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Need for Speed Underground 2 is racing-sports video game.Its one of the need for speed series game, the game very successful selling record in united kingdom. You can play via single-player and multiplayer.
Open "Need for Speed Underground 2 – ApunKaGames" folder, double click on "Setup" and install it.
Open folder, double click on "speed2" icon to play the game. Done!
Is this working game or just Demo?
i have installed the game but still i am not able to PLAY. Is this because i am using windows 7.
Are there cutscene and vynil?
The vynils won't show up. I can't create the Legendary Skyline without them.
password don't work. how it is?
thankyou bhai for highly compressed games.


text:
For Sale: 19 Frank Lloyd Wright Houses, Including His Last
If you're over six feet tall, you may need to duck to enter.
by Sabrina Imbler September 25, 2019
The Lykes House, nestled in the mountains of Phoenix, Arizona. Jaydec/CC by 3.0
This month, two very different American homes are up for sale. One, in Phoenix, is a curvy, sand-colored behemoth that seems to ripple out of the Arizona mountains. The other, in Manchester, New Hampshire, appears to be built mostly out of concrete bricks. The first is stunning; the second, well, an acquired taste. Both houses, however, bear the distinction of being designed by the legendary architect Frank Lloyd Wright.
Though his name is now associated with sprawling masterpieces such as the iconic Fallingwater or the Ennis House, a Mayan-revival abode that's seen many a film shoot, Frank Lloyd Wright designed many smaller, less famous houses in a range of styles. Approximately 380 Wright buildings remain standing today, most of which are still used as private residences, according to John Waters, the preservation programs manager at the Frank Lloyd Wright Building Conservancy. "You can look at his work and see a massive house in Los Angeles, a small house in Oak Park, and medium-sized houses dotted around the country," Waters says. But while his styles varied, Wright's goal was always a "thoughtful, intentional design," he says.
To his credit, Frank Lloyd Wright always had a flair for his personal brand. When the would-be architect turned 18, he swapped out his given middle name "Lincoln" for "Lloyd," giving his name a catchy three-meter beat, writes Thomas de Monchaux in the fall 2017 issue of the Frank Lloyd Wright Quarterly. Wright sometimes endowed homes with poetic names such as Hollyhock and Wingspread, betraying the usual convention of naming homes after clients. And he branded many of his buildings, which he considered "opuses," with a small red square that bore his initials, de Monchaux writes.
Wright's original blueprint for the Lykes house. Courtesy Heritage Auctions
Two Frank Lloyd Wright houses for sale simultaneously might seem like a big deal—an unusual glut in the market for buildings that are the brainchild of America's most famous architect. But at the moment of writing, there are actually 19 Frank Lloyd Wright homes for sale, according to a list assembled by Frank Lloyd Wright Building Conservancy. "At any one time, we have between 15 and 20 listings," Waters says. By that estimate, you can expect around five or six percent of Wright houses to be on the market.
The house in Phoenix, also known as the Lykes House, seems to be the more beautiful of the two recent listings. Built in 1967, it is said to be the last home Wright designed before his death, and its floorplan consists of concentric circles that seem only natural in the surrounding mountains, where it overlooks the Phoenix Mountains Preserve. It's the kind of house that lives up to Wright's reputation for both muted grandeur and harmony with the surrounding landscape.
Unlike certain other distinguished Wright homes, Lykes has never had a leaky roof. Wright famously responded to complaints of leaking roofs and skylights by telling the disgruntled party to simply move their desk, chair, or whatever was getting wet; in 2001, Fallingwater nearly fell into the water. "His greatest weakness is a monstrous ego: though he talks of democracy, he has the manners and beliefs of a Renascence prince: at heart he is a despot," Wright's former friend, the historian Lewis Mumford, once wrote in a letter to the poet John Gould Fletcher. Mumford also noted that Wright, who was five foot seven, deliberately designed short doorways so anyone over six feet tall would bump their heads.
The Kalil House, one of seven Wright-designed Usonian Automatic homes. Photos courtesy of Paula Martin Group at Keller Williams
Wright's house in Manchester may seem less architecturally impressive, but its sale is slightly more newsworthy. Known as the Kalil House, the building is one of only seven examples of the Usonian Automatic style, Wright's later-in-life attempt at crafting an economical home in the wake of the Great Depression. Wright devised a concrete masonry system that relied on cheap concrete blocks around one foot wide and two feet long, which, in theory, prospective owners could order to assemble houses themselves. In practice, however, building the structures was more difficult than expected. "It was meant to be a relatively inexpensive house, making fine architecture within the reach of the middle class," Waters says.
The Kalil House listing marks the first time a Wright-designed Usonian Automatic house has ever been listed on the market. It was built in 1955 for Toufic and Mildred Kalil, who lived just down the street from Isadore and Lucille Zimmerman. The Zimmermans had just commissioned a classic Usonian house from Wright, and the Kalils wanted one of their own, Waters says. The Zimmerman House is now the only Wright building in New England that's open for public tours.
A whole lotta concrete blocks. Photos courtesy of Paula Martin Group at Keller Williams
Though many of Wright's homes are listed for sale at any given time, it may seem surprising that some linger on the market for several years. In 2017, The New York Times reported on the knotty process of selling a Frank Lloyd Wright home, which can be due to anything from a remote location to their inherently unique design. But, Waters says, "that's true of all houses that are unique, and not necessarily what the market thinks it's looking for at this moment."
If you're looking for something unique and astonishing, the Lykes House recently dropped in price by nearly a quarter-million dollars, from $2,985,000 to $2,650,000. If, on the other hand, you're looking for something uniquely crafted from concrete legos, the Kalil house is listed for a more modest $850,000. Either way, if you're over six feet tall, be prepared to duck.
The Hidden French Tunnel Sculpture That Took 30 Years to Complete
Artist Daniel Monnier spent a couple of decades away before returning to complete his vision.
frank lloyd wrightbuildingsfor salehousesarchitecture
For Sale: An Off-the-Grid Castle Sitting on an Actual Gold Mine
Way, way out in the Nevada desert, there's a perfect house for the posh recluse in your life.
Jessica Leigh Hester May 10, 2019
Picturing Manhattan's Shortest Buildings From the Ground Floor
A photographer set out to document single-story buildings—the anti-skyscrapers.
Jessica Leigh Hester November 11, 2019
Liverpool's Crosby Beach Is a Mile of World War II Blitz Rubble
An undergraduate archaeologist is trying to make sure the prewar city isn't forgotten.
Isaac Schultz November 6, 2019
The Strange Second Life of Ohio's 'Big Basket' Building
In 2020, guests will be able to bask in the glory of a seven-story, basket-shaped hotel.
Sabrina Imbler November 5, 2019
Video • PinDrop
Explore a Rescued Frank Lloyd Wright Masterpiece
Sponsored by Visit Buffalo Niagara
Video • AO Docs
The Muslims Preserving Kolkata's Last Jewish Synagogues
California's Dazzling Mosaic Tile House
Step Inside a Surreal, Dizzying Italian Fortress
Step Inside Gaudí's First House in Barcelona
For Sale: A Home Built in an Old Train Depot
House-hunting rail enthusiasts, rejoice.

redpajama_set_name: RedPajamaCommonCrawl


text:
Mauritius Müller, auch Moritz Müller, geborener Kaspar Josef Müller (* 12. Februar 1677; † 16. September 1745) war ein Schweizer Geistlicher und mit Unterbrechungen von 1707 bis 1738 Bibliothekar des Klosters St. Gallen.

Leben
Pater Mauritius (oder Moritz) stammte aus Wil. 1693 ist er im Kloster St. Gallen bezeugt, wo er am 31. Mai 1696 die Profess ablegte. 1698 wurde er Subdiakon, 1699 Diakon. Die Priesterweihe empfing er am 21. Mai 1701, die Primiz folgte am 12. Juni des gleichen Jahres. Er war bekannt für seine Gelehrsamkeit.

Wirken
Am 15. August 1701 wurde Pater Mauritius von Abt Leodegar Bürgisser mit der Weiterführung der Klosterchronik beauftragt. Ein Jahr später wurde ihm die Herausgabe einer Festschrift anlässlich des Jubiläums der Schweizerischen Benediktinerkongregation übertragen. Daneben war er als Lehrer im Kloster St. Gallen tätig. Am 7. April 1707 ernannte ihn der Abt zum Stiftsbibliothekar. Es folgten weitere Lehrtätigkeiten sowohl in St. Gallen als auch in Pfäfers. Ab 1711 ist er wiederum als Stiftsbibliothekar bezeugt. Im Umfeld des Toggenburgerkrieges machte er sich um die Rettung der Buchbestände verdient. Letztlich musste aber auch er aus St. Gallen fliehen und war fortan als Theologieprofessor in Rheinau sowie als Lehrer in Bellinzona tätig.

Zurück aus dem Exil, stand Mauritius ab 1719 bis 1722 erneut der Stiftsbibliothek St. Gallen vor. Anschliessend übernahm er an verschiedenen Orten (Berg, Alt St. Johann, Peterzell) seelsorgerische Aufgaben. Wegen seiner rhetorischen Fähigkeiten und wegen des freundlichen Umgangs war er als Seelsorger sehr geschätzt. 1734 kehrte er nach St. Gallen zurück und trat dort wiederum als Bibliothekar in Erscheinung. Daneben verrichtete er weitere Seelsorgedienste, besonders bei Festanlässen wie der 1738 erfolgten Translation des Katakombenheiligen Pankratius in Wil. Ab 1739 ist er in verschiedenen amtlichen Funktionen in Rorschach, Wil und St. Gallen bezeugt. Pater Mauritius verfasste zahlreiche gedruckte und ungedruckte Schriften vorwiegend historiographischen, theologischen und katechetischen (Predigten) Inhalts.

Literatur
 Gall Heer: Johannes Mabillon und die Schweizer Benediktiner. Ein Beitrag zur Geschichte der historischen Quellenforschung im 17. und 18. Jahrhundert. St. Gallen 1938, S. 307–312.
 Rudolf Henggeler: Professbuch der fürstl. Benediktinerabtei der heiligen Gallus und Otmar zu St. Gallen. Einsiedeln 1929, S. 353–355, Nr. 439.
 Thomas Stockinger: Fidelis tametsi inutilis servus. P. Moritz Müller OSB (St. Gallen) in seiner historisch-literarischen und politisch-diplomatischen Tätigkeit im Spiegel seiner Korrespondenz 1709–1714. In: Studien und Mitteilungen zur Geschichte des Benediktinerordens und seiner Zweige. 118 (2007), S. 339–432, hier S. 344 f.
 Franz Weidmann: Geschichte der Bibliothek von St. Gallen seit ihrer Gründung um das Jahr 830 bis auf 1841. Aus den Quellen bearbeitet auf die tausendjährige Jubelfeier. St. Gallen 1841, S. 87, 150–153.

Benediktiner
Bibliothekar (St. Gallen)
Stiftsbibliothek St. Gallen
Schweizer
Geboren 1677
Gestorben 1745
Mann

redpajama_set_name: RedPajamaWikipedia


After the Nuclear Suppliers Group's approval of the nuclear deal, its opponents in India such as the Left and the Bharatiya Janata Party (BJP) have been at a loose end. Having fruitlessly expended so much energy on blocking the deal, they now seem to have run out of ideas about their next step.
The Left's threat to cancel the deal if it acquired a position of power after the next general election is clearly as much of an empty boast as the BJP's declared intention of renegotiating the measure all over again.
Apart from such meaningless rhetoric, what is evident is that they have been left high and dry because they never really expected the Manmohan Singh government to sail through the trust vote in parliament and acquire an issue which is highly popular with the middle class, India Inc. and the mainstream media.
Clearly, where these three segments are concerned, the Congress has a kind of electoral advantage which it didn't expect before July 22, the day of the trust vote.
The Left's current position is seemingly even more pathetic than the BJP's. At the national level, its hasty attempts to form an anti-Congress, anti-BJP Third Front to camouflage its setback in the trust vote have run into trouble. Two of its constituents - Om Prakash Chautala of the Indian National Lok Dal (INLD) and Brindaban Goswami of the Asom Gana Parishad (AGP) - appear to be more interested in a tie-up with the BJP than on remaining in the front.
That has left only the mercurial Mayawati of the Bahujan Samaj Party (BSP) and the undependable H.D.Deve Gowda of the Janata Dal (Secular) in the front along with the Telugu Desam's Chandrababu Naidu and, of course, the communists. But it is not a combination which can fire the imagination of the voters, as Prakash Karat of the Communist Party of India-Marxist (CPI-M) thinks.
Besides, it is worth remembering that all three - Mayawati, Deve Gowda and Naidu - have been allies of the BJP in the past and may be in the future if the situation so demands.
The Left's problems have been compounded by the deadlock in Singur in its own stronghold of West Bengal, where opposition leader Mamata Banerjee has very nearly won her battle to force the Tatas to close their small car factory and leave. Therefore, Nano, the famed car priced at a mere Rs.100,000, is unlikely to roll out from Singur, as was once expected.
It is a huge setback for the state's Left Front government for it will mark the beginning of the end of its plans for an industrial revival by undoing the damage the comrades themselves caused in the sixties via their militant trade unionism. The flight of capital from West Bengal, which started then, can be said to be still continuing with the non-communist Mamata Banerjee leading the charge this time against the investors.
The weakening of the Left in Kerala because of the unending squabbling between the CPI-M's rival factions led by Chief Minister V.S. Achuthanandan and party chief Pinarayi Vijayan as well as in West Bengal may make the communists a marginal force in Indian politics after the elections. At present, it commands a mere 8.3 percent of the national vote.
For many, the Left's setback will seem to be well deserved because of the negative role it played in the matter of economic reforms and the nuclear deal. But the problem is that the space created by the Left's decline is likely to be partly filled by the BJP, the very party which the commissars profess to oppose because of its anti-minority policies.
However, it is worth remembering that the BJP itself is not in the pink of health. Its difficulties arise from fears that the party may have alienated a section of its middle class constituency by opposing the nuclear deal. It was driven to do this by two of its members, Yashwant Sinha and Arun Shourie, neither of whom has a genuine saffron background. Sinha came into the BJP from one of the Janata outfits while Shourie was a media personality.
If they have got away with their posturing, it is because the party's prime ministerial candidate, L.K. Advani, did not seem to be able to gather either the willpower or the intellectual conviction to take charge of the debate. As a result, he drifted along with the others with their endless carping about the denial of the right to test and loss of sovereignty which seem as hackneyed as the Left's charge of surrendering to American imperialism.
For a time, the land dispute relating to the Amarnath shrine diverted attention from the nuclear deal and revived the BJP's morale since the party could focus on the pro-Pakistani demonstrations by Muslim separatists in the Kashmir valley. Advani and others became so enthused about being able to play the nationalist card that they thought of organising a rath yatra from Amarnath in the north and linking it up with another rath yatra on the Ram setu issue from the south. The setu is the mythical bridge said to have been built by the Hindu god in the Palk Strait separating India and Sri Lanka.
The two emotive issues were expected to sweep the BJP to power in the next general election. But there was an unexpected blow to Advani's hopes when the organisers of the agitation in Jammu, who were influenced by the Vishwa Hindu Parishad (VHP) and the Bajrang Dal, frowned on the BJP trying to make political capital of the issue and even warned party chief Rajnath Singn and Advani to stay away from Jammu.
The VHP and the Bajrang Dal have been sore with the BJP and particularly Advani for failing to construct the Ram temple, their favourite project, when the party was in power at the centre. It is this internal rift in the saffron brotherhood which may be as big a cause of worry to the BJP as the lack of an emotive issue before the elections.


