)R_U(1)=\overline{\rho^{-1/2}\xi},\ \text{so}\ \ (\zeta,\xi^\bullet)=R^*_U(\zeta\otimes\xi)\ \ \text{for all}\ \ \zeta\in H_{\bar U}.
$$
Define an anti-linear map $\tilde{\mathcal B}_F\to\tilde{\mathcal B}_F$, $x\mapsto x^\bullet$, by
$$
(\bar\xi\otimes X)^\bullet=\overline{\xi^\bullet}\otimes X^\bullet.
$$
For $x\in{\mathcal B}_F$ put $x^*=\pi(x^\bullet)$. On $A\subset{\mathcal B}_F$ this clearly coincides with the involution on $A$. Although we will not need this, we remark that it is not difficult to show that the particular choice of solutions $(R_U,\bar R_U)$ was not important for defining the involution on ${\mathcal B}_F$, in the sense that for every $x\in\tilde {\mathcal B}_F$ the element $\pi(x^\bullet)$ is independent of any choices.

\begin{lemma}
The map $x\mapsto x^*$ defines an involution on the algebra ${\mathcal B}_F$, and for every $x\in\tilde{\mathcal B}_F$ we have $\pi(x)^*=\pi(x^\bullet)$.
\end{lemma}

\bp We start by proving the second part. We have to show that $\pi(\pi(x)^\bullet)=\pi(x^\bullet)$. Take an element $x=\bar\xi\otimes X\in\bar H_U\otimes F(U)$. Choose isometries $w_i\in\Mor(U_{\alpha_i},U)$ defining a decomposition of $U$ into irreducibles. Write $R_i$ for $R_{U_{\alpha_i}}$ and $\bar R_i$ for $\bar R_{U_{\alpha_i}}$. Then $R_U=\sum_i(\bar w_i\otimes w_i)R_i$ and $\bar R_U=\sum_i(w_i\otimes \bar w_i)\bar R_i$. For any $Y\in F(\bar U)$ we have
$$
F(\bar R_U^*)F_2(X\otimes Y)=\sum_iF(\bar R_i^*)F_2(F(w_i^*)X\otimes F(\bar w_i^*) Y)
=\sum_i\langle (F(w_i^*)X)^\bullet,F(\bar w_i^*) Y\rangle,
$$
so $X^\bullet=\sum_i F(\bar w_i)(F(w_i^*)X)^\bullet$. We also have $\xi^\bullet=\sum_i \bar w_i(w_i^*\xi)^\bullet$. Therefore
$$
x^\bullet=\sum_i \bar{\bar w}_i\overline{(w_i^*\xi)^\bullet}\otimes F(\bar w_i)(F(w_i^*)X)^\bullet.
$$
Applying $\pi$ and using \eqref{epi} we get
$$
\pi(x^\bullet)=\sum_i\pi\big(\overline{(w_i^*\xi)^\bullet}\otimes(F(w_i^*)X)^\bullet\big)
=\pi(\pi(x)^\bullet).
$$

We next prove anti-multiplicativity of the map $*$ on ${\mathcal B}_F$. For this it suffices to check that for all $x,y\in\tilde{\mathcal B}_F$ we have $\pi((x\cdot y)^\bullet)=\pi(y^\bullet\cdot x^\bullet)$. Take $x=\bar\xi\otimes X\in\bar H_U\otimes F(U)$ and $y=\bar\zeta\otimes Y\in\bar H_V\otimes F(V)$. The unitary $\sigma\colon H_{\bar V}\otimes H_{\bar U}\to H_{\overline{U\times V}}$ mapping $\bar\eta\otimes\bar\vartheta$ into $\overline{\vartheta\otimes\eta}$ defines an equivalence between $\bar V\times\bar U$ and $\overline{U\times V}$, and we have
$$
R_{U\times V}=(\sigma\otimes\iota\otimes\iota)(\iota\otimes R_U\otimes\iota)R_V\ \ \text{and}\ \ \bar R_{U\times V}=(\iota\otimes\iota\otimes\sigma)(\iota\otimes \bar R_V\otimes\iota)\bar R_U.
$$
Assuming that $A$ is unital we compute:
\begin{align*}
F_2(X\otimes Y)^\bullet &=S_{F_2(X\otimes Y)}^*F(\bar R_{U\times V})(1)&\text{by Lemma~\ref{lbullet}}\\
&=S^*_YS_X^*F(\iota\otimes\iota\otimes\sigma)F(\iota\otimes \bar R_V\otimes\iota)F(\bar R_U)(1)&\text{as\ \ }S_{F_2(X\otimes Y)}=S_XS_Y \\
&=F(\sigma)S^*_YF(\bar R_V\otimes\iota)S^*_XF(\bar R_U)(1)&\text{by \eqref{enats}}\\
&=F(\sigma)S^*_YF(\bar R_V\otimes\iota)(X^\bullet)\\
&=F(\sigma)S^*_YF_2(F(\bar R_V)(1)\otimes X^\bullet)\\
&=F(\sigma)F_2(S^*_YF(\bar R_V)(1)\otimes X^\bullet )\\
&=F(\sigma)F_2(Y^\bullet\otimes X^\bullet).
\end{align*}
In the nonunital case we get the same identity by replacing $X$ and $Y$ by elements of the form $aX$ and $bY$, see the proof of Lemma~\ref{lbullet}. We also have $(\xi\otimes\zeta)^\bullet=\sigma(\zeta^\bullet\otimes\xi^\bullet)$. Therefore
$$
(x\cdot y)^\bullet=(\bar\sigma\otimes F(\sigma))\big(\overline{(\zeta^\bullet\otimes\xi^\bullet)}\otimes F_2(Y^\bullet\otimes X^\bullet)\big)=(\bar\sigma\otimes F(\sigma))(y^\bullet\cdot x^\bullet).
$$
Applying $\pi$ we get $\pi((x\cdot y)^\bullet)=\pi(y^\bullet\cdot x^\bullet)$.

\smallskip

It remains to show that the map $x\mapsto x^*$ on ${\mathcal B}_F$ is involutive. Equivalently, we have to show that $\pi(x^{\bullet\bullet})=\pi(x)$ for all $x\in\tilde{\mathcal B}_F$. Take an element $x=\bar\xi\otimes X\in\bar H_U\otimes F(U)$. Consider the unitary $u\colon H_U\to H_{\bar{\bar U}}$ mapping $\zeta$ into $\bar{\bar\zeta}$. Then $\bar R_{\bar U}=(\iota\otimes u)R_U$. Hence, for any $Y\in F(\bar{\bar U})$,
$$
\langle X^{\bullet\bullet},Y\rangle=F(\bar R_{\bar U}^*)F_2(X^\bullet\otimes Y)
=F(R^*_U)F_2(X^\bullet\otimes F(u^*)Y)=\langle X,F(u^*)Y\rangle,
$$
where the last equality follows from Lemma~\ref{lbullet}. Thus $X^{\bullet\bullet}=F(u)X$. We also have $\xi^{\bullet\bullet}=\bar{\bar\xi}=u\xi$. Therefore
$$
x^{\bullet\bullet}=(\bar u\otimes F(u))x,
$$
and applying $\pi$ we get $\pi(x^{\bullet\bullet})=\pi(x)$.
\ep

We next define a linear map $\theta_F\colon{\mathcal B}_F\to{\mathbb C}[G]\otimes{\mathcal B}_F$ by
$$
\theta_F(\bar\xi\otimes X)=(U^c_\alpha)^*_{21}(1\otimes\bar\xi\otimes X)\ \ \text{for}\ \ \bar\xi\otimes X\in\bar H_\alpha\otimes F(U_\alpha).
$$
In other words, if we fix an orthonormal basis $\{\xi_i\}_i$ in $H_\alpha$ and write $U_\alpha$ as a matrix $(u_{ij})_{i,j}$, then
$$
\theta_F(\bar\xi_i\otimes X)=\sum_ju_{ij}\otimes\bar\xi_j\otimes X.
$$

\begin{lemma}
The map $\theta_F$ defines a left algebraic action of $G$ on ${\mathcal B}_F$ with fixed point algebra $A$.
\end{lemma}

\bp Clearly, the map $\theta_F$ turns ${\mathcal B}_F$ into a comodule over $({\mathbb C}[G],\Delta)$ with fixed point subcomodule~$A$.

\smallskip

In order to show that $\theta_F$ is a homomorphism, observe first that we have a left comodule structure $\tilde\theta_F\colon\tilde{\mathcal B}_F\to {\mathbb C}[G]\otimes\tilde{\mathcal B}_F$ on $\tilde{\mathcal B}_F$ defined in the same way as for ${\mathcal B}_F$, so $\theta_F(\bar\xi\otimes X)=(U^c)^*_{21}(1\otimes\bar\xi\otimes X)$ for $\bar\xi\otimes X\in\bar H_U\otimes F(U)$. Then $\pi\colon\tilde{\mathcal B}\to{\mathcal B}_F$ is a comodule map, since if $w\in\Mor(U,V)$, then $U^{c*}(\bar w\otimes1)=(\bar w\otimes1)V^{c*}$. Using that $(U\times V)^{c*}=(U^{c})^*_{13}(V^{c})^*_{23}$, modulo identification of $\overline{H_U\otimes H_V}$ with $\bar H_U\otimes\bar H_V$, it is easy to see that $\tilde\theta_F$ is a homomorphism. Hence $\theta_F$ is also a homomorphism.

\smallskip

Next let us check that $\theta_F$ is $*$-preserving. It suffices to show that $\tilde\theta_F(x)^{*\otimes\bullet}=\tilde\theta_F(x^\bullet)$ for $x\in \bar H_U\otimes F(U)\subset\tilde{\mathcal B}_F$. Fixing an orthonormal basis $\{\xi_i\}_i$ in $H_U$ and identifying $\bar{\bar H}_U$ with $H_U$, we get
$$
\tilde\theta_F(\bar\xi_i\otimes X)^{*\otimes\bullet}=\sum_ju^*_{ij}\otimes\rho^{-1/2}\xi_j\otimes X^\bullet\ \ \text{and}\ \ \tilde\theta_F((\bar\xi_i\otimes X)^\bullet)=(\bar U^c)^{*}_{21}(1\otimes\rho^{-1/2}\xi_i\otimes X^\bullet).
$$
Since $\bar U^{c*}=(\rho^{-1/2}\otimes1)U^*(\rho^{1/2}\otimes1)$, these expressions coincide.

\smallskip

It remains to show that ${\mathcal B}_F$ is a right pre-Hilbert $A$-module with inner product $\langle x,y\rangle=E(x^*y)$, where  $E=(h\otimes\iota)\theta_F$, and the left action of $A$ on ${\mathcal B}_F$ by multiplication is bounded. This will follow immediately, if we can show that the spaces $\bar H_\alpha\otimes F(U_\alpha)$ are mutually orthogonal and
$$
\langle\bar\xi\otimes X,\bar\zeta\otimes Y\rangle=\frac{1}{\dim_qU_\alpha}(\rho^{-1}\xi,\zeta)\langle X,Y \rangle\ \ \text{for}\ \ \xi,\zeta\in H_\alpha,\ X,Y\in F(U_\alpha).
$$
Note that if $z=\bar\eta\otimes Z\in\bar H_U\otimes F(U)$, then $E(\pi(z))=\sum_i\overline{w^*_i\eta}\otimes F(w^*_i)Z$, where $w_i\in\Mor(\un,U)$ are isometries such that $\sum_iw_iw_i^*$ is the projection onto the isotypic component of $U$ corresponding to the trivial representation. This clearly implies mutual orthogonality of the spaces $\bar H_\alpha\otimes F(U_\alpha)$. If $U=\bar U_\alpha\times U_\alpha$, then the only isometry in $\Mor(\un,U)$, up to a phase factor, is $(\dim_qU_\alpha)^{-1/2}R_\alpha$, where $R_\alpha=R_{U_\alpha}$. Therefore for $\xi,\zeta\in H_\alpha$ and $X,Y\in F(U_\alpha)$ we have
$$
\langle\bar\xi\otimes X,\bar\zeta\otimes Y\rangle=\frac{1}{\dim_qU_\alpha}\overline{R^*_\alpha(\xi^\bullet\otimes\zeta)}F(R^*_\alpha)F_2(X^\bullet\otimes Y).
$$
By Lemma~\ref{lbullet} we have $F(R^*_\alpha)F_2(X^\bullet\otimes Y)=\langle X,Y\rangle$. Using that $\xi^\bullet=\overline{\rho^{-1/2}\xi}$ it is also straightforward to check that $R^*_\alpha(\xi^\bullet\otimes\zeta)=(\zeta,\rho^{-1}\xi)$. This finishes the proof of the lemma.
\ep

As we discussed in the previous section, an algebraic action of $G$ on ${\mathcal B}$ uniquely defines a completion~$B_F$ of~${\mathcal B}_F$ carrying a continuous action of the reduced form of $G$. Therefore the previous lemma finishes our construction of a continuous action from a weak unitary tensor functor.

\bp[Proof of Theorem~\ref{tmain}]
It is clear that isomorphic actions produce  naturally unitarily monoidally  isomorphic weak unitary tensor functors, and naturally unitarily  monoidally isomorphic weak unitary tensor functors produce isomorphic actions. It remains to show that up to isomorphisms the constructions are inverse to each other.

\smallskip

Assume $\theta$ is a continuous left action of $G$ on a C$^*$-algebra $B$ with fixed point algebra $A$. Let $F$ be the associated spectral functor and ${\mathcal B}\subset B$ be the subalgebra of regular elements. Consider the algebraic action $\theta_F$ of $G$ on ${\mathcal B}_F$ defined by $F$ as described above. We have a linear isomorphism
$$
{\mathcal B}_F\cong{\mathcal B}\ \ \text{mapping}\ \ \pi(\bar\xi\otimes X)\in{\mathcal B}_F\ \ \text{into}\ \ (\bar\xi\otimes\iota)(X)\in{\mathcal B}
$$
for $\bar\xi\otimes X\in\bar H_U\otimes (H_U\otimes B)^G$.
It is easy to see that this is a $G$-equivariant isomorphism of algebras. It is a bit less obvious that this isomorphism is $*$-preserving. In order to show this, fix an irreducible representation $U_\alpha$ and an orthonormal basis $\{\xi_i\}_i$ in $H_\alpha$. Consider an element $X=\sum_i\xi_i\otimes x_i\in (H_\alpha\otimes B)^G$. Writing $\bar R_\alpha$ for $\bar R_{U_\alpha}$, assuming for simplicity that $A$ is unital and using Lemma~\ref{lbullet} and identity~\eqref{esxstar} for $S_X^*$, we get
$$
X^\bullet=S^*_XF(\bar R_\alpha)(1)=S^*_X\left(\sum_j\rho^{1/2}\xi_j\otimes\bar\xi_j\otimes1\right)
=\sum_{i,j}(\rho^{1/2}\xi_j,\xi_i)\bar\xi_j\otimes x^*_i=\sum_i\overline{\rho^{1/2}\xi_i}\otimes x_i^*.
$$
From this we see that the image of the element $(\bar\xi\otimes X)^*=\pi\Big(\overline{\overline{\rho^{-1/2}\xi}}\otimes X^\bullet\Big)\in{\mathcal B}_F$ in ${\mathcal B}$ equals
$$
\sum_i(\rho^{-1/2}\xi,\rho^{1/2}\xi_i)x_i^*=\left(\sum_i(\xi_i,\xi)x_i\right)^*,
$$
so the isomorphism ${\mathcal B}_F\cong{\mathcal B}$ is indeed $*$-preserving.