A new whole-time and retained fire station on a brown-field site to replace existing obsolete accommodation and to support the Service's "Modernising for the Future" programme.
The building sets new standardsin fire station design and operation and has in-built flexibility to provide for a range ofscenarios for future expansion. Rainwater harvesting provides water for training purposes and the project includes a range of specialist training facilities. Community involvement in the project was a key feature and the design allows for the use of many of the facilities by the local community.


2017 Notice: Completed updates for the 2017 vacation season included new flooring throughout, flat-screen TVs in all bedrooms, new sofa and love seat set in living area, renovated master bathroom with tiled walk-in shower, fresh paint, and new vanity. The photo slideshow and virtual tour have been updated to reflect these changes.
Amazing condo with beautiful decor. Loved the floorplan.
We had a wonderful time staying at the Seasider 405. It was a very well maintained unit with beautiful décor. Would definitely stay here again. Such a short walk to the beach made it even better.
My wife and I recently rented this unit for the month of March. We were 100% satisfied with the condo and the location. The unit is well laid out, extremely clean and quite.The deck with a view of the ocean is a great place to relax with a morning cup of coffee or a leisure afternoon glass of wine. We are looking forward to visiting again next year.
We enjoyed our stay at this condo. I would rent this one again. The only complaint I have is not enough parking at this unit.


text:
\section{Introduction}\label{sec:intro}

The purpose of this paper is to prove a reduced equivalence theorem for cross-sectional algebras of
Fell bundles over groupoids, and to prove that the \ib{} which implements the equivalence between
the reduced $C^*$-algebras is a quotient of the Muhly-Williams equivalence bimodule between the
full $C^*$-algebras \cite{muhwil:dm08}.

An increasingly influential interpretation of Hilbert bimodules (or $C^*$-cor\-respon\-dences) is
to regard them as generalized endomorphisms of $C^*$-algebras. Imprimitivity bimodules represent
isomorphisms, and a Fell bundle over a groupoid $G$ is then the counterpart of an action of $G$ on
a $C_0(G^0)$-algebra $A$. The cross-sectional algebras of the bundle are analogues of groupoid
crossed products. For example, if $G$ is a group and each imprimitivity module is of the form
$_{\alpha}A$ for an automorphism $\alpha$ of $A$ (see, for example \cite{pim:fic87}), then the
cross-sectional algebras of such bundles are precisely those arising from group crossed products;
Fell and Doran called these semidirect products in their magnum opus \cite{fd:representations2}*{\S
VIII.4.2}. In particular, if $A = C_0(\go)$ and each fibre of the Fell-bundle is 1-dimensional,
then the cross-sectional algebras are the usual groupoid $C^*$-algebras.

The classical result which motivates this paper is that if groups $G$ and $H$ act freely, properly
and transitively on the same locally compact Hausdorff space $P$ and the actions commute, then the
groups are the same. To see why, fix $x \in P$. Then for each $g \in G$, there is a unique $h \in
H$ such that $g\cdot x = x\cdot h$, and since the actions commute, $g \mapsto h$ is an isomorphism
of $G$ with $H$. Hence $C^*(G) \cong C^*(H)$ and $C^*_r(G) \cong C^*_r(H)$. A particularly powerful
viewpoint on this is the following.  If $P^{\mathrm{op}}$ is a copy of the space $P$, but with the
actions reversed so that $G$ acts on the right and $H$ on the left, then $L = G \sqcup P \sqcup
P^{\mathrm{op}} \sqcup H$ is a groupoid, called the linking groupoid, with two units. The isotropy
at one unit is $G$ and the isotropy at the other is $H$, and conjugation in $L$ by any element of
$P$ determines an isomorphism from $G$ to $H$. At the level of $C^*$-algebras, we obtain the
following very nice picture: the actions of $G$ and $H$ on $P$ induce convolution-like products
$C_c(G) \times C_c(P) \to C_c(P)$ and $C_c(P) \times C_c(H) \to C_c(P)$, and $C_c(L)$ decomposes as
a block $2 \times 2$ matrix algebra
\[
C_c(L) \cong
\begin{pmatrix}
  C_c(G)  &  C_c(P) \\
  C_c(P^{\mathrm{op}}) & C_c(H)
\end{pmatrix}.
\]
Moreover, the universal norm on $C_c(L)$ restricts to the universal norm on each of $C_c(G)$ and
$C_c(H)$, and likewise for reduced norms. The characteristic function $1_P$ of $P$ is a partial
isometry in the multiplier algebra of each of $C^*(L)$ and $C^*_r(L)$ and conjugation by $1_P$
implements the isomorphisms $C^*(G) \cong C^*(H)$ and $C^*_r(G) \cong C^*_r(H)$.

When $G$ and $H$ do not act transitively, the actions of
$G$ and $H$ on $P$ induce actions of $G$ on $P/H$ and of $H$ on
$G\backslash P$. The picture at the level of groups is now somewhat
more complicated, but the $C^*$-algebraic picture carries over nicely:
replacing $C_c(G)$ with $C_c(G, C_c(P/H))$ and $C_c(H)$ with $C_c(H,
C_c(G \backslash P))$ in the matrix above, we obtain a $^*$-algebra
$L(P)$. The $C^*$-identity allows us to extend the norm on $C_0(P/H)
\rtimes G$ to a norm on $L(P)$. Moreover, this norm is
consistent with the norm on $C_0(G \backslash P) \rtimes H$, and the
completion of $L(P)$ in this norm contains $C_0(P/H) \rtimes G$ and
$C_0(G \backslash P) \rtimes H$ as complementary full
corners. Further, this whole apparatus descends under
quotient maps to reduced crossed products.

To prove an analogue of this equivalence theorem in the context of
Fell bundles, one uses the notion of an \emph{equivalence} of Fell
bundles specified in \cite{muhwil:dm08}. The concept is closely
modeled on the situation of groups; but the natural objects on which
Fell bundles act are Banach bundles in which the fibres are
equivalence bimodules. That is, given Fell bundles $\B$ and $\CC$ over
groupoids $G$ and $H$, an equivalence between the two is, roughly
speaking, an upper semicontinuous Banach bundle $\E$ over a space $Z$
such that $Z$ admits actions of $G$ and $H$ making it into an
equivalence of groupoids in the sense of Renault, each fibre $E_z$ of
$\E$ is an {\ib} from the fibre $B_{r(z)}$ of $\B$ over $r(z)$ to
$C_{s(z)}$, and there are fibred multiplication operations $\B * \E
\to \E$ and $\E * \CC \to \E$ which are compatible with the bundle
maps, and which implement isomorphisms $B_x \otimes_{B_u} E_z \cong
E_{x\cdot z}$. Muhly and Williams show in
\cite{muhwil:dm08}*{Theorem~6.4} that given such an equivalence, the
full cross-sectional algebras $C^*(G, \B)$ and $C^*(H, \CC)$ are
Morita equivalent (Kumjian proves the corresponding statement for
reduced $C^*$-algebras in the $r$-discrete situation in
\cite{kum:pams98}).

In this paper, we show that Muhly and Williams's Morita equivalence passes to reduced algebras. We
do so by constructing a linking bundle $\linke = \B \sqcup \E \sqcup \eop \sqcup \CC$ and showing
that $\sa_{c}(L, \linke)$ has a matrix decomposition as above. We then prove that the completion of
$\linke$ in the universal norm is a linking algebra for a Morita equivalence between $C^*(G,\B)$
and $C^*(H,\CC)$, and likewise for reduced $C^*$-algebras.

We conclude by showing how to recover a generalisation of Quigg and
Spielberg's theorem \cite{quispi:hjm92} which says that the symmetric
imprimitivity bimodule arising in Raeburn's symmetric imprimitivity
theorem \cite{rae:ma88} passes under the quotient map to an
imprimitivity bimodule for reduced crossed products.

Our reduced equivalence theorem itself is not new: late in the development of this paper, we
learned that Moutou and Tu also prove that equivalent Fell bundles have Morita equivalent reduced
cross-sectional algebras \cite{moutu:xx11}. It appears that Moutou and Tu deal only with Fell
bundles whose underlying Banach bundles are required to be continuous rather than just upper
semicontinuous.  (Upper semicontinuous bundles turn out to be the more natural object in the
context of $C^*$-algebras --- see \cite{muhwil:dm08} and especially
\cite{wil:crossed}*{Appendix~C}). Moreover Moutou and Tu restrict attention to \emph{principle}
$G$-spaces for their groupoid equivalences. But these are minor points and the arguments of
\cite{moutu:xx11} would surely go through unchanged to our setting. The main new contribution in
this article that we develop the linking bundle technology to show explicitly that the full
cross-sectional algebras of the linking bundle is a linking algebra for the full cross sectional
algebras of $\B$ and $\CC$, and that the quotient map from the full to the reduced cross-sectional
algebra of the linking bundle implements the quotients $C^*(G,\B) \to C^*_r(G, \B)$ and $C^*(H,
\CC) \to C^*_r(H, \CC)$. In particular, if $I^{\CC}_r$ is the ideal of $C^*(H, \CC)$ consisting of
elements whose reduced norm is zero, then the equivalence bimodule $X_r$ which we obtain between
reduced cross-sectional algebras is the quotient of the equivalence bimodule $X$ between full
algebras by $X \cdot I^{\CC}_r$. Consequently, induction over $X$ carries $I^\CC_r$ to the
corresponding ideal $I^{\B}_r$ of $C^*(G, \B)$.

\section{Background}\label{sec:background}

Recall that for second-countable locally compact Hausdorff groupoids
$G$ and $H$, a $G\sme H$ equivalence is a locally compact Hausdorff
space $Z$ which is simultaneously a free and proper left $G$-space and
a free and proper right $H$-space (with continuous \emph{open} fibre
maps) such that the actions of $G$ and $H$ on $Z$ commute, the map
$r_Z$ induces a homeomorphism of $Z/H$ with $\go$ and the map $s_Z$
induces a homeomorphism of $G\backslash Z$ with $\ho$. Then $G$ acts
on $Z *_r Z$ by $g \cdot (y,z) = (g\cdot y, g\cdot z)$, and the
formula $h \cdot \h[g,h] = g$ defines a homeomorphism $\h[\cdot,\cdot]
: G\backslash (Z *_r Z) \to H$; and $\g[\cdot, \cdot] : Z*_s Z \to G$
is defined similarly (see \cite{mrw:jot87}*{Definition~2.1} for
details).