\smallskip

Now conversely, assume we start with a weak unitary tensor functor $F$, consider the action $\theta_F$ of $G$ on $B_F$, and define the corresponding spectral functor $F'$. It is easy to see that if we fix an irreducible representation $U_\alpha$ and an orthonormal basis $\{\xi_i\}_i$ in $H_\alpha$, then the dense subspace $(H_\alpha\otimes{\mathcal B}_F)^G$ of $F'(U_\alpha)=(H_\alpha\otimes B_F)^G$ consists of vectors of the form $\sum_i(\xi_i\otimes\bar\xi_i\otimes X)$, with $X\in F(U_\alpha)$. We have the obvious $A$-bilinear map $F(U_\alpha)\to F'(U_\alpha)$ with dense image, mapping $X$ into $\sum_i(\xi_i\otimes\bar\xi_i\otimes X)$. Let us check that this map is isometric. Taking vectors $X'=\sum_i(\xi_i\otimes\bar\xi_i\otimes X)$ and $Y'=\sum_i(\xi_i\otimes\bar\xi_i\otimes Y)$ in $F'(U_\alpha)$, and writing $R_\alpha$ for $R_{U_\alpha}$, we compute:
$$
\langle X',Y'\rangle=\sum_i(\bar\xi_i\otimes X)^*(\bar\xi_i\otimes Y)
=\pi\left(\sum_i(\overline{\overline{\rho^{-1/2}\xi_i}}\otimes X^\bullet)\cdot(\bar\xi_i\otimes Y)\right)
=\pi\big(\overline{R_\alpha(1)}\otimes F_2(X^\bullet\otimes Y)\big).
$$
Since, up to a scalar factor, $R_\alpha$ is an isometry in $\Mor(\un,\bar U_\alpha\times U_\alpha)$, the last expression equals
$$
F(R^*_\alpha)F_2(X^\bullet\otimes Y)=\langle X,Y\rangle
$$
by Lemma~\ref{lbullet}. Thus we get unitary isomorphisms $F(U_\alpha)\cong F'(U_\alpha)$. These isomorphisms for all $\alpha$ extend uniquely to a natural unitary isomorphism between the functors $F$ and $F'$. It is straightforward to check that this isomorphism is monoidal.
\ep

\bigskip

\section{Module categories}

In this section we will give a different categorical description of actions in terms of module categories. Recall that given a C$^*$-tensor category ${\mathcal C}$, a right ${\mathcal C}$-module C$^*$-category is a C$^*$-category~${\mathcal M}$ equipped with a bilinear unitary functor $\otimes\colon {\mathcal M}\times{\mathcal C}\to{\mathcal M}$ together with natural unitary isomorphisms $\phi\colon (M\otimes U)\otimes V\to M\otimes (U\otimes V)$ and $e\colon M\otimes\un\to M$ satisfying certain coherence relations, see~\cite{DCY} for details. If ${\mathcal C}$ is strict, a module category ${\mathcal M}$ is called strict if $\phi$ and $e$ are the identity morphisms. Any module category over a strict C$^*$-tensor category is equivalent to a strict one.

In the following discussion we will tacitly assume that the C$^*$-categories that we consider have subobjects, meaning that for every projection $p$ in $\End(M)$ there exists an object $N$ and an isometry $v\in \Mor(N,M)$ such that $vv^*=p$. This is a very mild assumption, as we can always complete a C$^*$-category with respect to subobjects.

\smallskip

Assume we are given a continuous left action of a reduced compact quantum group $G$ on a {\em unital} C$^*$-algeb\-ra~$B$. Following~\cite{DCY}, consider the category ${\mathcal D}_B$ of unitary $G$-equivariant finitely generated right Hilbert $B$-modules. By definition, the morphisms in ${\mathcal D}_B$ are $G$-equivariant maps of Hilbert $B$-modules. Since we consider only finitely generated Hilbert modules, such maps are automatically adjointable, so ${\mathcal D}_B$ is a C$^*$-category. It is a strict right $(\Rep G)$-module C$^*$-category: given a right Hilbert $B$-module $M$ with the action of~$G$ given by an isometry $\delta_M\colon M\to C(G)\otimes M$, we define $M\otimes U$ as the Hilbert $B$-module $M\otimes H_U$ with the action of $G$ given by $x\otimes \xi\mapsto U^*_{31}(\delta_M(x)\otimes\xi)$. Note that for $M=B$ the module $B\otimes U$ is, up to identification of $H_U\otimes B$ with $B\otimes H_U$, the same equivariant module $H_U\otimes B$ that we considered in Section~\ref{s1}. The module $B$ generates the category~${\mathcal D}_B$, in the sense that any object in ${\mathcal D}_B$ is a subobject of $B\otimes U$ for some $U$. In other words, any $G$-equivariant finitely generated right Hilbert $B$-module $M$ is isomorphic to a direct summand of $B\otimes H_U$ for some $U$, see \cite[Section~3.2]{Ver} or \cite[Lemma~3.2]{NTK}.

Let $F$ be the spectral functor associated with the action of $G$ on $B$. We have canonical isomorphisms
$$
F(U)=(H_U\otimes B)^G\cong\Mor(B,B\otimes U)
$$
that map $\sum_i(\xi_i\otimes x_i)\in(H_U\otimes B)^G$ into the morphism $x\mapsto \sum_i(x_ix\otimes\xi_i)$. For ergodic actions, these isomorphisms, modulo some identifications in terms of Frobenius reciprocity, were already used in \cite[Section~6]{DCY} to identify algebras constructed in~\cite{DCY} with those defined by Pinzari and Roberts~\cite{PR}. In other words, the key relation between the results in \cite{DCY} and \cite{PR} can be described by saying that given a $(\Rep G)$-module C$^*$-category ${\mathcal M}$ and a simple object $M$ in ${\mathcal M}$, the functor $U\mapsto\Mor(M,M\otimes U)$ has all the properties of a spectral functor.  With our characterization of spectral functors this becomes almost immediate. Specifically, and more generally, we have the following.

\begin{proposition}
Assume ${\mathcal M}$ is a strict right module C$^*$-category over a strict C$^*$-tensor category~${\mathcal C}$. Take an object $M\in{\mathcal M}$ and consider the unital C$^*$-algebra $A=\End(M)$. Then the following defines a weak unitary tensor functor ${\mathcal C}\to\Hilb A$:
$$
F(U)=\Mor(M,M\otimes U),\ \
$$
with the right $A$-module structure on $F(U)$ given by composition of morphisms, the left $A$-module structure by $aX=(a\otimes\iota)X$ and the inner product by $\langle X,Y\rangle=X^*Y$, the action of $F$ on morphisms is defined by $F(T)X=(\iota\otimes T)X$, and
$$
F_2\colon F(U)\otimes_A F(V)\to F(U\otimes V)
$$
is given by $X\otimes Y\mapsto (X\otimes\iota)Y$.
\end{proposition}

\bp This is a routine verification. We only remark that the adjoint of the map
$$
S_X\colon F(V)\to F(U\otimes V),\ \ Y\mapsto (X\otimes\iota)Y,
$$
is obviously given by $S^*_XZ=(X^*\otimes\iota)Z$.
\ep

If the object $M$ happens to be generating, then we can reconstruct the whole category ${\mathcal M}$ from the functor $F$. For ${\mathcal C}=\Rep G$ this gives the following result.

\begin{proposition}
Assume $G$ is a reduced compact quantum group and ${\mathcal M}$ is a strict right $(\Rep G)$-module C$^*$-category generated by an object $M$. Put $A=\End(M)$ and consider the weak unitary tensor functor $F\colon\Rep G\to\Hilb A$ defined by the object $M$ as described in the previous proposition. Let $\theta\colon B\to C(G)\otimes B$ be the continuous action corresponding to this functor by Theorem~\ref{tmain}. Then~${\mathcal D}_B$ is unitarily equivalent, as a $(\Rep G)$-module C$^*$-category, to ${\mathcal M}$, via an equivalence that maps the generator $B\in{\mathcal D}_B$ into $M$.
\end{proposition}

\bp Consider the functor $F'\colon{\mathcal D}_B\to\Hilb A$ defined by the object $B\in{\mathcal D}_B$. By the above discussion, it is naturally unitarily monoidally   isomorphic to the spectral functor associated with the action of $G$ on $B$, hence to $F$. Let $\psi\colon F'\to F$ be such an isomorphism. Note that we automatically have that $\psi\colon A=F'(\un)\to F(\un)=A$ is the identity map, since it is a bimodule map such that $\psi F_2=F_2'(\psi\otimes\psi)$.

\smallskip

Consider the full subcategories $\tilde{\mathcal D}_B\subset{\mathcal D}_B$ and $\tilde{\mathcal M}\subset{\mathcal M}$ consisting of objects $B\otimes U$ and $M\otimes U$, respectively. We want to define a functor $E\colon\tilde{\mathcal D}_B\to\tilde{\mathcal M}$. On objects we put $E(B\otimes U)=M\otimes U$. For morphisms $T\in\Mor(B,B\otimes U)$ we put $E(T)=\psi(T)$. More generally, given two finite dimensional unitary representations $U$ and $V$, we have Frobenius reciprocity isomorphisms
$$
\Mor(B\otimes U,B\otimes V)\to\Mor(B,B\otimes V\otimes\bar U),\ \ T\mapsto(T\otimes\iota)(\iota\otimes\bar R_U),
$$
with inverse $S\mapsto(\iota\otimes\iota\otimes R^*_U)(S\otimes\iota)$. We also have similar isomorphisms in $\tilde{\mathcal M}$. Hence we can define linear isomorphisms
$$
E\colon\Mor(B\otimes U,B\otimes V)\to\Mor(M\otimes U,M\otimes V)
$$
by $E(T)=(\iota\otimes\iota\otimes R^*_U)\big(\psi\big((T\otimes\iota)(\iota\otimes\bar R_U)\big)\otimes\iota\big)$.

Before we turn to the proof that $E$ is indeed a functor, let us make two observations. The first one is that given a morphism $T\colon B\otimes U\to B\otimes V$ and a finite dimensional unitary representation~$W$ of~$G$, for the morphism $T\otimes\iota_W=T\otimes\iota\colon B\otimes U\otimes W\to B\otimes V\otimes W$ we have
\begin{equation} \label{ee1}
E(T\otimes\iota)=E(T)\otimes\iota.
\end{equation}
The second observation is that given a morphism $T\colon B\otimes U\to B\otimes V$ and a morphism $S\colon V\to W$, we have
\begin{equation} \label{ee2}
E((\iota\otimes S)T)=(\iota\otimes S)E(T).
\end{equation}
Both claims follow easily from naturality of $\psi$, which means that for any morphisms $T\colon B\to B\otimes U$ and $S\colon U\to V$ we have $\psi((\iota\otimes S)T)=(\iota\otimes S)\psi(T)$.

Consider now morphisms $R\colon B\otimes U\to B\otimes V$ and $T\colon B\otimes V\to B\otimes W$, and define the morphisms $P=(R\otimes\iota)(\iota\otimes\bar R_U)\colon B\to B\otimes V\otimes\bar U$ and $S=(T\otimes\iota)(\iota\otimes\bar R_V)\colon B\to B\otimes W\otimes\bar V$. We then have
\begin{align*}
TR&=(\iota_B\otimes\iota_W\otimes R^*_V)(S\otimes\iota_V)(\iota_B\otimes\iota_V\otimes R^*_U)(P\otimes\iota_U)\\
&=(\iota_B\otimes\iota_W\otimes R^*_V)(\iota_B\otimes\iota_W\otimes\iota_{\bar V}\otimes\iota_V\otimes R^*_U)( (S\otimes\iota\otimes\iota)P\otimes\iota_U)\\
&=(\iota_B\otimes\iota_W\otimes R^*_V\otimes R^*_U)(F'_2(S\otimes P)\otimes\iota_U),
\end{align*}
where $F'_2(S\otimes P)=(S\otimes\iota\otimes\iota)P\colon B\to B\otimes W\otimes\bar V\otimes V\otimes\bar U$. A similar computation gives
$$
E(T)E(R)=(\iota_M\otimes\iota_W\otimes R^*_V\otimes R^*_U)(F_2(\psi(S)\otimes \psi(P))\otimes\iota_U).
$$
From this we immediately get that $E(TR)=E(T)E(R)$ using \eqref{ee2}, \eqref{ee1} and monoidality of $\psi$, which means that $\psi F'_2(S\otimes P)=F_2(\psi(S)\otimes \psi(P))$. Therefore $E$ is a functor. Since it is surjective on objects and fully faithful, it is an equivalence of the linear categories $\tilde{\mathcal D}_{\mathcal B}$ and $\tilde{\mathcal M}$.

\smallskip

Let us show next that the equivalence $E$ is unitary, that is, $E(T)^*=E(T^*)$ on morphisms. Let us check this first for $T\in\Mor(B,B\otimes U)$. Since $\psi$ is unitary, for any $S\in\Mor(B,B\otimes U)$ we have
$$
E(T)^*E(S)=\psi(T)^*\psi(S)=\langle\psi(T),\psi(S)\rangle=\langle T,S\rangle=T^*S=E(T^*S)=E(T^*)E(S).
$$
Since this is true for all $S$, we conclude that $E(T)^*=E(T)^*$. By virtue of \eqref{ee1} we then also get $E(T\otimes\iota)^*=E((T\otimes\iota)^*)$. But any morphism in $\tilde{\mathcal D}_B$ is a composition of such a morphism $T\otimes\iota_W$ and a morphism of the form $\iota_M\otimes S$ for some morphism $S$ in $\Rep G$. Since as a particular case of~\eqref{ee2} we have
$
E(\iota\otimes S)^*=\iota\otimes S^*=E((\iota\otimes S)^*),
$
it follows that $E$ is unitary.

\smallskip

Next, from \eqref{ee1} and \eqref{ee2} we see that if we define
$$
E_2=E_{2,B\otimes U,V}\colon E(B\otimes U)\otimes V\to E(B\otimes U\otimes V)
$$
to be the identity maps, then we get a natural isomorphism of bilinear functors $E(\cdot)\otimes\cdot$ and~$E(\cdot\otimes\cdot)$. Therefore the pair $(E,E_2)$ defines a unitary equivalence of $(\Rep G)$-module categories $\tilde{\mathcal D}_B$ and $\tilde{\mathcal M}$.

Finally, since ${\mathcal D}_B$ and ${\mathcal M}$ are completions of these categories with respect to subobjects, the equivalence between $\tilde{\mathcal D}_B$ and $\tilde{\mathcal M}$ extends uniquely, up to a natural unitary isomorphism, to a unitary equivalence between the $(\Rep G)$-module C$^*$-categories ${\mathcal D}_B$ and ${\mathcal M}$.
\ep

This leads to the main theorem of this section, a generalization of results of De Commer and Yamashita~\cite{DCY} to the nonsemisimple/nonergodic case.

\begin{theorem}\label{tmain2}
Assume $G$ is a reduced compact quantum group. Then by associating to an action of~$G$ on a unital C$^*$-algebra $B$ the $(\Rep G)$-module category ${\mathcal D}_B$ with generator $B$, we get a bijection between isomorphism classes of continuous left actions of $G$ on unital C$^*$-algebras and unitary equivalence classes of pairs $({\mathcal M},M)$, where ${\mathcal M}$ is a right $(\Rep G)$-module C$^*$-category and $M$ is a generating object in ${\mathcal M}$.
\end{theorem}

\bp In view of the above proposition we only have to show that two actions of $G$ on unital C$^*$-algebras $B$ and $C$ are isomorphic if and only if the pairs $({\mathcal D}_B,B)$ and $({\mathcal D}_C,C)$ are unitarily equivalent. Given such an equivalence, we first of all get an isomorphism $B^G=\End_{{\mathcal D}_B}(B)\cong\End_{{\mathcal D}_C}(C)=C^G$. Modulo the identification of $B^G$ with $C^G$ using this isomorphism, we then also get a natural unitary isomorphism between the spectral functors associated with our actions. Hence the actions are isomorphic by the easy part of Theorem~\ref{tmain}. Conversely, it is clear that isomorphic actions produce unitarily equivalent pairs.
\ep

As in \cite{DCY}, this result can also be formulated in terms of Morita equivalent actions.

\begin{corollary}
For any reduced compact quantum group $G$, there is a bijection between Morita equivalence classes of continuous left actions of $G$ on unital C$^*$-algebras and unitary equivalence classes of singly generated right $(\Rep G)$-module C$^*$-categories.
\end{corollary}

\bp It suffices to show that two actions of $G$ on unital C$^*$-algebras $B$ and $C$ are Morita equivalent if and only if the $(\Rep G)$-module C$^*$-categories ${\mathcal D}_B$ and ${\mathcal D}_C$ are unitarily equivalent. In one direction this is obvious:
if a $C$-$B$-bimodule $M$ defines the Morita equivalence, then ${\mathcal D}_C$ and ${\mathcal D}_B$ are unitarily equivalent, via an equivalence that maps $N\in{\mathcal D}_C$ into $N\otimes_C M$. Conversely, assume we have a unitary equivalence $E\colon{\mathcal D}_C\to{\mathcal D}_B$ of $(\Rep G)$-module categories. Consider the right Hilbert $B$-module $M=E(C)$. Since it is a generating object in ${\mathcal D}_B$, and therefore the right Hilbert $B$-module~$B$ can be isometrically embedded into $M\otimes H_U$ for some representation $U$, the module $M$ must be full. Thus the action of $G$ on $B$ is Morita equivalent to the action of $G$ on $C'=\End_B(M)$. The $(\Rep G)$-module C$^*$-categories ${\mathcal D}_B$ and ${\mathcal D}_{C'}$ are unitarily equivalent, via an equivalence that maps $M\in{\mathcal D}_B$ into $C'\in{\mathcal D}_{C'}$. It follows that the $(\Rep G)$-module C$^*$-categories~${\mathcal D}_C$ and~${\mathcal D}_{C'}$ are unitarily equivalent, via an equivalence that maps~$C$ into~$C'$. By Theorem~\ref{tmain2} this implies that the actions of $G$ on $C$ and $C'$ are isomorphic, so the actions of $G$ on $B$ and $C$ are Morita equivalent.
\ep