Recall that an upper semicontinuous Banach bundle over a locally compact Hausdorff space $Z$ is a
topological space $\B$ together with a continuous open surjection $q : \B \to Z$ such that each
$\B_z := q^{-1}(z)$ is a Banach space and: $b \mapsto \|b\|$ is upper semicontinuous; addition is
continuous from $\B *_q \B \to \B$; scalar multiplication is continuous on $\B$; and $\|b_i\| \to
0$ and $q(b_i) \to z$ implies $b_i \to 0_z \in q^{-1}(z)$.  The concept of an upper semicontinuous
Banach bundle goes back to \cite{dg:banach}, where they were called $(H)$-bundles, and the work of
Hofmann \citelist{\cite{hofkei:lnm79} \cite{hof:lnm77}\cite{hof:74}\cite{dauhof:mams68}}. Fell
calls such bundles loose in \cite{fd:representations1}*{Remark~C.1}.  Further details and comments
concerning upper semicontinuous Banach bundles are given in \cite{muhwil:dm08}*{Appendix~A} and in
the \cs-case in \cite{wil:crossed}*{Appendix~C}. As in \cite{muhwil:dm08}, a \emph{Fell bundle}
over a locally compact Hausdorff groupoid $G$ is an upper semicontinuous Banach bundle $q : \B \to
G$ endowed with a continuous bilinear associative map $(a,b) \mapsto ab$ from $\B^{(2)} :=
\set{(a,b) \in \B \times \B : s(q(a)) = r(q(b))}$ to $\B$ such that
\begin{enumerate}
\item $q(ab) = q(a)q(b)$ for all $(a,b) \in \B^{(2)}$;
\item $q(a^*) = q(a)^{-1}$ for all $a \in \B$;
\item $(ab)^* = b^*a^*$ for all $(a,b) \in \B^{(2)}$;
\item for each $u \in \go$, the fibre $A_u := q^{-1}(u)$ is a
  $\cs$-algebra under these operations; and
\item for each $g \in G \setminus \go$, the fibre $B_g := q^{-1}(g)$
  is an $A_{r(g)}\sme A_{s(g)}$-{\ib} with actions determined by
  multiplication in $\B$ and inner products $\lip<a,b> = a^*b$ and
  $\rip<a,b> = ab^*$.
\end{enumerate}
As a notational convenience, we define $r,s : \B \to \go$ by $r(a) :=
r_G(q(a))$ and $s(a) := s_G(q(a))$. See \cite{muhwil:dm08} for more
details regarding Fell bundles over groupoids.

\begin{remark} In the context of bundles over groups, the
fibres in a Fell bundle are not always assumed to be imprimitivity bimodules (they are not assumed
to be full --- see \cite{kum:pams98}*{2.4}). Bundles in which all the fibres are indeed
imprimitivity bimodules are then called \emph{saturated}. We take this condition as part of our
definition.  It should also be observed that the underlying Banach bundle of a Fell bundle over a
\emph{group} is always continuous \cite{bmz}*{Lemma~3.30}.
\end{remark}

\begin{remark}\label{rmk:why notation}
In our notation the fibre of $\B$ over a unit $u$ can be denoted either $A_u$ or $B_u$. The dual
notation allows us to emphasise its dual roles. We write $A_u$ to emphasise its role as a
$C^*$-algebra, and $B_u$ to emphasise its role as an {\ib}.  The \cs-algebra $A:=\sa_{0}(\go,\B)$
is called the \emph{\cs-algebra of the Fell bundle $\B$ over $\go$}.
\end{remark}

We recall from \cite{muhwil:dm08} the definition of an equivalence of
Fell bundles. First, fix a second-countable locally compact Hausdorff
groupoid $G$, a left $G$-space $Z$, a Fell bundle $q_G : \B \to G$,
and a Hausdorff space $\E$ together with a continuous open surjection
$q : \E \to Z$. Again, as a notational convenience, we shall write $r$
for the composition $r_Z \circ q : \E \to \go$. we say that $\B$
\emph{acts on the left of $\E$} if there is a pairing $(b, e) \mapsto
b\cdot e$ from $\B * \E = \set{(b,e) \in \B \times \E : s(b) = r(e)}$
to $\E$ such that
\begin{enumerate}
\item $q(b\cdot e) = q_G(b)q(e)$ for $(b,e) \in \B * \E$;
\item $a\cdot(b\cdot e) = (ab)\cdot e$ whenever $(a,b) \in \B^{(2)}$
  and $(b,e) \in \B * \E$;
\item $\|b \cdot e\| \le \|b\| \|e\|$ for $(b,e) \in \B *
  \E$.\footnote{The equality appearing in the corresponding item in
    \cite{muhwil:dm08} is a typographical error.}
\end{enumerate}
If $\E$ is a right $H$-space, and $q_H : \CC \to H$ is a Fell bundle,
then a right action of $\CC$ on $\E$ is defined similarly.

Now fix second-countable locally compact Hausdorff groupoids $G$ and
$H$ and a $G\sme H$ equivalence $Z$. Suppose that $q_G : \B \to G$ and
$q_H : \CC \to H$ are Fell bundles. Fix a Banach bundle $q : \E \to
Z$. We write $\E *_s \E$ for $\set{(e,g) \in \E \times \E : s(e) =
  s(g)}$ and we define $\E *_r \E$ similarly. We call $\E$ a $\B\sme
\CC$ equivalence if:
\begin{enumerate}
\item\label{it:FBE actions} there are a left action of $\B$ on $\E$
  and a right action of $\CC$ on $\E$ which commute;
\item\label{it:FBE ips} there are sesquilinear maps
  $\lip\B<\cdot,\cdot> : \E *_s \E \to \B$ and $\rip\CC<\cdot,\cdot> :
  \E *_r \E \to \CC$ such that the relations
  \begin{enumerate}
  \item\label{it:FBE ips1} $q_G(\lip\B<e,f>) = \g[q(e),q(f)]$ and
    $q_H(\rip\CC<e,f>) = \h[q(e),q(f)]$,
  \item\label{it:FBE ips2} $\lip\B<e,f>^* = \lip\B<f,e>$ and
    $\rip\CC<e,f>^* = \rip\CC<f,e>$,
  \item\label{it:FBE ips3} $b \lip\B<e,f> = \lip\B<b\cdot e, f>$ and
    $\rip\CC<e,f> c = \rip\CC<e,f\cdot c>$ and
  \item\label{it:FBE ips4} $\lip\B<e,f>\cdot g = e \cdot \rip\CC<f,g>$
  \end{enumerate}
  are satisfied whenever they make sense; and
\item\label{it:FBE imprim} under the actions described in~(\ref{it:FBE
    actions}) and the inner-products defined in~(\ref{it:FBE ips}),
  each $\E_z := q^{-1}(z)$ is an $A_{r(z)}\sme D_{s(z)}$-{\ib}.
\end{enumerate}

As in \cite{simwil:jot11}, if $G,H$ are second-countable locally
compact Hausdorff groupoids with Haar systems $\lg, \lh$ and $Z$ is a
$G \sme H$ equivalence, we write $\zop$ for the ``opposite
equivalence'' $\zop = \set{\bar{z} : z \in Z}$ with $r(\bar{z}) =
s(z)$, $s(\bar{z}) = r(z)$, $h \cdot \bar z := \overline{z \cdot
  h^{-1}}$ and $\bar z \cdot g := \overline{g^{-1} \cdot z}$. Then $L
:= G \sqcup Z \sqcup \zop \sqcup H$ with $\lo := \go \sqcup \ho
\subseteq L$ is a groupoid containing $G$ and $H$ as subgroupoids: we
extend the inverse map to $Z$ and $\zop$ by setting $z^{-1}
:= \bar{z}$; and multiplication between $Z$ and $G,H$ is implemented
by the left and right actions, while multiplication between $Z$ and
$\zop$ is implemented by $\g[\cdot,\cdot]$ and $\h[\cdot,\cdot]$. See
\cite{simwil:jot11}*{Lemma~5} for details. There is a Haar system on
$L$ determined by
\[
\ll^w(F) := \begin{cases}
  \textstyle\int_{G} F(g)\,d\lg^{w}(g) + \int_{H} F(z\cdot
  h)\,d\lh^{s(z)}(h) &\text{ if $w \in \go$}\vadjust{\vskip0.5em}\\
  \textstyle\int_{G} F(\bar{y}\cdot g)\,d\lg^{s(\bar{y})}(g) +
  \int_{H} F(h)\,d\lh^{w}(h) &\text{ if $w \in \ho$}
\end{cases}
\]
for $F \in C_c(L)$ and $w \in \lo$ (see \cite{simwil:jot11}*{Lemma~6}). For $u \in \go$ and $v \in
\ho$, we write $\sigma^u_Z$ and $\sigma^v_{\zop}$ for the restrictions of $\ll^u$ to $Z$ and of
$\ll^v$ to $\zop$. The main results of \cite{simwil:jot11} say that $\cs(L,\ll)$ contains
$\cs(G,\lg)$ and $\cs(H,\lh)$ as the complementary full corners determined by the multiplier
projections $1_{\go}$ and $1_{\ho}$, and that this Morita equivalence passes under the quotient map
$\cs(L,\ll) \to \cs_r(L,\ll)$ to reduced groupoid \cs-algebras. Our goal in this article is to
establish the corresponding statement for Fell bundles. As a first step, we show in the next
section how to construct from an equivalence of Fell bundles a linking bundle over the linking
groupoid.

\section{Linking bundles}\label{sec:linking bundle}

Let $G$ and $H$ be locally compact Hausdorff groupoids, let $Z$ be a
$G\sme H$ equivalence, and let $L$ be the linking groupoid as
above. Suppose that $\pg : \B \to G$ and $\ph : \CC \to H$ are
upper-semicontinuous Fell bundles, and that $q : \E \to Z$ is a bundle
equivalence. We denote by $A$ the \cs-algebra $\sa_{0}(\go,
p_G^{-1}(\go))$ of the bundle $\B$, and by $D$ the \cs-algebra
$\sa_{0}(\ho, p_H^{-1}(\ho))$ of $\CC$; so the fibre over $u \in \go$
is $A_u$, the fibre over $v \in \ho$ is $D_v$, and each $\E_z$ is an
$A_{r(z)}\sme D_{s(z)}$-{\ib}.

Let $\eop=\set{\bar e:e\in\E}$ be a copy of the topological space $\E$
endowed with the conjugate Banach space structure $\alpha \bar e +
\bar f = \overline{(\overline{\alpha} e + f)}$ on each fibre. Then
$\qop:\eop\to\zop$ is an upper-semicontinuous Banach bundle with
$\qop(\bar e)=\overline{q(e)}$.  We have $s(\bar e) = s(\qop(\bar e))
= r(e)$ and likewise $r(\bar e) = s(e)$, so we obtain a right
$\B$-action and a left $\CC$-action on $\eop$ by
\begin{equation}\label{eq:eop actions}
  \bar e\cdot b=\overline{b^{*}\cdot e} \quad\text{ and }\quad c\cdot \bar e =\overline{e\cdot c^{*}}.
\end{equation}
The inner products on $\eop*_{r}\eop$ and $\eop *_{s} \eop$ are given
by $\rip\B<\bar e,\bar f>=\lip\B<e,f>$ and $\lip\CC<\bar e,\bar
f>=\rip\CC<e,f>$.  Routine calculations show that each $E^{\op}(\bar
z)$ is the dual {\ib} $E(z)^{\sim}$ of $E(z)$. Since $s(z)=r(\bar z)$
and $r(z)=s(\bar z)$, axioms (a), (b)~and~(c) of
\cite{muhwil:dm08}*{Definition~6.1} hold, so $\eop$ is a
$\CC\sme\B$-equivalence.

Let $\linke=\B\sqcup\E\sqcup\eop\sqcup\CC$ and define
$\linkq:\linke\to L$ by
\[
\linkq|_{\B} = \pg, \quad \linkq|_{\CC} = \ph,\quad \linkq|_{\E} =
q\quad\text{ and }\quad \linkq|_{\eop} = \qop.
\]
Since $e \mapsto \bar e$ is a fiberwise-isometric homeomorphism from
$\E$ to $\eop$ and since $z \mapsto \bar z$ is a homeomorphism from
$Z$ to $\zop$, the bundle $\linke$ is an upper semicontinuous Banach
bundle. Let
\begin{equation*}
  \linke^{(2)}=\set{(\ea,\eb)\in \linke\times\linke:
    s(\linkq(\ea))=r(\linkq(\eb))}.
\end{equation*}
Define $m:\linke^{(2)}\to\linke$ to coincide with the given
multiplications on $\B$ and $\CC$ and with the actions of $\B$ and
$\CC$ on $\E$ and $\eop$, and to satisfy
\begin{equation*}
  m(e, \bar f)=\lip\B<e,f>\text{ for
      $(e,f)\in\E*_{s}\E$}
    \quad\text{and}\quad
    m(\bar e, f)=\rip\CC<e,f>\text{ for $(e,f)\in\E*_{r}\E$.}
\end{equation*}
We define $\ea\mapsto \ea^{*}$ on $\linke$ to extend the given
involutions on $\B$ and $\CC$ by setting $e^{*}=\bar e$ on $\E$ and
${\bar e}^{*}=e$ on $\eop$.

\begin{lemma}
  \label{lem:linking bundle}
  With notation as above, the bundle $\linke$ is a Fell bundle over
  $L$. Moreover, the \cs-algebra $\sa_0(\lo, \linkq^{-1}(\lo))$ is
  isomorphic to $A \oplus D$.
\end{lemma}
\begin{proof}
  We know already that $\linke$ is an upper-semicontinuous Banach
  bundle, that each $\linke_u$ is a \cs-algebra and each $\linke_x$ is
  a $\linke_{r(x)}\sme \linke_{s(x)}$-{\ib}. The fibre map $q$
  preserves multiplication and involution by definition of these
  operations. The operations are continuous because they are
  continuous on each component of $\linke$ and of $\linke * \linke$,
  and the components are topologically disjoint. That $(\ea\eb)^* =
  \eb^*\ea^*$ is clear on $\B * \B$ and $\CC * \CC$, follows from the
  inner-product axioms on $\E * \eop$ an $\eop * \E$, and follows
  from~\eqref{eq:eop actions} for the remaining
  pairings. Associativity for triples from $\E * \eop * \E$ and $\eop
  * \E * \eop$ follows from the {\ib} axiom $\Lip<e,f> g = e
  \Rip<f,g>$, and is clear for all other triples.