\begin{remark}
In the above proof we used that if an equivariant right Hilbert $B$-module $M$ is a generating object in ${\mathcal D}_B$, then it is full. The proof implies that the converse is also true, since if~$M$ is full, then $M$ is the image of the generating object $C$ in ${\mathcal D}_C$, where $C=\End_B(M)$, under the equivalence of categories ${\mathcal D}_C$ and ${\mathcal D}_B$ defined by $M$, hence $M$ is a generating object in ${\mathcal D}_B$. Somewhat more explicitly this can also be proved as follows. It suffices to show that $B\in{\mathcal D}_B$ is a subobject of $M\otimes V$ for some finite dimensional unitary representation~$V$ of~$G$. Replacing $M$ by $M^n$ we may assume that there exists a vector $X\in M$ such that the element $\langle X,X\rangle\in B$ is invertible. Furthermore, since the union of spectral subspaces of~$M$ is dense in $M$, we may assume that $X$ lies in a spectral subspace of $M$ corresponding to some representation~$U$. In other words, there exist an orthonormal basis  $\{\xi_i\}_i$ in $H_U$ and vectors $X_i\in M$ such that $\delta_M(X_i)=\sum_ju^*_{ij}\otimes X_j$ and one of the inner products  $\langle X_i,X_i\rangle\in B$ is invertible. Consider the vector
$$
Y=\sum_iX_i\otimes\overline{\rho^{1/2}\xi_i}\in M\otimes H_{\bar U}.
$$
Then $Y$ is invariant and
$\langle Y,Y\rangle=\sum_{i,j}\langle X_i,X_j\rangle(\rho\xi_i,\xi_j)$. Since the matrix $((\rho\xi_i,\xi_j))_{i,j}$ is positive and invertible, and the matrix $(\langle X_i,X_j\rangle)_{i,j}$ is positive, there exists a constant $c>0$ such that
$$
\langle Y,Y\rangle\ge c\sum_i\langle X_i,X_i\rangle.
$$
It follows that $\langle Y,Y\rangle$ is invertible, so the map $B\ni x\mapsto Y\langle Y,Y\rangle^{-1/2}x$ gives an equivariant isometric embedding of $B$ into $M\otimes H_{\bar U}$.
\end{remark}

\bigskip

\section{Examples and applications}

The data provided by a weak unitary tensor functor and the construction of the corresponding C$^*$-algebra are reminiscent of various crossed product type constructions. To make the connection more explicit, let us give an equivalent description of weak unitary tensor functors in terms of collections of Hilbert module maps satisfying a system of quadratic relations.

\smallskip

Assume $G$ is a compact quantum group and $F\colon\Rep G\to\Hilb A$ is a weak unitary tensor functor. As in Section~\ref{s2}, fix representatives $U_\alpha$ of irreducible unitary representations of $G$, and assume $U_e$ is the trivial representation. Consider the correspondences $M_\alpha=F(U_\alpha)$ and the linear maps $\varphi^\gamma_{\alpha,\beta}$ from $\Mor(U_\alpha\times U_\beta,U_\gamma)$ into the space of bounded $A$-bilinear maps $M_\alpha\otimes_AM_\beta\to M_\gamma$ defined by $\varphi^\gamma_{\alpha,\beta}(T)=F(T)F_2$. We then have the following:
\begin{itemize}
\item[(i)] $M_e=A$;
\item[(ii)] if a morphism of the form $(T_1,\dots,T_n)\colon U_\alpha\times U_\beta\to\oplus^n_{i=1}U_{\gamma_i}$ is unitary, then the map $(\varphi(T_1),\dots,\varphi(T_n))\colon M_\alpha\otimes_A M_\beta\to\oplus^n_{i=1}M_{\gamma_i}$ is isometric;
\item[(iii)] the image of the identity map $U_\beta\to U_\beta$ under $\varphi^\beta_{e,\beta}$ is the map $A\otimes_A M_\beta\to M_\beta$ such that $a\otimes X\mapsto aX$, and similarly the image of the identity map $U_\alpha\to U_\alpha$ under $\varphi^\alpha_{\alpha,e}$ is the map $M_\alpha\otimes_AA\to M_\alpha$ such that $X\otimes a\mapsto Xa$;
\item[(iv)] if a morphism $U_\alpha\times U_\beta\times U_\gamma\to U_\delta$ is written as $\sum_iS_i(T_i\otimes\iota)=\sum_jS_j'(\iota\otimes T'_j)$ for some morphisms $T_i\colon U_\alpha\times U_\beta\to U_{\alpha_i}$, $S_i\colon U_{\alpha_i}\times U_\gamma\to U_\delta$, $T'_j\colon U_\beta\times U_\gamma\to U_{\beta_j}$ and $S'_j\colon U_{\alpha}\times U_{\beta_j}\to U_\delta$, then
$$
\sum_i\varphi(S_i)(\varphi(T_i)\otimes\iota)=\sum_j\varphi(S_j')(\iota\otimes\varphi(T'_j)) \ \ \text{as maps}\ \ M_\alpha\otimes_A M_\beta\otimes_AM_\gamma\to M_\eta;
$$
\item[(v)] for every vector $X\in M_\alpha$ and every morphism $T\colon U_\alpha\times U_\beta\to U_\gamma$, the right $A$-linear map $S_X[T]\colon M_\beta\to M_\gamma$ mapping $Y$ into $\varphi(T)(X\otimes Y)$ is adjointable, and if a morphism $U_\gamma\times U_\delta\to U_\alpha\times U_\eta$ is written as $\sum_i(\iota\otimes S_i)(T^*_i\otimes \iota)=\sum_jP_j^*R_j$ for some morphisms $T_i\colon U_{\alpha}\times U_{\beta_i}\to U_{\gamma}$, $S_i\colon U_{\beta_i}\times U_\delta\to U_\eta$, $R_j\colon U_\gamma\times U_\delta\to U_{\gamma_j}$ and $P_j\colon U_{\alpha}\times U_{\eta}\to U_{\gamma_j}$, then
$$
\sum_i\varphi(S_i)(S_X[T_i]^*\otimes\iota)=\sum_jS_X[P_j]^*\varphi(R_j) \ \ \text{as maps}\ \ M_\gamma\otimes_A M_\delta\to M_\eta.
$$
\end{itemize}
Properties (i)-(iv) follow immediately by definition. As will become clear from the proof of the following proposition, the last property, in the presence of the other four, is equivalent to condition~(v) in Definition~\ref{dwutf}. In particular, if in (ii) we have unitary maps instead of isometric maps, then (v) is a consequence of properties (i)-(iv).

\begin{proposition}
Assume we are given correspondences $M_\alpha\in\Hilb A$ and linear maps $\varphi^\gamma_{\alpha,\beta}$ from $\Mor(U_\alpha\times U_\beta,U_\gamma)$ into the space of bounded $A$-bilinear maps $M_\alpha\otimes_AM_\beta\to M_\gamma$ such that the above conditions (i)-(v) are satisfied. Then there exists a unique, up to a natural unitary  monoidal  isomorphism, weak unitary tensor functor $F\colon\Rep G\to\Hilb A$ such that $F(U_\alpha)=M_\alpha$ and $F(T)F_2=\varphi^\gamma_{\alpha,\beta}(T)$ for $T\in\Mor(U_\alpha\times U_\beta,U_\gamma)$.
\end{proposition}

\bp By virtue of semisimplicity of $\Rep G$, there exists a unique, up to a natural unitary isomorphism, unitary functor $F\colon\Rep G\to\Hilb A$ such that $F(U_\alpha)=M_\alpha$. We then define
$$
F_2\colon F(U_\alpha)\otimes_AF(U_\beta)\to F(U_\alpha\times U_\beta) \ \ \text{by}\ \ F_2=\sum_iF(w_i^*)\varphi(w_i),
$$
where $w_i\colon U_\alpha\times U_\beta\to U_{\gamma_i}$ are coisometric morphisms such that $\sum_iw_i^*w_i=\iota$. It is easy to see that this definition does not depend on the choice of $w_i$. By condition (ii) the map $F_2$ is isometric. Note also that $F(w_i)F_2=\varphi(w_i)$, whence $F(T)F_2=\varphi(T)$ for all $T\in\Mor(U_\alpha\times U_\beta,U_\gamma)$. By semisimplicity of $\Rep G$ the isometries $F_2\colon F(U_\alpha)\otimes_AF(U_\beta)\to F(U_\alpha\times U_\beta)$ uniquely define a family of natural isometries $F_2\colon F(U)\otimes_AF(V)\to F(U\times V)$. The only not entirely obvious property left to check is commutativity of two diagrams in Definition~\ref{dwutf}.

\smallskip

For the first diagram, we have to check that $F_2(F_2\otimes\iota)=F_2(\iota\otimes F_2)$ as maps $F(U_\alpha)\otimes_A F(U_\beta)\otimes_AF(U_\gamma)\to F(U_\alpha\times U_\beta\times U_\gamma)$. It suffices to check that $F(w)F_2(F_2\otimes\iota)=F(w)F_2(\iota\otimes F_2)$ for any morphism $w\colon U_\alpha\times U_\beta\times U_\gamma\to U_\delta$. Any such morphism can be written as $\sum_iS_i(T_i\otimes\iota)=\sum_jS_j'(\iota\otimes T'_j)$. By naturality of $F_2$ we then have
$$
F(w)F_2(F_2\otimes\iota)=\sum_iF(S_i)F_2(F(T_i)F_2\otimes\iota) =\sum_i\varphi(S_i)(\varphi(T_i)\otimes\iota),
$$
and similarly $F(w)F_2(\iota\otimes F_2)=\sum_j\varphi(S_j')(\iota\otimes\varphi(T'_j))$. By condition (iv) these expressions are equal.

\smallskip

It remains to show that for every $X\in F(U)$ the maps $S_X=S_{X,V}\colon F(V)\to F(U\otimes V)$ are adjointable and $F_2(S^*_X\otimes\iota)=S_X^*F_2$. For the adjointability it suffices to show that the map $S_X\colon F(U_\beta)\to F(U_\alpha\times U_\beta)$ is adjointable for every $X\in F(U_\alpha)$. Decomposing $U_\alpha\times U_\beta$ into irreducible representations, we see that adjointability of $S_X$ is equivalent to adjointability of $F(T)S_X$ for all morphisms $T\colon U_\alpha\times U_\beta\to U_\gamma$. Since $F(T)S_X(Y)=F(T)F_2(X\otimes Y)=\varphi(T)(X\otimes Y)$, we have $F(T)S_X=S_X[T]$, so adjointability of $F(T)S_X$ is part of condition (v).

Finally, we have to show that $F_2(S^*_X\otimes\iota)=S_X^*F_2$ as maps $F(U_\alpha\times U_\beta)\otimes_A F(U_\delta)\to F(U_\beta\times U_\delta)$ for $X\in F(U_\alpha)$. This is equivalent to $$F(S)F_2(S^*_X\otimes\iota)(F(T^*)\otimes\iota)=F(S)S_X^*F_2(F(T^*)\otimes\iota)\ \ \text{as maps}\ \ F(U_\gamma)\otimes_A F(U_\delta)\to F(U_\eta)$$ for all morphisms $S\colon U_\beta\times U_\delta\to U_\eta$ and $T\colon U_\alpha\times U_\beta\to U_\gamma$. The left hand side of the above identity equals $\varphi(T)(S_X[T]^*\otimes\iota)$, while the right hand side, by \eqref{enats}, equals
$$
S^*_XF(\iota\otimes S)F_2(F(T^*)\otimes\iota)=S^*_XF((\iota\otimes S)(T^*\otimes\iota))F_2.
$$
Writing the morphism $(\iota\otimes S)(T^*\otimes\iota)$ as $\sum_jP_j^*R_j$ for some $R_j\colon U_\gamma\times U_\delta\to U_{\gamma_j}$ and $P_j\colon U_{\alpha}\times U_{\eta}\to U_{\gamma_j}$, we can write the last expression as
$$
\sum_j S^*_XF(P_j^*)F(R_j)F_2=\sum_jS_X[P_j]^*\varphi(R_j).
$$
We thus see that the identity $F_2(S^*_X\otimes\iota)=S_X^*F_2$ follows from condition (v). Furthermore, from the proof we see that it is equivalent to that condition, since any morphism $U_\gamma\times U_\delta\to U_\alpha\times U_\eta$ can be written as $\sum_i(\iota\otimes S_i)(T^*_i\otimes \iota)$ for appropriate morphisms $T_i\colon U_{\alpha}\times U_{\beta_i}\to U_{\gamma}$ and $S_i\colon U_{\beta_i}\times U_\delta\to U_\eta$ using Frobenius reciprocity.
\ep

Given data $\{M_\alpha\}_\alpha$ and $\{\varphi^\gamma_{\alpha,\beta}\}_{\alpha,\beta,\gamma}$ as above, the construction of the corresponding C$^*$-algebra~$B_F$ from Section~\ref{s2} goes as follows. For every $\alpha$ define a new scalar product on $\bar H_\alpha=\bar H_{U_\alpha}$~by
$$
(\bar\xi,\bar\zeta)=\frac{1}{\dim_qU_\alpha}(\zeta,\rho^{-1}\xi).
$$
Consider the right Hilbert $A$-module
$$
M=\ell^2\text{-}\bigoplus_\alpha \bar H_\alpha\otimes M_\alpha.
$$
For every $x=\bar\xi\otimes X\in\bar H_\alpha\otimes M_\alpha$ define an operator $L_x$ on $M$ by
$$
L_x(\bar\zeta\otimes Y)=\sum_i\overline{w_i(\xi\otimes\zeta)}\otimes\varphi^{\gamma_i}_{\alpha,\beta}(w_i)(X\otimes Y)\ \ \text{for}\ \ \bar\zeta\otimes Y\in \bar H_\beta\otimes M_\beta,
$$
where $w_i\in\Mor(U_\alpha\times U_\beta,U_{\gamma_i})$ are coisometries such that $\sum_i w_i^*w_i=\iota$. Then the results of Section~\ref{s2} can be summarized by saying that the operators $L_{\bar\xi\otimes X}$ for all $\xi\in H_\alpha$, $X\in M_\alpha$ and all indices $\alpha$, span a $*$-algebra of bounded operators on the Hilbert $A$-module $M$, and~$B_F$ is the norm closure of this algebra.

\begin{example}
Assume $G$ is the dual of a discrete group $\Gamma$. We identify the set of isomorphism classes of irreducible representations of $G$ with $\Gamma$. Then, up to equivalence, a weak unitary tensor functor $F\colon\Rep G\to\Hilb A$ is the same as a collection of C$^*$-correspondences $M_\alpha$, $\alpha\in\Gamma$, over $A$, together with $A$-bilinear isometries $\varphi_{\alpha,\beta}\colon M_\alpha\otimes_A M_\beta\to M_{\alpha\beta}$ such that
\begin{itemize}
\item[(a)] $M_e=A$;
\item[(b)] $\varphi_{e,\alpha}\colon A\otimes_A M_\alpha\to M_\alpha$ and $\varphi_{\alpha,e}\colon M_\alpha\otimes_A A\to M_\alpha$ are the maps $a\otimes X\mapsto aX$ and $X\otimes a\mapsto Xa$, respectively;
\item[(c)] $\varphi_{\alpha\beta,\gamma}(\varphi_{\alpha,\beta}\otimes\iota) =\varphi_{\alpha,\beta\gamma}(\iota\otimes\varphi_{\beta,\gamma})$;
\item[(d)] for every vector $X\in M_\alpha$ and $\beta\in\Gamma$, the map $S_X\colon M_\beta\to M_{\alpha\beta}$, $Y\mapsto\varphi_{\alpha,\beta}(X\otimes Y)$, is adjointable, and $\varphi_{\beta,\gamma}(S^*_X\otimes\iota)=S^*_X\varphi_{\alpha\beta,\gamma}$ as maps $M_{\alpha\beta}\otimes_A M_\gamma\to M_{\beta\gamma}$.
\end{itemize}
This is similar to the definition of product systems of C$^*$-correspondences~\cite{Fow}. The difference is that instead of semigroups we consider groups, the maps $\varphi_{\alpha,\beta}$ are not assumed to be unitary, but then the additional assumption (d) is required. We remind again that if the maps $\varphi_{\alpha,\beta}$ are unitary, condition~(d) is not needed.