  The map $f \mapsto (f|_{\go}, f|_{\ho})$ is a surjection $\sa_0(\lo,
  \linkq^{-1}(\lo)) \to A \oplus D$, and the inverse makes sense
  because $\go$ and $\ho$ are topologically disjoint. Hence
  $\sa_0(\lo, \linkq^{-1}(\lo)) \cong A \oplus D$
\end{proof}

Resume the hypotheses of Lemma~\ref{lem:linking bundle}. It is routine
to check that $(\Pg \phi)(g) := \chi_{\go}(r(g)) \phi(g)$ determines a
bounded self-adjoint map on $\sacgb$ under the inner-product
$(\phi,\psi) \mapsto \phi^*\psi$, and hence extends to a multiplier
projection\label{page:multiplier}, also denoted $\Pg$, of
$\cs(G;\B)$. Taking adjoints, $(\phi \Pg)(\ea) = \chi_{\go}(s(\ea))
\phi(\ea)$. The corresponding projection $\Ph$ for $H$ is defined
similarly.

\begin{remark}
  As in \cite{simwil:jot11}, we think of $\varphi \in \Gamma_c(L,
  \linke)$ as a matrix
  \[
  \begin{pmatrix} \varphi_{G} & \varphi_{Z} \\ \varphi_{\zop} &
    \varphi_{H}\end{pmatrix}
  \]
  where $\varphi_{G}$ is the restriction of $\varphi$ to $G \subseteq
  L$ and similarly for the other terms. With respect to this
  decomposition, we have
  \[
  \varphi\psi =
  \begin{pmatrix} \varphi_G \psi_G + \varphi_Z \psi_{\zop} & \varphi_G
    \psi_Z + \varphi_Z \psi_H \\
    \varphi_{\zop} \psi_G + \varphi_H \psi_Z & \varphi_{\zop} \psi_Z +
    \varphi_H \psi_H \end{pmatrix},
  \]
where we have used juxtaposition for the convolution product restricted to the various
corners.\footnote{In fact, the products in the matrix can be expressed in terms of the
inner-products and module actions from \cite{muhwil:dm08}*{Theorem~6.4}.} Moreover $\varphi_G = \Pg
\varphi \Pg$, $\varphi_Z = \Pg \varphi \Ph$, $\varphi_{\zop} = \Ph \varphi \Pg$, and $\varphi_H =
\Ph \varphi \Ph$.
\end{remark}

\begin{lemma}
\label{lem:full corners}
Resume the hypotheses of Lemma~\ref{lem:linking bundle}. Then $\Pg$
and $\Ph$ are full multiplier projections of $\cs(L; \linke)$.
\end{lemma}
\begin{proof}
We just show that $\Pg$ is full; the corresponding statement for $\Ph$ follows by symmetry. Fix
$\varphi, \psi \in \Gamma_c(L;\linke)$. Using the matrix notation established above, we have
\[
\varphi \Pg \psi =
\begin{pmatrix}
  \varphi_G \psi_G & \varphi_G \psi_Z \\
  \varphi_{\zop} \psi_G & \varphi_{\zop}\psi_Z
\end{pmatrix}.
\]
That elements of the form $\varphi_G \psi_G$ span a dense subalgebra
of $\Gamma_c(G; \B)$ is clear.  That elements of the form $\varphi_G
\psi_Z$ span a dense subspace of $\Gamma_c(Z; \E)$ and likewise that
elements of the form $\varphi_{\zop} \psi_G$ span a dense subspace
of $\Gamma_c(\zop; \eop)$ follows from
\cite{muhwil:dm08}*{Proposition~6.10}. That elements of the form
$\varphi_{\zop}\psi_Z$ span a dense subspace of $\Gamma_c(H;\CC)$
follows from the argument which establishes axiom~(IB2) in
\cite{muhwil:dm08}*{Section~7}.
\end{proof}

Recall that the inductive-limit topology on $C_c(X)$ for a locally compact Hausdorff space $X$ is
the unique finest locally convex topology such that for each compact $K \subseteq X$, the inclusion
of $C_c(X)^K = \{f \in C_c(X) : \supp(f) \subseteq K\}$ into $C_c(X)$ is continuous (see for
example \cite{fd:representations1}*{II.14.3} or \cite{rw:morita}*{\S D.2}). In particular,
\cite{rw:morita}*{Lemma~D.10} says that to check that a linear map $L$ from $C_{c}(X)$ into any
locally convex space $M$ is continuous, it suffices to see that if $f_{n}\to f$ uniformly and if
all the supports of the $f_{n}$ are contained in the same compact set $K$, then $L(f_{n})\to L(f)$.

\begin{remark}
  \label{rem:ourproof}
  We are now in a situation analogous to that of
  \cite{simwil:jot11}*{Remark~8}.  By the Disintegration Theorem for
  Fell bundles, \cite{muhwil:dm08}*{Theorem~4.13}, any pre-$\cs$-norm
  $\|\cdot\|_\alpha$ on $\Gamma_c(L;\linke)$ which is continuous in
  the inductive-limit topology is dominated by the universal
  norm. Hence the argument of \cite{simwil:jot11}*{Remark~8} shows that
  $\Pg \cs_\alpha(L;\linke) \Ph$ is a $\cs_\alpha(G;\B) \sme
  \cs_\alpha(H;\CC)$-imprimitivity bimodule. So to prove that
  $\cs(G;\B)$ is Morita equivalent to $\cs(H, \CC)$ we just need to
  show that for $F \in \Pg \sa_{c}(L,\linke) \Pg$, the universal norms
  $\|F\|_{\cs(L,\linke)}$ and $\big\|F|_G\big\|_{\cs(G;\B)}$ coincide,
  and similarly for the reduced algebras (the corresponding statements
  for $H$ hold by symmetry).
\end{remark}

\section{The reduced norm}

In this section we recall the construction of the reduced
cross-sectional algebra of a Fell bundle.  We first discuss how to
induce representations from \cs-algebra of the restriction of a Fell
bundle to a closed subgroupoid up to representations of the
\cs-algebra of the whole bundle. We then apply this construction to
the closed subgroupoid $\go$ of $G$ to induce representations of the
$C^*$-algebra $A = \sa_{0}(G^{(0)}, \B|_{G{(0)}})$ up to
representations of $\cs(G;\B)$. These are, by definition, the regular
representations whose supremum determines the reduced norm.

\subsection{Induced representations}\label{sec:induc-repr}

Let $G$ be a second countable locally compact Hausdorff groupoid with Haar system
$\sset{\lambda^{u}}_{u\in\go}$.  Let $q : \B\to G$ be a separable Fell bundle as described in
\cite{kmqw:nyjm10}*{\S1.3}. Assume that $H$ is a closed subgroupoid of $G$ with Haar system
$\sset{\alpha^{u}}_{u\in\ho}$.  We write $q_{H}:\B\restr H\to H$ for the Fell bundle obtained by
restriction to $H$.  We want to induce representations of $\cs(H,\B\restr H)$ to $\cs(G;\B)$ using
the Equivalence Theorem \cite{muhwil:dm08}*{Theorem~6.4} for Fell bundles.  We will use the set-up
and notation from \cite{ionwil:pams08}*{\S2}.  In particular, we recall that $\gho=s^{-1}(\ho)$ is
a $(\hg,H)$-equivalence where $\hg$ is the \emph{imprimitivity groupoid} $(\gho*_{s}\gho)/H$.  Let
$\sigma:\hg\to G$ be the continuous map given by $\sigma\bigl([x,y]\bigr) =xy^{-1}$.  The pull-back
Fell bundle $\sigma^{*}q:\sigma^{*}\B\to \hg$ is the Fell bundle $\sigma^{*}\B = \{([x,y], b) :
[x,y] \in \hg, b \in \B, \sigma([x,y]) = q(b)\}$ with bundle map $\sigma^*([x,y], b) = [x,y]$ over
$\hg$.

Let $\E=q^{-1}(\gho)$; then $q$ restricts to a map $q:\E\to \gho$. We
wish to make this Banach bundle into a $\ssb\sme\bh$-equivalence (see
\cite{muhwil:dm08}*{Definition~6.1}).  It is clear how $\bh$ acts on
the right of $\E$, and we get a left action of $\ssb$ via
\begin{equation*}
  \bigl([x,y],b\bigr)\cdot e:=be\quad\text{for $q(e)=yh$.}
\end{equation*}
(Since $q(b)=xy^{-1}$, $q(be)=xh$ as required.) The ``inner products''
on $\E*_{r}\E$ and $\E*_{s}\E$ are given by
\begin{equation*}
  \rip\bh<e,f>=e^{*}f\quad\text{and}\quad \lip\ssb<e,f>=\bigl([q(e),
  q(f)], ef^{*}\bigr),
\end{equation*}
respectively.  It now straightforward to check that $\E$ is a
$\ssb\sme\bh$-equivalence.  By \cite{muhwil:dm08}*{Theorem~6.4},
$\sa_{c}(\gho,\E)$ is a pre-{\ib} with actions and inner products
determined by
\begin{align}
  F\cdot \phi(z)&=\int_{G}
  F\bigl([z,y]\bigr)\phi(y)\,d\lambda_{s(z)}(y),\label{eq:1}
  \\
  \phi\cdot g(z)&=\int_{H}
  \phi(zh)g(h^{-1})\,d\alpha^{s(z)}(h),\label{eq:2} \\
  \Rip<\phi,\psi>(h)
  &=\int_{G}\phi(y)^{*}\psi(yh)\,d\lambda_{r(h)}(y),\label{eq:3}
  \\
  \Lip<\phi,\psi>\bigl([x,y]\bigr)&=
  \int_{G}\phi(xh)\psi(yh)^{*}\,d\alpha^{s(x)}(h)\label{eq:4}
\end{align}
for $F\in\sa_{c}(\hg,\ssb)$, $\phi,\psi\in \sa_{c}(\gho,\E)$ and $g\in
\sa_{c}(H, \bh)$.  The completion $\X=\X_{H}^{G}$ is a
$\cs(\hg,\ssb)\sme\cs(H,\bh)$-{\ib}.
\begin{remark}
  \label{rem-oops}
  It is pleasing to note that the formalism of Fell bundles is such
  that equations \eqref{eq:1}--\eqref{eq:4} are virtually identical to
  those in the scalar case: see
  \cite{ionwil:pams08}*{Eq. (1)--(4)}.\footnote{Well, they would be if
    it weren't for the typos in equations (1)~and (4) in
    \cite{ionwil:pams08}.}  The only difference is that complex
  conjugates in the scalar case are replaced by adjoints.
\end{remark}

To construct induced representations using the machinery of
\cite{rw:morita}*{Proposition~2.66}, we need a nondegenerate
homomorphism $V : \cs(G;\B) \to \L(\X)$ which will make
$\X$ into a right Hilbert $\cs(G;\B)\sme\cs(H,\bh)$-bimodule (the data
needed to induce representations \emph{a la} Rieffel.)  Define
$V:\sacgb \to \Lin(\sa_{c}(\gho,\E))$ by
\begin{equation}\label{eq:V-def}
  V(f)(\phi):=\int_{G}f(y)\phi(y^{-1}z)\,d\lambda^{r(z)}(y).
\end{equation}
By the Tietz Extension Theorem for upper semicontinuous Banach bundles
\cite{muhwil:dm08}*{Proposition~A.5}, each $\phi \in \sa_{c}(\gho,\E)$
is the restriction of an element of $\sacgb$. So the argument of
\cite{ionwil:pams08}*{Remark~1} and the paragraph which follows
yields\begin{equation*} \Rip<V(f)\phi,\psi>=\Rip<\phi,V(f^{*})\psi>.
\end{equation*}
The map $f\mapsto \Rip<V(f)\phi,\psi>$ is continuous in the
inductive-limit topology, and the existence of approximate units in
$\sacgb$ implies that
\[
\set{V(f)\phi:\text{$f\in \sacgb$ and $\phi\in\sa_{c}(\gho,\E)$}}
\]
spans a dense subspace of $\sa_{c}(\gho,\E)$.  Then
\cite{kmqw:nyjm10}*{Proposition~1.7} implies that $V$ is bounded and
extends to a nondegenerate homomorphism as required.

Now if $L$ is a representation of $\cs(H,\bh)$, then the induced
representation $\Ind_{H}^{G}L$ of $\cs(G;\B)$ acts on the completion
of $\X\odot \H_{L}$ with respect to
\begin{equation*}
  \ip(\phi\tensor h|\psi\tensor k)=
  \bip(L(\Rip<\psi,\phi>)h|k)_{\H_{L}}.
\end{equation*}
Fix $\phi\in\sa_{c}(\gho,\E)$. Writing $f \cdot \phi$ for $V(f)\phi$,
we have
\begin{equation*}
  (\Ind_{H}^{G}L)(f)(\phi\tensor h)=f\cdot\phi\tensor h,
\end{equation*}
and, as in \cite{ionwil:pams08}*{Remark~1}, $f\cdot \phi=f*\phi$.

\subsection{Regular Representations and the reduced
  \texorpdfstring{\cs}{C*}-algebra}
\label{sec:regul-repr-cs_rg}

\emph{Regular representations} are, by definition, those induced from $A=\sa_{0}(\go,\B)$.  Thus,
in the notation of Section~\ref{sec:induc-repr}, $H=\go$, $\gho=G$, $\E=\B$, and we write $\A$ in
place of $\B\restr{\go}$ (see Remark~\ref{rmk:why notation}); in particular each $B(x)$ is a
$A(r(x))\sme A(s(x))$-{\ib}. We also have $\Rip<\phi,\psi>={\phi^{*}\psi\restr{\go}}$, with the
product being computed in $\sa_{c}(L,\linke)$.