Since conditions (a)-(d) describe spectral subspaces of an arbitrary coaction of $\Gamma$, our results for $G=\hat\Gamma$ simply mean that these conditions give an equivalent characterization of Fell bundles over~$\Gamma$~\cite{FD2}. Explicitly, the $*$-structure on the bundle $\{M_\alpha\}_{\alpha\in\Gamma}$ is given by the operation $\bullet$ defined in Lemma~\ref{lbullet}, so $X^\bullet=S^*_X(1)\in M_{\alpha^{-1}}$ for $X\in M_\alpha$ if $A$ is unital, and in general~$X^\bullet$ is characterized by $\langle X^\bullet,Y\rangle=\varphi_{\alpha,\alpha^{-1}}(X\otimes Y)$ for $Y\in M_{\alpha^{-1}}$. Clearly, $B_F$ is nothing else than the cross-sectional C$^*$-algebra of this bundle.\ee
\end{example}

\begin{example}
Assume $G=\T$. Let $M$ be a Hilbert $A$-bimodule, meaning that $M$ carries the structures of a right Hilbert $A$-module with inner product $\langle\cdot,\cdot\rangle_R$ and of a left Hilbert $A$-module with inner product $_L\langle\cdot,\cdot\rangle$, and $X\langle Y,Z\rangle_R={}_L\langle X,Y\rangle Z$ for all $X,Y,Z\in M$. Consider the complex conjugate Hilbert $A$-bimodule~$\bar M$, so $a\bar X=\overline{Xa^*}$, $\bar Xa=\overline{a^*X}$, $_L\langle \bar X,\bar Y\rangle=\langle X,Y\rangle_R$ and $\langle \bar X,\bar Y\rangle_R={_L}\langle X,Y\rangle$. Define C$^*$-correspondences $M_n$, $n\in\Z$, over $A$ by $M_0=A$, $M_n=M^{\otimes_A n}$ for $n\ge1$ and $M_n=\bar M^{\otimes_A |n|}$ for $n\le-1$. We have obvious isometries $\varphi_{m,n}\colon M_m\otimes_A M_n\to M_{m+n}$. In order to show that they define a weak unitary tensor functor $F\colon\Rep\T\to\Hilb A$, we have to check conditions (a)-(d) from the previous example. Conditions (a) and (b) are obviously satisfied. A moment's reflection shows that since the maps $\varphi_{m,n}$ are surjective for $m$ and $n$ of the same sign, it suffices to check the other two conditions only for $\alpha=\pm1$. For such $\alpha$ conditions (c) and (d) easily follow from the identity $X\langle Y,Z\rangle_R={}_L\langle X,Y\rangle Z$.

The corresponding C$^*$-algebra $B_F$ is the algebra $A\rtimes_M\Z$, the crossed product of $A$ by the Hilbert $A$-bimodule $M$, defined in~\cite{AAE}, where it was shown directly that $\{M_n\}_{n\in\Z}$ forms a Fell bundle over~$\Z$. Recall that the C$^*$-algebra $A\rtimes_M\Z$ is canonically isomorphic to the Cuntz-Pimsner algebra $\mathcal O_M$.\ee
\end{example}

Let us return to the case of a general compact quantum group $G$. Recall that a unitary $2$-cocycle on the dual discrete quantum group $\hat G$ is a unitary element
$$
\Omega\in W^*(G)\bar\otimes W^*(G)\subset({\mathbb C}[G]\otimes{\mathbb C}[G])^*
$$
such that $(\Omega\otimes1)(\Dhat\otimes\iota)(\Omega)=(1\otimes\Omega)(\iota\otimes\Dhat)(\Omega)$. Any such cocycle defines a unitary fiber functor $E_\Omega\colon\Rep G\to\Hilbf$ that is identity on objects and morphisms, while the tensor structure $E_\Omega(U)\otimes E_\Omega(V)\to E_\Omega(U\times V)$ is given by the action of $\Omega^*$. By Woronowicz's Tannaka-Krein duality,
this functor defines a new deformed compact quantum group $G_\Omega$. More concretely, we have $W^*(\hat G_\Omega)=W^*(\hat G)$ as von Neumann algebras, while the new coproduct is given by $\Dhat_\Omega(\omega)=\Omega\Dhat(\omega)\Omega^*$. Equivalently, ${\mathbb C}[G_\Omega]={\mathbb C}[G]$ as coalgebras, while the new product and involution are obtained by dualizing $(W^*(G_\Omega),\Dhat_\Omega)$.

Assume now that we have a continuous left action $\theta$ of the reduced form of $G$ on a C$^*$-algebra~$B$ with fixed point algebra $A$. Consider the corresponding spectral functor $F\colon\Rep G\to\Hilb A$. Since by construction the categories $\Rep G$ and $\Rep G_\Omega$ are equivalent, we can consider $F$ as a weak unitary tensor functor $\Rep G_\Omega\to\Hilb A$. It defines a C$^*$-algebra $B_\Omega$ carrying a continuous left action of the reduced form of $G_\Omega$. These algebras were defined and studied in greater generality in~\cite{DRV} and~\cite{NT} (and in various special cases by many authors earlier; in particular, see \cite{Yam} for the case of the dual of a discrete group). But as we will see in a moment, the categorical picture provides a very simple and concrete approach.

In order to formulate the result we need to introduce more notation. First of all we will need a special element $u\in\U(G)={\mathbb C}[G]^*$ defined by $u=m(\iota\otimes\hat S)(\Omega)$, where $m\colon ({\mathbb C}[G]\otimes{\mathbb C}[G])^*\to{\mathbb C}[G]^*$ is the product map, which is by definition dual to the coproduct on~${\mathbb C}[G]$. The element $u$ is invertible, with inverse given by
$$
u^{-1}=m(\hat S\otimes\iota)(\Omega^*)=\hat S(u^*).
$$
The antipode on the dual of $G_\Omega$ is given by $\hat S_\Omega=u\hat S(\cdot)u^{-1}$, and correspondingly the involution $\dagger$ on~${\mathbb C}[G_\Omega]$ is given by
$$
a^\dagger=\big[(u^{-1})^*\otimes\iota\otimes u^*)\Delta^{(2)}(a)\big]^*
=(u^*\otimes\iota\otimes (u^{-1})^*)\Delta^{(2)}(a^*),
$$
see for instance~\cite[Example~2.3.9]{NTbook}. It is easy to check that the element $u$ can also be characterized by the identities
\begin{equation}\label{eu}
\Omega R_U=(u\otimes\iota)R_U\ \ \text{as maps}\ \ {\mathbb C}\to H_{\bar U}\otimes H_U,
\end{equation}
see~\cite[Example~2.2.23]{NTbook}.

Next, consider the subalgebra ${\mathcal B}\subset B$ of regular elements. Then the map ${\mathcal B}\otimes\U(G)\to{\mathcal B}$, $x\otimes\omega\mapsto x\lhd\omega=(\omega\otimes\iota)\theta(x)$, defines a right $\U(G)$-module structure on ${\mathcal B}$.

\begin{proposition}
With the above notation, the following formulas define a new $*$-algebra ${\mathcal B}_\Omega$ with underlying space ${\mathcal B}$, product $\star$ and involution $\dagger$:
$$
x\star y=m((x\otimes y)\lhd\Omega),\ \ x^\dagger=x^*\lhd u^*,
$$
where $m\colon{\mathcal B}\otimes{\mathcal B}\to{\mathcal B}$ is the original product map. Furthermore, the map $\theta$, considered as a map ${\mathcal B}_\Omega\to{\mathbb C}[G_\Omega]\otimes{\mathcal B}_\Omega$, defines a left algebraic action of $G_\Omega$ on ${\mathcal B}_\Omega$.
\end{proposition}

\bp By multiplying $\Omega$ by a phase factor we may assume that $\Omega$ is counital, that is, $(\hat\eps\otimes\iota)(\Omega)=(\iota\otimes\hat\eps)(\Omega)=1$.

For every finite dimensional  unitary representation $U\in B(H_U)\otimes{\mathbb C}[G]$ denote by $U^\Omega$ the same element $U$ considered as an element of $B(H_U)\otimes{\mathbb C}[G_\Omega]$. Then we have a unitary monoidal equivalence of categories $E^\Omega\colon \Rep G_\Omega\to \Rep G$ such that $E^\Omega(U^\Omega)=U$, $E$ is the identity map on morphisms, and $E^\Omega_2\colon E^\Omega(U^\Omega)\otimes E^\Omega(V^\Omega)\to E^\Omega(U^\Omega\times V^\Omega)$ is given by $\Omega\colon H_U\otimes H_V\to H_U\otimes H_V=H_{U\times V}$.