Let $\pi$ be a representation of $A$ on $H_\pi$. Let $\tilde{\pi}$ be
the extension of $\pi$ to $M(A)$, and let $i: C_0(\go) \to M(A)$ be
the map characterised by $\big(i(f)a\big)(u) = f(u)a(u)$ for $u \in
\go$. Then $\phi := \tilde{\pi} \circ i$ is a representation of
$C_0(\go)$ on $H_\pi$ which commutes with $\pi$. Example~F.25 of
\cite{wil:crossed} shows that there is a Borel Hilbert bundle $\go *
\HH$ and a finite Radon measure $\mu$ on $\go$ such that $\pi$ is
equivalent to a direct integral $\int^\oplus_{\go} \pi_u \,d\mu$, and
such that if $L : f \to L_f$ is the diagonal inclusion of $C_0(\go)$
in $B(L^2(\go * \HH, \mu))$, then $\pi(i(f)a) = L_f \pi(a)$ for $a \in
A$. So each $\pi_u$ factors through $A_u$. We will usually write
$\pi_{u}\bigl(a(u)\bigr)$ in place of $\pi_{u}(a)$ for $a\in A$. See
\cite{muhwil:nyjm08}*{p.~46} for more details. The regular
representation $\Ind \pi=\Ind_{\go}^{G} \pi$ then acts on the
completion of $\sacgb\odot L^{2}(\go*\HH,\mu)$ with respect to
$\bip(\phi\tensor h|\psi\tensor k) =
\bip(\pi\bigl(\psi^{*}*\phi\bigr)h|k)$, and a quick calculation yields
\begin{equation}
  \label{eq:6}
  \bip(\phi\tensor h|\psi\tensor k)
  = \int_{\go}\int_{G} \bip(\pi_{u}\bigl(\psi(x)^{*}\phi(x)\bigr)h(u) |{k(u)})\,d\lambda_{u}(x)\,d\mu(u).
\end{equation}
Then $\Ind\pi$ acts by:
\begin{equation}
  \label{eq:7}
  (\Ind\pi)(f)(\phi\tensor h)=V(f)(\phi)\tensor h= f\cdot\phi\tensor h.
\end{equation}

We next define the reduced algebra of a Fell bundle. We define the
reduced norm by analogy with the one-dimensional case as the supremum
of the norms determined by induced representations of $A$. We then
show that this agrees, via $V$, with the operator norm on
$\L(\X)$. This is equivalent to Definition~2.4 and Lemma~2.7 of
\cite{moutu:xx11}, though the roles of definition and lemma are
interchanged.

\begin{definition}
  We define the \emph{reduced norm} on $\sacgb$ by
  \begin{equation*}
    \|f\|_{r}:=\sup\set{\|(\Ind\pi)(f)\|:\text{$\pi$ is a
        representation of $A$}}.
  \end{equation*}
\end{definition}

Since the kernel of $\Ind\pi$ depends only on the kernel of $\pi$ (see
\cite{rw:morita}*{Corollary~2.73}), we have $\|f\|_{r}=\|(\Ind\pi)(f)\|$ for any faithful
representation $\pi$ of $A$. We define the \emph{reduced \cs-algebra} $\cs_{r}(G;\B)$ of $\B$ to be
the quotient of $\cs(G;\B)$ by $I_{\cs_{r}(G,\B)}:=\set{a\in\cs(G,B):\|a\|_{r}=0}$.

\begin{lemma}
  \label{lem-x-red}
  Let $\X=\X_{\go}^{G}$ and $V:\cs(G;\B)\to \L(\X)$ the homomorphism
  determined by~\eqref{eq:V-def}.  Then $\ker V=I_{\cs_{r}(G;\B)}$ and
  $V$ factors through an injection of $\cs_{r}(G;\B)$ into $\L(\X)$.
  In particular, $\|V(f)\|=\|f\|_{r}$.
\end{lemma}
\begin{proof}
  Let $\pi$ be a faithful representation of $A$.  Then for any $x\in
  \X$, $h\in\H_{\pi}$ and $f\in\cs(G;\B)$, we have
  \begin{equation}
    \label{eq:12}
    \bigl\|(\Ind\pi)(f)(x\tensor h)\bigr\|^{2}=\bigl\|V(f)(x)\tensor
    h\bigr\|^{2} = \bip(\pi\bigl(\Rip<V(f)(x),V(f)(x)>\bigr)h|h).
  \end{equation}
  Thus if $V(f)=0$, then $(\Ind\pi)(f)=0$.  On the other hand, given
  $x$ and $f$, we can find a unit vector $h$ such that the right-hand
  side of \eqref{eq:12} is at least
  \begin{equation*}
    \frac12\bigl\|\pi\bigl(\Rip<V(f)(x),V(f)(x)>\bigr)\bigr\|=\frac12
    \bigl\|V(f)(x)\bigr\|^{2}.
  \end{equation*}
  Therefore $\Ind\pi(f)=0$ implies that $V(f)=0$.  We have shown that
  $\ker V=\ker(\Ind\pi)$, and hence $V$ factors through an injection
  of $\cs_{r}(G;\B)$ into $\L(X)$ as claimed.
\end{proof}

We digress briefly to check that the definition of the reduced
$C^*$-algebra which we have given is compatible with existing
definitions on some special cases.

\begin{example}[The Scalar Case: Groupoid \cs-Algebras]
  \label{ex-scalar}
  Let $\B=G\times\C$ so that $\cs(G;\B)=\cs(G)$.  So $A=C_{0}(\go)$,
  and $\pi$ defined by multiplication on $L^{2}(\go,\mu)$ is a
  faithful representation of $A$. Then $\Ind\pi$ acts on the
  completion $H$ of $C_{c}(G) \odot L^{2}(\go)$ and, if we let $\nu =
  \mu \circ \lambda$, then \eqref{eq:6} becomes
  \begin{equation*}
    \bip(\phi\tensor h|\psi\tensor k)= \int_{G}
    \bip(\phi(x)h(s(x))|{\psi(x)k(s(x))} )\,d\nu^{-1}(x).
  \end{equation*}
  Hence there is a unitary $U$ from $H$ onto $L^{2}(G,\nu^{-1})$ defined by $U(\phi\tensor h)(x)=\phi(x)h(s(x))$,
  and $U$ intertwines $\Ind\pi$ with the representation
  $(\Ind\mu)(f)\xi(x)=\int_{G}f(y)\xi(y^{-1}x)\,d\lambda^{r(x)}(y)$. Hence
  our definition of the reduced norm agrees with the usual definition
  (see \cite{simwil:jot11}*{\S3}, for example), and
  $\cs_{r}(G\times\C)=\cs_{r}(G)$.
\end{example}

\begin{example}[Groupoid Crossed Products]
  \label{ex-crossed-prod}
  Suppose that $(\A,G,\alpha)$ is a dynamical system and form the
  associated semidirect product Fell bundle $\B=r^{*}\A$ as in
  \cite{muhwil:dm08}*{Example~2.1}.  Working with the appropriate
  $\A$-valued functions, as in \cite{muhwil:dm08}*{Example~2.8}, a
  quick calculation starting from~\eqref{eq:6} gives
  \begin{multline}
    \bip(f\tensor h|g\tensor k)
    = \\ \int_{\go}\int_{G} \bip(\pi_{u}\bigl(\alpha_{x}^{-1}
    \bigl(f(x)\bigr)\bigr)h(u) |
    {\pi_{u}\bigl(\alpha_{x}^{-1}\bigl(g(x)\bigr)\bigr) k(u)} ) \,
    d\lambda_{u}(x)\,d\mu(u)\label{eq:10}.
  \end{multline}
  (Since $\lambda_{u}$ is supported on $G_u$, each
  $\alpha_x^{-1}(g(x)) \in A_u$, so the integrand makes sense.)  Let
  $G*\HH_{s}$ be the pull back of $\go*\HH$ via $s$. Given a
  representation $\pi$ of $A$, there is a unitary $U$ from the space
  of $\Ind\pi$ to $L^{2}(G*\HH_{s},\nu^{-1})$ defined by $U(f\tensor
  h)=\pi_{s(x)}\bigl(\alpha_{x}^{-1}\bigl(f(x)\bigr)\bigr)h\bigl(s(x)\bigr)$. This
  $U$ intertwines $\Ind\pi$ with the representation $L^{\pi}$ given by
  \begin{equation}\label{eq:9}
    L^{\pi}(f)\xi(x)=\int_{G}
    \pi_{s(x)}\bigl(\alpha_{x}^{-1}\bigl(f(y)\bigr)\bigr)
    \xi(y^{-1}x)\,d\lambda^{r(x)} (y).
  \end{equation}
  Applying this with a faithful representation $\pi$ of $A$, we deduce
  that $\A \rtimes_{\alpha,r} G \cong \cs_{r}(G, r^*\A)$.
\end{example}

\begin{remark}
\label{rem-problem}
In Examples \ref{ex-scalar}~and \ref{ex-crossed-prod}, the essential
step in finding a concrete realization of the space of $\Ind\pi$ is
to ``distribute the $\pi_{u}$'' in the integrand in \eqref{eq:6} to
both sides of the inner product.  But for general Fell bundles,
$\pi_{u}(\phi(x))$ makes no sense for general $x\in G_{u}$.  This
often makes analyzing regular representations of Fell bundle
\cs-algebras considerably more challenging.
\end{remark}

\begin{example}
  \label{ex-pointmass}
  Any representation $\pi_u$ of $A_u$ determines a representation
  $\pi_u \circ \epsilon_u$ of $A$ by composition with evaluation at
  $u$ (in the direct-integral picture, $\pi = \pi_u \circ \epsilon_u$
  is a direct integral with respect to the point-mass $\delta_u$). We
  abuse notation slightly and write $\Ind\pi_{u}$ for $\Ind(\pi_u
  \circ\epsilon_u)$ which acts on the completion of
  $\sacgb\odot\H_{\pi_{u}}$ under
  \begin{equation}\label{eq:8}
    \ip(\phi\tensor h|\psi\tensor
    k)=\int_{G}\bip(\pi_{u}\bigl(\psi(x)^{*}\phi(x) \bigr)h(u)|{k(u)})
    \,d\lambda_{u}(x).
  \end{equation}
  Equation~\eqref{eq:8} depends only on $\phi\restr{G_{u}}$ and
  $\psi\restr{G_{u}}$; and conversely each element of
  $\sa_{c}(G_{u},\B)$ is the restriction of some $\phi\in\sacgb$ by
  the Tietz Extension Theorem for upper semicontinuous Fell bundles
  \cite{muhwil:dm08}*{Proposition~A.5}.  So we can view the space of
  $\Ind\pi_{u}$ as the completion of $\sa_{c}(G_{u},\B)\odot
  \H_{\pi_{u}}$ with respect to~\eqref{eq:8}.
\end{example}

\section{The equivalence theorem}\label{sec:equivalence theorem}

Fix for this section second-countable locally compact Hausdorff
groupoids $G$ and $H$ with Haar systems $\lg$ and $\lh$, a $G\sme H$
equivalence $Z$, Fell bundles $\pg : \B \to G$ and $\ph : \CC \to H$
and a $\B\sme\CC$ equivalence $q : \E \to Z$.  Let $\ll$ denote the
Haar system on $L$ obtained from \cite{simwil:jot11}*{Remark~11}, and
let $\linkq : \linke \to L$ be the linking bundle of
Section~\ref{sec:linking bundle}.

\begin{thm}
  \label{thm-main-reduced}
  Suppose that $F \in \sa_{c}(L,\linke)$ satisfies $f(\zeta) = 0$ for
  all $\zeta \in L \setminus G$. Let $f := F|_G \in \sacgb$.  Then
  $\|F\|_{\cs(L,\linke)}=\|f\|_{\cs(G;\B)}$ and
  $\|F\|_{\cs_{r}(L;\linke)}=\|f\|_{\cs_{r}(G;\B)}$. Moreover, $\Pg
  \cs(L,\linke) \Ph$ is a $\cs(G;\B) \sme \cs(H;\CC)$-{\ib}, and $\Pg
  \cs_{r}(L,\linke) \Ph$ is a $\cs_{r}(G;\B) \sme
  \cs_{r}(H,\CC)$-{\ib} which is the quotient module of $\Pg
  \cs(L,\linke) \Ph$ by the kernel $I_r$ of the canonical homomorphism
  of $\cs(H,\CC)$ onto $\cs_{r}(H,\CC)$.
\end{thm}

\begin{remark}
  \label{rmk-scalar-case}
Recall the set-up of Example~\ref{ex-scalar}. It is not difficult to
see that if $G$ and $H$ are groupoids and $Z$ is a $G\sme H$
equivalence, then the trivial bundle $Z \times \C$ is a $(G \times
\C)\sme(H \times \C)$ equivalence. Hence we recover Theorem~13,
Proposition~15 and Theorem~17 of \cite{simwil:jot11} from
Theorem~\ref{thm-main-reduced}.
\end{remark}

To prove Theorem~\ref{thm-main-reduced}, we first establish some
preliminary results. Our key technical result is a norm-estimate for
the representations of $\sacgb \subseteq \sa_{c}(L,\linke)$ coming
from elements of $H^0$.