We identify the algebra ${\mathcal B}$ with the algebra ${\mathcal B}_F$ defined by a weak unitary tensor functor $F$. Now we claim that with the above setup the $*$-algebra ${\mathcal B}_{FE^\Omega}$ corresponding to the weak unitary tensor functor $FE^\Omega$ is exactly ${\mathcal B}_\Omega$, and the map $\theta_{FE^\Omega}$ coincides with $\theta$. Note that counitality of $\Omega$ is needed for condition (iii) in Definition~\ref{dwutf} to be satisfied by the functor $FE^\Omega$.

\smallskip

As linear spaces, we have
$$
{\mathcal B}_{FE^\Omega}=\bigoplus_\alpha\bar H_{U^\Omega_\alpha}\otimes FE^\Omega(U^\Omega_\alpha)=\bigoplus_\alpha\bar H_\alpha\otimes F(U_\alpha)={\mathcal B}.
$$
Denote by $\star$ the product on ${\mathcal B}_{FE^\Omega}$. Note that if $w\colon H_\gamma\to H_\alpha\otimes H_\beta$ is a morphism $U_\gamma\to U_\alpha\times U_\beta$, then $\Omega w$ is a morphism $U^\Omega_\gamma\to U^\Omega_\alpha\times U^\Omega_\beta$. From this we get that if $x=\bar\xi\otimes X\in\bar H_\alpha\otimes F(U_\alpha)$ and $y=\bar\zeta\otimes Y\in\bar H_\beta\otimes F(U_\beta)$, then
$$
x\star y=\sum_i \overline{w_i^*\Omega^*(\xi\otimes\zeta)}\otimes F(w_i^*)F_2(X\otimes Y),
$$
where $w_i\in\Mor(U_{\gamma_i},U_\alpha\times U_\beta)$ are isometries such that $\sum_iw_iw_i^*=\iota$. Since the right $\U(G)$-module structure on ${\mathcal B}={\mathcal B}_F$ is given by $(\bar\eta\otimes Z)\lhd\omega=\overline{\omega^*\eta}\otimes Z$, the above identity means exactly that
$$
x\star y=m((x\otimes y)\lhd\Omega).
$$

\smallskip

Denote the involution on ${\mathcal B}_{FE^\Omega}$ by $\dagger$. Take $x=\bar\xi\otimes X\in\bar H_\alpha\otimes F(U_\alpha)$. We may assume that $\bar U_\alpha=U_{\bar\alpha}$ for some index $\bar\alpha$. Then, by definition,
$$
x^\dagger=\overline{\xi^\#}\otimes X^\#,
$$
where $\xi^\#\in H_{\bar\alpha}$ is such that $(R^{\Omega}_\alpha)^*(\zeta\otimes\xi)=(\zeta,\xi^\#)$ for all $\zeta\in H_{\bar\alpha}$ and $X^\#\in F(U_{\bar\alpha})$ is such that $(FE^\Omega)(\bar R^{\Omega}_\alpha)^*(FE^\Omega)_2(X\otimes Y)=\langle X^\#,Y\rangle$ for all $Y\in F(U_{\bar\alpha})$, where $R^\Omega_\alpha$ and $\bar R^\Omega_\alpha$ solve the conjugate equations for $U^\Omega_\alpha$ and $U^\Omega_{\bar\alpha}$. Note that by irreducubility the operation $\#$ depends on the choice of such a solution, but $\overline{\xi^\#}\otimes X^\#$ does not. Taking the solutions $R_\alpha$ and $\bar R_\alpha$ of the conjugate equations for $U_\alpha$ and $U_{\bar\alpha}$ defined by \eqref{econj}, we can take $R^\Omega_\alpha=\Omega R_\alpha$ and $\bar R^\Omega_\alpha=\Omega\bar R_\alpha$. In this case $X^\#=X^\bullet$, while for $\xi^\dagger$, using~\eqref{eu}, we get
$$
(R^{\Omega}_\alpha)^*(\zeta\otimes\xi)=R^*_\alpha(u^*\zeta\otimes\xi)=(u^*\zeta,\xi^\bullet),
$$
so $\xi^\#=u\xi^\bullet$. Therefore
$$
x^\dagger=\overline{u\xi^\bullet}\otimes X^\bullet=(\overline{\xi^\bullet}\otimes X^\bullet)\lhd u^*=x^*\lhd u^*.
$$

\smallskip

Finally, the maps $\theta=\theta_F$ and $\theta_{FE^\Omega}$ coincide on ${\mathcal B}_{FE^\Omega}$, since they both define the same right $\U(G)$-module structure, given by $(\bar\eta\otimes Z)\lhd\omega=\overline{\omega^*\eta}\otimes Z$.
\ep

The construction of a new product on a module algebra using a cocycle on a Hopf algebra is, of course, well-known. The point of the above proposition is that it effortlessly gives not only the new product, but also the $*$-structure and the existence of a unique C$^*$-completion of the algebra carrying an action of the reduced deformed quantum group.

\bigskip\bigskip

redpajama_set_name: RedPajamaArXiv


The unrelenting Raiders offense went a long way in deciding Saturday's semi-final against the Panthers, dominating the attacking stats.
Standout performances by Jack Wighton, Elliot Whitehead and Josh Papalii gave the Raiders tremendous go-forward, resulting in the team making an extra 375 run metres over their opponents.
Whitehead led from the front with another standout performance, making the team's second highest number of runs whilst leading in tackles.
Jordan Rapana equaled one of the Raiders' longest standing records when he scored his 22nd try of the season, putting his name alongside club greats Noa Nadraku, Jason Croker and Brett Mullins.
Perhaps the match's defining stat was the Raiders' 34 extra tackle busts, which allowed them to create numerous chances out wide.
The Raiders registered 15 offloads, while the Panthers registered 11.
The Raiders made 6 linebreaks, while the Panthers made 2.
The Raiders had 57% possession, while the Panthers had 43%.
The Raiders completed 29 of 39 (74%) sets. The Panthers completed 22 of 33 (67%) sets.
The Raiders made 12 errors, while the Panthers made 11.
The Raiders had 1 line dropout, while the Panthers had 3.
The Raiders missed 16 tackles, while the Panthers missed 50.
The Raiders conceded 7 penalties, while the Panthers conceded 9.


text:
TRANSLATIONS FROM THE EDGE OF EUROPE
© 2016-2020 PICA PICA PRESS
Saulius Šaltenis
Saulius Šaltenis's career includes working as an editor, playwright, and screenwriter, and writing novels, short stories, and musicals. He helped found the cultural weekly Šiauras Atėnas in 1989 and actively took part in Lithuania's transition to independence. He then served as the Minister of Culture from 1996 to 1999. Read an extract from Baees in the Snow here.
Marius Katiliškis
Over the course of his life, Marius Katiliškis (Albinas Marius Vaitkus, 1915–1980), worked on a farm, logged in the forests, built roads, was a librarian in the town of Pasvalys and, after World War II, labored in the steel mills of Chicago. And he wrote. He published his first collection of short stories after emigrating to Germany. His novels Užuovėja, Išėjusiems negrįžti, and other notable works came out in the U.S.
Jurgis Kunčinas
The Lithuanian master of the macabre, Ričardas Gavelis (1950-2002) is best known for his novel Vilnius Poker, first published in English by Open Letter in 2009, and included in the list of Best Translated Books for that year. Gavelis's works include seven novels, five short story collections, and several plays and film scenarios; his work has been translated into English, French, Swedish, Polish and German. Read an extract from our forthcoming translation of his last novel The Life of Sun-Tzu in the Sacred City of Vilnius here.
Jurgis Kunčinas (1947-2002) was born in Alytus, Lithuania and studied German at Vilnius University. An extremely prolific writer and translator, Kunčinas wrote novels, short stories, plays, children's stories, essays, satires and radio plays. He has won many awards and honors in Lithuania, including an award from the Lithuanian Writers Union for the best novel of the year for Tūla in 1993. His works have been translated into Russian, Italian, Latvian, Swedish, Estonian and German. Read an extract from Tūla here.
Petras Cvirka
Twenty-six years old when his comic novel Frank Kruk raised a firestorm on both sides of the Atlantic, Petras Cvirka (1909-1947) went on to live an equally controversial life. Honored and glorified during the Soviet years, some now see him as a traitor to the Lithuanian nation. But his statue in Vilnius still stands, and many of his books for children are still in print. Read an extract from Frank Kruk here.