Let $\sset{\rho_{Z}^{v}}_{v\in\ho}$ be the Radon measures on $Z$
introduced in \cite{simwil:jot11}*{Theorem~13}.  For each $v\in\ho$,
fix $\zeta \in Z$ with $s(\zeta) = v$, and define a $D(v)$-valued form
on $\Y_{0}=\sa_{c}(Z,\E)$ by
\begin{equation}
  \label{eq:13}
  \ripd<\phi,\psi>(v)
  = \int_{G}\rip\CC<\phi(x^{-1}\cdot \zeta), \psi(x^{-1}\cdot
  \zeta)>\,d\lambda^{r(\zeta)} (x).
\end{equation}
Left-invariance of $\rho$ implies that this formula does not depend on
the choice of $\zeta\in Z$ such that $s(\zeta)=v$. The map
$(\phi,\psi) \mapsto \ripd<\phi,\psi>(v)$ is the restriction to $\lo$
of the product $\phi^*\psi$ computed in $\sa_c(L, \linke)$.

The following lemma constructs what is essentially an ``integrated
form'' of the modules used in \cite{moutu:xx11}*{Proposition~4.3}.

\begin{lemma}
  \label{lem-preD}
  With respect to the pre-inner product~\eqref{eq:13}, $\Y_{0}$ is a
  pre-Hilbert $D$-module whose completion, $\Y$ is a full right
  Hilbert $D$-module.
\end{lemma}
\begin{proof}
  That~\eqref{eq:13} takes values in $D$ follows from the observation
  above that $\ripd<\phi,\psi> = (\phi^*\psi)|_{\lo}$. Since each
  $E_z$ is an imprimitivity bimodule, the range of
  $\ripd<\cdot,\cdot>(v)$ is all of $D_v$. Since $D$ is a
  $C_0(\ho)$-algebra, to see that $X := \clsp\set{\ripd<\phi,\psi> :
    \phi,\psi \in \Y_{0}} = D$, it therefore suffices to show that $X$
  is a $C_0(\ho)$-module (see, for example
  \cite{wil:crossed}*{Proposition~C.24}), for which one uses the right
  action of $C_0(\ho)$ on $\Y_{0}$ to check that
  $(\ripd<\phi,\psi>\cdot f)(v) := f(v) \ripd<\phi,\psi>(v)$ is
  bilinear from $X \times C_0(\ho)$ to $X$. An argument like that of
  page~\pageref{page:multiplier} shows that for $f \in \sa_{c}(\lo,
  \linkq^{-1}(\lo))$, $M_f(\psi)(g) := f(r(g))\psi(g)$ determines a
  multiplier of $\cs(L; \linke)$, so $\Y_{0}$ is a pre-Hilbert
  $D$-module which is full since $X = D$.
\end{proof}

\begin{remark}
  \label{rem-norms}
  Since $D$ is a $C_{0}(\ho)$-algebra, to each $v\in\ho$ there
  corresponds a quotient module
  \begin{equation*}
    \Y(v):=Y/Y\cdot I_{v},
  \end{equation*}
  where $I_{v}=\set{d\in D:d(v)=0}$.  As in
  \cite{rw:morita}*{Proposition~3.25}, $\Y(v)$ is a right Hilbert
  $D(v)$-module: if we denote by $x(v)$ the image of $x$ in $\Y(v)$,
  then we have $\brip D(v)<x(v),y(v)>=\ripd <x,y>(v)$. Since
  $\|y\|^{2}=\|\ripd <y,y>\|$, we obtain
  $\|y\|=\sup_{v\in\ho}\|y(v)\|$. Indeed, the $\Y(v)$ are isomorphic
  to the modules used in \cite{moutu:xx11}*{Proposition~4.3}.

  Fix $T \in \L(\Y)$. Since $T$ is $D$-linear, for each $v \in \ho$
  there is an adjointable operator $T_{v}$ on $\Y(v)$ satisfying
  $T_{v}(x(v))=(Tx)(v)$ for all $x \in \Y$. Since $\brip
  D(v)<T_{v}(x(v)),y(v)> = \ripd <Tx,y>(v)$, we have
  $\|T\|=\sup_{v\in\ho}\|T_{v}\|$.
\end{remark}

\begin{prop}[\cite{moutu:xx11}*{Proposition~4.3}]
  \label{prop-key}
  There is a homomorphism $M$ from $\cs(G;\B)$ to $\L(\Y)$ such that if
  $f\in\sacgb$ and $\phi\in\Y_{0}$, then
  \begin{equation}
    \label{eq:14}
    M(f)\phi(\zeta)=\int_{G}f(x)\phi(x^{-1}\cdot
    \zeta)\,d\lambda^{r(\zeta)}(x) .
  \end{equation}
  We have $I_{\cs_{r}(G;\B)} \subset \ker M$, and $M$ factors through
  $\cs_{r}(G;\B)$.  In particular, $\|M(f)\|\le\|f\|_{r}$.
\end{prop}
\begin{proof}
  Direct computation shows that
  \begin{equation*}
    \ripd<M(f)\phi,\psi>=\ripd<\phi,M(f^{*})\psi>\quad\text{for
      $f\in\sacgb$ and $\phi,\psi\in\sa_{c}(Z,\E)$}.
  \end{equation*}
  A calculation using Remark~\ref{rem-norms}, the Cauchy-Schwartz
  inequality for Hilbert modules (\cite{rw:morita}*{Lemma~2.5}) and
  the characterization of inductive-limit topology
  continuous maps
  out of $C_{c}(G)$ in terms of eventually compactly supported uniform
  convergence
  shows that $f \mapsto \ripd<M(f) \phi,\psi>$ is continuous in the
  inductive-limit topology. The existence of approximate identities as
  in \cite{muhwil:dm08}*{Proposition~6.10} then implies that
  $\operatorname{span}\set{M(f)\phi:\text{$f\in\sacgb$ and
      $\phi\in\Y_{0}$}}$ is dense in $\Y$ in the inductive limit
  topology, so \cite{kmqw:nyjm10}*{Proposition~1.7} implies that $M$
  is bounded and extends to $\cs(G;\B)$.

  Since $\|M(f)\|=\sup_{v\in\ho}\|M_{v}(f)\|$, it now suffices to show
  that
  \begin{equation}
    \label{eq:15}
    \|M_{v}(f)\|\le\|f\|_{r}\quad\text{for all $v\in\ho$.}
  \end{equation}

  Fix $v\in\ho$ and choose $\zeta\in Z$ such that $s(\zeta)=v$.  Let
  $u=r(\zeta)$.  For any $e\in q^{-1}(\zeta)$ we have $\blip
  \B<\phi(x\cdot \zeta),e> \in B(\g[x\cdot \zeta,\zeta])=B(x)$.  Thus
  we can define
  \begin{equation*}
    U_{e}:\sa_{c}(Z,\E)\to \sa_{c}(G_{u},\B)\quad
    \text{by}\quad
    U(\phi)(x)=\blip \B<\phi(x\cdot \zeta),e>.
  \end{equation*}
  Just as in Remark~\ref{rem-norms}, we can form the quotient module
  $\X(u)$, and the map $V:\cs(G;\B)\to\L(\X)$ from
  Lemma~\ref{lem-x-red} gives operators $V_{u}(f)\in\L(\X(u))$ such
  that $\|V_{u}(f)\|\le\|f\|_{r}$.  The inner product
  $\Rip<\phi,\psi>(u)$ depends only on the $\phi\restr{G_{u}}$ and
  $\psi\restr{G_{u}}$, and every element of $\sa_{c}(G_{u},\B)$
  extends to an element of $\sacgb$ by the Tietz Extension Theorem for
  upper semicontinuous Banach bundles
  \cite{muhwil:dm08}*{Proposition~A.5}. So we can view $U_{e}$ as a
  map from $\sa_{c}(Z,\E)$ to $\X(u)$.

  Using that $\blip\B<\phi(x^{-1}\zeta),e>^{*}
  \blip\B<\phi(x^{-1}\cdot\zeta),e> = \blip\B<e\cdot
  {\rip\CC<\phi(x^{-1}\cdot \zeta),\phi(x^{-1}\cdot \zeta)>},e>$, one
  computes to see that
  \begin{equation}\label{eq:16}
    \Rip<U_{e}(\phi),U_{e}(\phi)>(u) = \brip\B<e\cdot \ripd<\phi,\phi>(v),e>.
  \end{equation}
  So if $\|e\| \le 1$, the Cauchy-Schwartz inequality for Hilbert
  modules (\cite{rw:morita}*{Lemma~2.5}) implies that
  $\|U_{e}(\phi)\|_{\X(u)}\le \|\phi\|_{\Y(v)}$.

  For $x\in G_{u}$, that the pairing $\blip\B<\cdot,\cdot>$ is
  $A$-linear in the first variable gives
  \[
  U_{e}(M(f)\phi)(x) = \int_{G}\;\blip\B<f(y)\phi(y^{-1}x\cdot
  \zeta),e>\,d\lambda^{r(x)}(y) = V_{u}(f)U(\phi)(x).
  \]

  Fix $f\in\sacgb$ and $\epsilon>0$. Fix $\phi\in\sa_{c}(Z,\E)$ such
  that $\|\phi\|_{\Y(v)}=1$ and such that
  $\|M(f)\phi\|_{\Y(v)}>\|M_{v}(f)\|-\epsilon$. By~\eqref{eq:16},
  there exists $e\in q^{-1}(\zeta)$ with $\|e\|=1$ such that
  \begin{equation*}
    \|U_{e}(M(f)\phi)\|_{\X(u)}>\|M(f)\phi\|_{\Y(v)}-\epsilon.
  \end{equation*}
  Hence
  \[
  \|M_{v}(f)\|-2\epsilon < \|M(f)\phi\|_{\Y(v)}-\epsilon <
  \|U_{e}(M(f)\phi)\|_{\X(u)} = \|V_{u}(f)U_{e}(\phi)\|_{\X(u)}
  \le\|f\|_{r}.
  \]
  Letting $\varepsilon \to 0$ gives~\eqref{eq:15}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-main-reduced}]
  Since every representation of $\cs(L;\linke)$ restricts to a
  representation of $\cs(G;\B)$, we have $\|F\|_{\cs(L;\linke)} \le
  \|f\|_{\cs(G;\B)}$, so we just have to establish the reverse
  inequality. The argument for this is nearly identical to that of
  \cite{simwil:jot11}*{Proposition~15}. The key differences are that:
  \cite{muhwil:dm08}*{Theorem~6.4} is used in place of
  \cite{muhwil:nyjm08}*{Theorem~5.5}; and
  \cite{muhwil:dm08}*{Proposition~6.10} is used to obtain an
  approximate identity for both $\sacgb$ and $\sa_{c}(Z,\E)$ which can
  be used in place of the approximate identity in $\bundlefont{C}(L)$
  to establish the analogue of \cite{simwil:jot11}*{Equation~(10)} and
  to complete the norm approximation at the end of the proof.

  We now turn to the proof that the reduced norms agree. Fix faithful
  representations $\pi_u$ of the $A(u)$ and $\tau_v$ of the
  $D(v)$. Then
  \begin{equation*}
    \|F\|_{\cs_{r}(L;\linke)}
    = \max\Bigl\{\sup_{u\in\go}\|(\Ind^{L}\pi_{u})(F)\|, \sup_{v\in\ho}\|(\Ind\tau_{v})(F)\| \Bigr\}.
  \end{equation*}

  Fix $u\in\go$. Let $\H_{1}$ be the space of $\Ind^{G}\pi_{u}$; that
  is, the completion of $\sa_{c}(G_{u},\B)\odot\H_{\pi_{u}}$ as in
  Example~\ref{ex-pointmass}. The representation $\Ind^{L}\pi_{u}$
  acts on the completion of $\sa_{c}(L_{u},\linke)\odot \H_{\pi_{u}}$
  which decomposes as $\H_{1}\oplus\H_{2}$ where $\H_{1} =
    \overline{\sa_{c}(G_{u},\linke)\odot \H_{\pi_{u}}}$ and $\H_{2} =
    \overline{\sa_{c}(Z_{u}, \linke) \odot \H_{\pi_u}}$. Moreover, the
  restriction of $\Ind^L\pi_u$ to $H_2$ is the zero
  representation. Hence
  \begin{equation*}
    \|F\|_{\cs_{r}(L;\linke)} \ge \sup_{u \in \go}
    \|\Ind^{L}\pi_{u}(f)\| = \|f\|_{\cs_{r}(G;\B)},
  \end{equation*}
  and it suffices now to establish that $\|\Ind^L\tau_{v}(F)\| \le
  \sup_u\|\Ind^{L}\pi_{u}(F)\|$ for all $F \in \sa_{c}(L,\linke)$ and
  $v \in \ho$.

  Fix $v\in\ho$. Then $\Ind^{L}\tau_{v}$ acts on the completion of
  $\sa_{c}(L_{v},\linke)\odot \H_{\tau_{v}}$ which again decomposes as
  a direct sum $\H_{3}\oplus \H_{4}$ (here $\H_3 =
    \overline{\sa_c(\zop_{v}, \linke) \odot \H_{\tau_{v}}}$ and $\H_4 =
    \overline{\sa_c(H_v, \linke) \odot \H_{\tau_{v}}}$). The
  restriction to $\H_{4}$ is zero, and $\H_{3}$ is the completion of
  $\sa_{c}(Z,\E)$ under
  \begin{equation*}
    \ip(\phi\tensor h|\psi\tensor k)=\int_{Z}
    \bip(\tau_{v}\bigl(\brip\CC<\psi(\zeta), \phi(\zeta)>\bigr)h|k) \,
    d\rho^{v}(\zeta).
  \end{equation*}
  An inner-product computation shows that if $\Y$ is the Hilbert
  $D$-module of Lemma~\ref{lem-preD}, then $\H_{3}$ is isomorphic to
  the completion of $\Y\odot \H_{\tau_{v}}$ under
  \begin{equation*}
    \ip(\phi\tensor h|\psi\tensor k)
    =\bip(\tau_{v}\bigl(\ripd<\psi,\phi>(v)\bigr) h|k),
  \end{equation*}
  and then the restriction of $(\Ind^{L}\tau_{v})(F)$ to $\H_{3}$ is
  $\yind\tau_{v}$. Hence $\|\Ind^{L}\tau_{v}(F)\| =
  \|\yind\tau_{v}(f)\|$. Since $\yind\tau_{v}[x\tensor h] =
  [M(f)x\tensor h]$ for all $x$, we have $\ker M\subset \ker
  \yind\tau_{v}$. Hence Proposition~\ref{prop-key}, implies that
  $\|\yind\tau_{v}(f)\|\le\|f\|_{\cs_{r}(G;\B)}$ as required.

  The final statement follows from \cite{rw:morita}*{Theorem~3.22}.
\end{proof}

\section{The reduced symmetric imprimitivity theorem}

Suppose that $K$ and $H$ are locally compact groups acting freely and
properly on the left and right, respectively, of a locally compact
space $P$.  Suppose also that we have commuting actions $\alpha$ and
$\beta$ of $K$ and $H$, respectively, on a \cs-algebra $D$.  Then we
can form the induced algebras $\Ind_{H}^{P}(D,\beta)$ and
$\Ind_{K}^{P}(D,\alpha)$ and get dynamical systems
\begin{equation*}
  \check\sigma:K\to\Aut\bigl(\Ind_{H}^{P}(D,\beta)\bigr)
  \quad\text{and} \quad
  \check\tau:H\to\Aut\bigl(\Ind_{K}^{P}(D,\alpha)\bigr)
\end{equation*}
for the diagonal actions as in \cite{wil:crossed}*{Lemma~3.54}.  Then
Raeburn's Symmetric Imprimitivity Theorem says that the crossed
products
\begin{equation*}
  \Ind_{H}^{P}(D,\beta)\rtimes_{\check\sigma}K\quad\text{and}\quad
  \Ind_{K}^{P}(D,\alpha)\rtimes_{\check\tau} H
\end{equation*}
are Morita equivalent.  In \cite{quispi:hjm92}, Quigg and Spielberg
proved that Raeburn's Morita equivalence passed to the reduced crossed
products.  (Kasparov had a different proof in
\cite{kas:im88}*{Theorem~3.15} and an Huef and Raeburn gave a
different proof of the Quigg and Spielberg result in
\cite{huerae:mpcps00}*{Corollary~3}.)

We consider the corresponding statements for groupoid dynamical
systems. Let $(\A,G,\alpha)$ be a groupoid dynamical system as in
\cite{muhwil:nyjm08}*{\S4}. Recall that the associated crossed product
$\A\rtimes_{\alpha}G$ is a completion of $\sa_{c}(G,r^{*}\A)$.  If
$\pi$ is a representation of $A:=\sa_{0}(\go,\A)$, then the associated
regular representation of $\A\rtimes_{\alpha}G$ is the representation
$L^{\pi}:=\Ind^{\A}\pi$ acting on $L^{2}(G*\H_{s},\nu^{-1})$ as in
\eqref{eq:9}. The reduced crossed product, $\A\rtimes_{\alpha,r}G$ is
the quotient of $\A\rtimes_{\alpha} G$ by the common kernel of the
$L^{\pi}$ with $\pi$ faithful. Let $\B := r^{*}\A$ with the semidirect
product Fell bundle structure so that $\cs(G,\B)$ is isomorphic to
$\A\rtimes_{\alpha}G$, then it follows from
Example~\ref{ex-crossed-prod} that $\cs_{r}(G,\B)$ is isomorphic to
the reduced crossed product $\A\rtimes_{\alpha,r}G$.

Now let $H$ be a locally compact group and let $X$ be a locally
compact Hausdorff $H$-space. Let $G$ be the transformation groupoid $G
= H \times X$. As in \cite{muhwil:nyjm08}*{Example~4.8}, suppose that
$A=\sa_{0}(X,\A)$ is a $C_{0}(X)$-algebra, and define $\lt : H \to
\Aut(C_{0}(X))$ by $\lt_{h}(\phi)(x) := \phi(h^{-1}\cdot x)$. Suppose
that $\beta:H\to \Aut A$ is a \cs-dynamical system such that
\begin{equation*}
  \beta_{h}(\phi\cdot a)=\lt_{h}(\phi)\cdot \beta_{h}(a)\quad\text{for
    $h\in H$, $\phi\in C_{0}(X)$  and $a\in A$.}
\end{equation*}
Then, following \cite{muhwil:nyjm08}*{Example~4.8}, we obtain a
groupoid dynamical system $(\A,G,\alpha)$ where
\begin{equation*}
  \alpha_{(h,x)}\bigl(a(h^{-1}\cdot x)\bigr) =\beta_{h}(a)(x).
\end{equation*}
Let $\Delta$ be the modular function on $H$. Then the map
$\Phi:C_{c}(H,A)\to \sa_{c}(G,r^{*}\A)$ given by
\begin{equation*}
  \Phi(f)(h,x)=\Delta(h)^{\half}f(h)(x)
\end{equation*}
extends to an isomorphism of $A\rtimes_{\beta}H$ with
$\A\rtimes_{\alpha}G$.

Fix a representation $\pi$ of $A$. By decomposing $\pi$ as a direct
integral over $X$ one checks that $(f \tensor_\pi \xi | g \tensor_\pi
\eta) = (\Phi(f) \tensor_\pi \xi | \Phi(g) \tensor_\pi \eta)$ for $f,g
\in C_{c}(G,A)$ and $\xi,\eta \in \H_{\pi}$.  We use this to show that
$U(f\tensor h)=\Phi(f)\tensor h$ determines a unitary from the space
of the regular representation $\Ind^{A}\pi$ of $A\rtimes_{\beta}H$ to
the space of the regular representation $\Ind^{\A}\pi$ of
$\A\rtimes_{\alpha}G$ which intertwines $\Ind^{A}\pi(f)$ and
$\Ind^{\A} \pi(\Phi(f))$ for all $f$. Therefore $\Phi$ factors through
an isomorphism $A\rtimes_{\beta,r}H \cong \A\rtimes_{\alpha,r}G$.

Now, back to the set-up of Raeburn's Symmetric Imprimitivity
Theorem. Since $\Ind_{H}^{P}(D,\beta)$ is a $C_{0}(P/H)$-algebra, it
is the section algebra of a bundle $\B$ over $P/H$. It is shown in
\cite{muhwil:nyjm08}*{Example~5.12} that there is a groupoid action
$\sigma$ of the transformation groupoid $K\times P/H$ on $\B$ such
that
\begin{equation*}
  \Ind_{H}^{P}(D,\beta)\rtimes_{\check\sigma}K\cong
  \B\rtimes_{\sigma}(K\times P/H).
\end{equation*}
Similarly,
\begin{equation*}
  \Ind_{K}^{P}(D,\alpha)\rtimes_{\check\tau}H\cong
  \A\rtimes_{\tau}(K\backslash P\rtimes H)
\end{equation*}
for an appropriate bundle $\A$ over $K\backslash P$ and action $\tau$.
Furthermore, the trivial bundle $\E:=P\times A$ is an equivalence
between $(\B,K\times P/H,\sigma) $ and $(\A,K\backslash P\times
H,\tau)$ in the sense of \cite{muhwil:nyjm08}*{Definition~5.1}.  (Thus
Raeburn's Symmetric Imprimitivity Theorem is a special case of
\cite{muhwil:nyjm08}*{Theorem~5.5}.)  Therefore the Quigg-Spielberg
result follows from following corollary of our main theorem.

\begin{cor}
  \label{cor--redcrossprod}
  Suppose that $q:\E\to Z$ is an equivalence between the groupoid
  dynamical systems $(\B,H,\beta)$ and $(\A,G,\alpha)$.  Then the
  Morita equivalence of \cite{muhwil:nyjm08}*{Theorem~5.5} factors
  through a Morita equivalence of the reduced crossed products
  $\B\rtimes_{\beta,r}H$ and $\A\rtimes_{\alpha,r}G$.
\end{cor}
\begin{proof}
  Recall that $r^*\A := \{(a,x) \in \A \times G : r(a) = r(x)\}$ is a
  Fell bundle over $G$ with bundle map $(a,x) \mapsto x$,
  multiplication $(a,x)(b,y) = (a\alpha_x(b), xy)$ and involution
  $(a,x)^* = (\alpha_{x^{-1}}(a), x^{-1})$ (see
  \cite{muhwil:dm08}*{Example~2.1}), and similarly for $r*\B$. Define
  maps $r^*\A * \E \to \E$ and $\E * r^*\B \to \E$ by
  \begin{equation*}
    (a,g)\cdot \ea := a \cdot\alpha_{g}(\ea)\quad\text{ and }\quad \ea
    \cdot (b,h) := \beta_h(\ea)\cdot b.
  \end{equation*}
  Define pairings $\lip{r^*\A}<\cdot,\cdot> : \E *_s \E \to \A$ and
  $\rip{r^*\B}<\cdot,\cdot> : \E *_r \E \to \B$ by
  \begin{align*}
    \lip{r^*\A}<\ea,\eb> &=
    \big(\rip{A_{r(\ea)}}<\ea,\alpha_{\g[q(\ea),q(\eb)]}(\eb)>,
    \g[q(\ea),q(\eb)]\big)\quad\text{ and } \\
    \rip{r^*\B}<\ea,\eb> &=
    \big(\rip{A_{r(\ea)}}<\ea,\beta_{\h[q(\ea),q(\eb)]}(\eb)>,
    \h[q(\ea),q(\eb)]\big).
  \end{align*}
  It is routine though tedious to show that $\E$ is an $r^*\A\sme
  r^*\B$ equivalence.

  The Morita equivalence $X$ of \cite{muhwil:nyjm08}*{Theorem~5.5} and
  the Morita equivalence $\Pg C^*(L,\linke) \Ph$ of
  Theorem~\ref{thm-main-reduced} are both completions of
  $\sa_{c}(Z,\E)$. From the formulae for the actions of $r^*\A$ on
  $\E$, we see that the identity map on $\sa_{c}(Z,E)$ determines a
  left-module map from $X$ to $\Pg C^*(L,\linke) \Ph$, and similarly
  on the right. So it suffices to show that the norms on $X$ and on
  $\Pg C^*(L,\linke) \Ph$ coincide. For this, observe that the formula
  \cite{muhwil:nyjm08}*{Equation~(5.1)} for the $\A\times_\alpha
  G$-valued inner-product on $\sa_{c}(Z,\E)$ is precisely the
  convolution formula for multiplication of the corresponding elements
  of $\sa_{c}(L,\linke)$ with respect to the Haar system $\ll$
  described in \cite{simwil:jot11}.
\end{proof}

\def\noopsort#1{}\def\cprime{$'$} \def\sp{^}
\begin{bibdiv}
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\bib{fd:representations1}{book}{
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\bib{fd:representations2}{book}{
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\bib{hof:74}{unpublished}{
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}

\bib{hof:lnm77}{incollection}{
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        date={1977},
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  {G}eorgia, {A}thens, {G}a., 1975)},
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   publisher={Springer},
     address={Berlin},
       pages={53\ndash 69},
      review={\MR{58 \#7117}},
}

\bib{hofkei:lnm79}{incollection}{
      author={Hofmann, Karl~Heinrich},
      author={Keimel, Klaus},
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  {B}anach spaces, {B}anach {$C(X)$}-modules},
        date={1979},
   booktitle={Applications of sheaves ({P}roc. {R}es. {S}ympos. {A}ppl. {S}heaf
  {T}heory to {L}ogic, {A}lgebra and {A}nal., {U}niv. {D}urham, {D}urham,
  1977)},
      series={Lecture Notes in Math.},
      volume={753},
   publisher={Springer},
     address={Berlin},
       pages={415\ndash 441},
      review={\MR{MR555553 (81f:46085)}},
}

\bib{huerae:mpcps00}{article}{
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      author={Raeburn, Iain},
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        date={2002},
        ISSN={0305-0041},
     journal={Math. Proc. Cambridge Philos. Soc.},
      volume={133},
      number={2},
       pages={249\ndash 259},
      review={\MR{2003g:46080}},
}

\bib{ionwil:pams08}{article}{
      author={Ionescu, Marius},
      author={Williams, Dana~P.},
       title={Irreducible representations of groupoid {$C\sp *$}-algebras},
        date={2009},
        ISSN={0002-9939},
     journal={Proc. Amer. Math. Soc.},
      volume={137},
      number={4},
       pages={1323\ndash 1332},
      review={\MR{MR2465655}},
}

\bib{kmqw:nyjm10}{article}{
      author={Kaliszewski, Steven},
      author={Muhly, Paul~S.},
      author={Quigg, John},
      author={Williams, Dana~P.},
       title={Coactions and {F}ell bundles},
        date={2010},
     journal={New York J. Math.},
      volume={16},
       pages={315\ndash 359},
}

\bib{kas:im88}{article}{
      author={Kasparov, Gennadi~G.},
       title={Equivariant {$KK$}-theory and the {N}ovikov conjecture},
        date={1988},
        ISSN={0020-9910},
     journal={Invent. Math.},
      volume={91},
      number={1},
       pages={147\ndash 201},
      review={\MR{88j:58123}},
}

\bib{kum:pams98}{article}{
      author={Kumjian, Alex},
       title={Fell bundles over groupoids},
        date={1998},
        ISSN={0002-9939},
     journal={Proc. Amer. Math. Soc.},
      volume={126},
      number={4},
       pages={1115\ndash 1125},
      review={\MR{MR1443836 (98i:46055)}},
}

\bib{moutu:xx11}{unpublished}{
      author={Moutuou, El-Ka\"ioum~M.},
      author={Tu, Jean-Louis},
       title={Equivalence of fell systems and their reduced {$C^*$}-algebras},
        date={2011},
        note={(arXiv:math.OA.1101.1235v1)},
}

\bib{mrw:jot87}{article}{
      author={Muhly, Paul~S.},
      author={Renault, Jean~N.},
      author={Williams, Dana~P.},
       title={Equivalence and isomorphism for groupoid {$C^*$}-algebras},
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        ISSN={0379-4024},
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      volume={17},
      number={1},
       pages={3\ndash 22},
      review={\MR{88h:46123}},
}

\bib{muhwil:dm08}{article}{
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  their {$C\sp *$}-algebras},
        date={2008},
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     journal={Dissertationes Math. (Rozprawy Mat.)},
      volume={456},
       pages={1\ndash 57},
      review={\MR{MR2446021}},
}

\bib{muhwil:nyjm08}{book}{
      author={Muhly, Paul~S.},
      author={Williams, Dana~P.},
       title={Renault's equivalence theorem for groupoid crossed products},
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     address={Albany, NY},
        date={2008},
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        note={Available at http://nyjm.albany.edu:8000/m/2008/3.htm},
}

\bib{pim:fic87}{incollection}{
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      series={Fields Inst. Commun.},
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       pages={189\ndash 212},
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\bib{wil:crossed}{book}{
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}

\end{biblist}
\end{bibdiv}
\end{document}

redpajama_set_name: RedPajamaArXiv


Anyone looking to write a history of the literary journal in Flanders and the Netherlands will undoubtedly discover that 2011 was a pivotal year. That was the year when the Dutch State Secretary for Culture scrapped all subsidies for literary magazines. Sadly, a number of journals have indeed disappeared. But in that accursed year 2011several new journals were actually founded, including Das Magazin. This journal, in particular, has grown into a phenomenon within a short space of time.


text:
Woman fends off attacker in Basalt
News News | Aug 7, 2006
Aspen Times Staff
A Basalt woman fought off a man who allegedly assaulted her early Saturday in a gas station parking lot, and police want the public's help in apprehending the suspect.
The woman had just gotten off the downvalley-bound bus near the Basalt Store when the man approached her at about 3 a.m.
"A struggle took place, and the victim was able to fight the attacker off while calling 911," a Basalt Police Department press release says. "The victim sustained minor facial injuries after being struck by her attacker several times."
Police Chief Keith Ikeda said the woman managed to inflict a severe bite on the suspect's hand.
"She fought back. I think that's what saved her," he said. "She had a lot of courage."
Ikeda said the victim's boyfriend came on the scene and tussled with the suspect.
The alleged attacker, described as Latino with a dark complexion, 5-foot-8, 165 pounds, brown eyes and black hair, fled on foot. He was still at large Sunday night and is considered dangerous.
"We are requesting that citizens walking home at night exercise more caution and report any suspicious activity," the police release says.
"We really want the community's help in identifying this guy," Ikeda said. "This is a very serious crime, and we want to get this guy off the streets."
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text:
district du congressionnel de l'Illinois est un district de l'État américain de l'Illinois, il couvre des parties des comtés de Cook et de DuPage. Il est représenté par le Démocrate Sean Casten depuis 2019

Géographie

Redécoupage de 2011
À partir du redécoupage de 2011 qui a suivi le recensement de 2010. Tout ou partie d'Algonquin, Barrington, Barrington Hills, Bartlett, Burr Ridge, Carol Stream, Carpentersville, Cary, Clarendon Hills, Crystal Lake, Darien, Deer Park, Downers Grove, Elgin, East Dundee, Forest Lake, Fox River Grove, Gilberts, Glen Ellyn, Hawthorn Woods, Hinsdale, Hoffman Estates, Inverness, Kildeer, Lake Barrington, Lake in the Hills, Lake Zurich, Lakewood, Lisle, Lombard, Long Grove, Naperville, North Barrington, Oak Brook, Oakbrook Terrace, Oakwood Hills, Palatine, Port Barrington, Rolling Meadows, Sleepy Hollow, South Barrington, South Elgin, St. Charles, Tower Lakes, Trout Valley, Warrenville, Wayne, West Chicago, West Dundee, Westmont, Wheaton, Willowbrook et Winfield sont inclus.

Redécoupage de 2021

En raison du redécoupage de 2020, le district passera du nord du Comté de Cook et du sud du Comté de Lake à un district principalement du sud du Comté de Cook et à une partie du Comté de DuPage, ainsi qu'à une partie de l'extrême sud-ouest de Chicago.

Le  district comprend les quartiers de Beverly à Chicago ; la majeure partie de Mount Greenwood; et l'ouest de Garfield Ridge et Clearing.

En dehors de Chicago, le  district comprend les communautés du Comté de Cook d'Orland Hills, Western Springs, Orland Park, Palos Hills, Hickory Hills, Chicago Ridge, Bridgeview, Willow Springs et Indian Head Park ; le nord de Tinley Park ; et les parties ouest et est d'Evergreen Park.

Le Comté de DuPage est divisé entre ce district, le  district, le  district et le  district. Les  et  districts sont divisés par 59th St, Illinois Highway 83, 55th St, Walker Ave, Park Ave, Golf Ave, Jane Ct, Prospect Ave, Chicago Ave, Middaugh Rd, Naperville Rd, Hinsdale Golf Course, Illinois Highway 34, Robert Kingery Highway, Oak Brook Rd, Regent Dr, 22nd St, Castle Dr, Illinois Highway 38, Fillmore St, Adams St, Madison St, Euclid Ave, York St et Illinois Highway 64.

Les  et  districts sont divisés par Grand Ave, Frontage Ave, Fullerton Ave, Harvard Ave, Armitage Ave, Addison Rd, Illinois Highway 64, Westmore Ave/Berman Ave, Plymouth St, Lincoln St, Vermont St, Westwood Ave, Le Moyne Ave/Illinois Highway 64, Highway 355, Union Pacific Railroad, North Path, President St et Naperville Rd.

Les  et  districs sont divisés par Illinois Highway 23, Highway 88, Fender Rd, Ogden Ave, Beau Bren Blvd, Eugenia Dr, Arlington Ave, Oak Hill Park, Oak Hill Dr, Yackley Ave, Maple Ave, Abbey Dr, Four Lakes Ave, River Bend Golf Course, Riverview Dr, Kohl Rd, Illinois Highway 53, 61st St, Essex Rd, Summerhill Park, Prentiss Creek, 59th St, Chase Ave, 63rd St, Highway 355, Wheeler St, Woodward Ave, 71st Ave, Illinois Highway 33, Illinois Highway 9, 87th St, Meyer Woods Park, Wards Creek, Highway 55, Cass Ave et 91st St. Le  district englobe les municipalités de Downers Grove, Lombard, Villa Park, Westmont; la plupart d'Elmhurst, Lisle et Darien; la moitié de Wheaton à l'est de l'Illinois Highway 23 et au sud de l'Union Pacific Railroad ; et la partie de Glen Ellyn au sud de l'Union Pacific Railroad.

Historique de vote

Représentants connus

Liste des Représentants du district

Résultats des récentes élections

2002

2004

2006

2008

2010

2012

2014

2016

2018

2020

2022

Frontières historiques du district

Références

 Martis, Kenneth C. (1989). The Historical Atlas of Political Parties in the United States Congress. New York: Macmillan Publishing Company
 Martis, Kenneth C. (1982). The Historical Atlas of United States Congressional Districts. New York: Macmillan Publishing Company
 Congressional Biographical Directory of the United States 1774–present

Liens externes

 Washington Post page on the 6th District of Illinois
 U.S. Census Bureau - 6th District Fact Sheet

6


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text:
How to get my friend back?
I dated somebody he liked -- but he never had a chance with her anyway!
Check out this article! https://www.salon.com/2013/01/15/how_to_get_my_friend_back/
January 15, 2013 6:00AM (UTC)
Hey Cary,
I'm writing to you with hopes of salvaging a friendship that seems all but lost.
I remained close with my friend Tommy after graduating college in 2008. About a year ago, he reconnected with an old high school friend named Laura, whom he had messaged out of the blue on Facebook after seeing that she had become single. He asked if she wanted to hang out, and she suggested they go to a hockey game that she already had tickets for. For Tommy, getting dates was a rare occurrence, and he was excited for what he considered to be a date with a very pretty girl.
But for Laura, the night out was no date, just a fun hockey game with an old friend. Here's where I come in. After the game, Tommy and Laura swung by a restaurant where I was having dinner with a friend. The four of us talked only for a brief time, but I had a feeling Laura was flirting with me. I didn't think anything of it, though, and Tommy and Laura left to get drinks at an upscale bar where Tommy had made reservations and footed the bill.
The next night, I was with Tommy at a sports bar. Laura happened to be there with some friends. This time, her flirtation with me was unmistakable: She hung around me instead of Tommy, making any excuse she could to touch me. I didn't respond to her advances at first because I thought she was with Tommy, but after an hour, I decided she was fair game and asked for her number. Back at my place later that night, Tommy casually brought it up: "Hey, I think Laura might have a crush on you." I felt bad for him and changed the subject.
Two days later, I went out with Laura for ice cream and drinks. "So what's up with you and Tommy?" I asked. "Nothing," she said bluntly. "Is he saying something is up?"
A couple days passed, and out of obligation I asked Tommy if it would be OK if I were to take Laura out. He didn't give a clear answer and was visibly uncomfortable with the situation. "What do you think you should do?" he asked. I waited a moment, then responded with a shrug: "I think if she's interested in me, then I should ask her out."
Laura and I eventually became exclusive, and though Tommy never told me, it was clear that he felt betrayed. He had considered me his best friend, and one by one our mutual friends chose sides, and most landed on his side. Chief among those who chose Tommy was Ian, my closest friend from college.
A year later, Ian still sides with Tommy, and it hurts me. Losing Tommy was surprising but not devastating; although I may not have been as sensitive to his feelings as I could've been, I consider his behavior childish, and, as equally childish as some might feel this is, I won't apologize for taking advantage of an opportunity that wasn't ever available to Tommy.
Ian feels that I was in the wrong for two reasons: 1) Although Laura is adamant that she felt nothing for Tommy, there was no way to know what would have happened between them had I not stepped in. 2) When it became apparent that I would lose Tommy as friend, I had an obligation to him to do whatever it took to preserve the friendship, even if it meant leaving Laura (at this point we had been together for months).
While I understand Ian's feelings, he has trouble empathizing with mine. Further complicating things is that his girlfriend of six years insists that I am absolutely wrong and should do "whatever it takes to get Tommy back," even if this involves ending things with Laura. I considered Ian to be a top-tier friend, but he has increasingly treated me as a second-tier friend.
Laura and I are still together and we're in love, so she's not going anywhere. My question is this: Taking into consideration how much Ian values his girlfriend's opinion and how committed he is to Tommy, is it possible to return our friendship to what it was before I met Laura?
Missing My Pal
Dear Missing Pal,
If you really want your friendship with Tommy back you can have it. It won't be the way it was but it can be a friendship. You're just going to have to work for it, and it's going to cost you. What it's going to cost you is something you don't need anyway: your self-righteousness. If you can let go of the need to be right, you can regain your friendship. That doesn't mean you admit to being wrong, necessarily. It just means you admit that you have hurt your friend and that you're sorry about hurting your friend and that you are willing to take certain steps to rebuild the friendship.
Here's what I suggest you do. Go to your friend Tommy and tell him that you value him as a friend and ask him if you and he can spend some time together. That's all. If he says no, well, give it a month and try again. Keep trying.
Forget about these rules that supposedly govern who can date whom.
It's silly to put too much stock in rules for personal relationships. Deciding who was right and who was wrong won't help. Rules work well when they are codified as laws, with enforcement mechanisms, because then once an established authority decides who was right and who was wrong, somebody can go to jail or pay a fine and then it's finished. But having rules in a realm where there's no authority and no punishments is silly. The only sanction available is this refusal to be friends, which hurts both parties. Rules about who can date whom are useful as basic guidelines, but once a romantic entanglement has set in, it's silly for parties to insist on freezing each other out.
That is true unless the bare facts of the matter just make things uncomfortable. But you ought to at least give it a try. Sometimes people are afraid to see each other after an embarrassing entanglement because they don't know what to say or they'll feel uncomfortable. But if you just do it anyway, the discomfort will ease up.
Friendship and romance are the realm of anarchy. That's what friendship is: It's anarchy. If Tommy wants you to apologize, go ahead and apologize. What's that going to hurt? Do you want to be right or do you want to be happy?
Dating Friendship Since You Asked
Bad dates on the way to finding love