text:
\section{Introduction}

In this paper, $\Gamma$ is a discrete group, $K$ is a (complex) Hilbert space and $\pi$ is a unitary representation of $\Gamma$ on $K$.
We denote by $1_\Gamma$ the one dimensional trivial representation of $\Gamma$, and $K^\pi$ the set of $\pi$-invariant vectors in $K$.
Recall that (see \cite{BC}) $\pi$ is said to have a \emph{spectral gap} if the restriction of $\pi$ on the orthogonal complement $(K^\pi)^\bot$ does not weakly contain $1_\Gamma$ (in the sense of Fell), i.e.\ there does not exist a net of unit vectors $\xi_i\in (K^\pi)^\bot$ satisfying $\|\pi_t\xi_i -\xi_i\| \to 0$ for every $t\in \Gamma$.

The following result on group actions, which relates the existence of spectral gap of certain representation and the uniqueness of the invariant states (means), is crucial to the solution of the Banach-Ruziewicz problem (see \cite{Marg},\cite{Sull},\cite{Drin}).

\begin{thm2}\label{motivation}
Let $\Gamma$ be a countable group with a measure preserving ergodic action on a standard non-atomic probability space $(X, \mu)$, and $\rho$ be the associated unitary representation of $\Gamma$ on $L^2(X, \mu)$.

\smallskip\noindent
(a) (\cite{dR}, \cite{Sch}, \cite{LH}) If $\rho$ has a spectral gap, then the integration with respect to $\mu$ is the unique $\Gamma$-invariant state on $L^\infty(X,\mu)$.

\smallskip\noindent
(b) (\cite{Sch}) If there is a unique $\Gamma$-invariant state on $L^\infty(X,\mu)$, then $\rho$ has a spectral gap.
\end{thm2}

Motivated by this result we will define spectral gap actions of discrete groups on von Neumann algebras, and study the corresponding invariant states on the von Neumann algebras.

From now on, $N$ is a von Neumann algebra  with standard form $(N, \mathfrak{H}, \mathfrak{J}, \mathfrak{P})$, i.e.\ $\mathfrak{H}$ is a Hilbert space, $\mathfrak{J}:\mathfrak{H} \to \mathfrak{H}$ is a conjugate linear bijective isometry, and $\mathfrak{P}$ is a self-dual cone in $\mathfrak{H}$ such that there is a (forgettable) faithful representation of $N$ on $\mathfrak{H}$ satisfying some compatibility conditions with $\mathfrak{J}$ and $\mathfrak{P}$.
The readers may find in \cite{Haag-st-form} more information on this topic.
As proved in \cite[Theorem 3.2]{Haag-st-form}, there exists a unique unitary representation $\mathcal{U}$ of the $^*$-automorphism group ${\rm Aut}(N)$ on $\mathfrak{H}$ such that
$$g(x) = \mathcal{U}_{g} x \mathcal{U}_{g}^*, \quad \mathcal{U}_g\mathfrak{J}=\mathfrak{J}\mathcal{U}_g \quad \text{and} \quad \mathcal{U}_g(\mathfrak{P})\subseteq \mathfrak{P}\qquad (x\in N; g\in{\rm Aut}(N)).$$
Hence, an action $\alpha$ of $\Gamma$ on $N$ gives rise to a unitary representation $\mathcal{U}_{\alpha}=\mathcal{U}\circ\alpha$ of $\Gamma$ on $\mathfrak{H}$.

\begin{defn2}
Let $\alpha$ be an action of $\Gamma$ on $N$.

\smallskip\noindent
(a) $\alpha$ is said to have a \emph{spectral gap} if the representation $\mathcal{U}_{\alpha}$ has a spectral gap.

\smallskip\noindent
(b) $\alpha$ is said to be \emph{standard} if every $\alpha$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $\alpha$-invariant states.
\end{defn2}

We will study the relationship between spectral gap actions and standard actions of discrete groups on von Neumann algebras. This is the main topic of Section 2. Our first main result generalizes Theorem \ref{motivation}(a) to the situation of discrete group actions on von Neumann algebras.

\begin{thm2}\label{thm:main}Let $\alpha$ be an action of a (possibly uncountable) discrete group $\Gamma$ on a von Neumann algebra $N$. If $\alpha$ has a spectral gap, then $\alpha$ is standard.
\end{thm2}

In contrast to Theorem \ref{motivation}(b), the converse statement of Theorem \ref{thm:main} is not true in general (see Example \ref{eg:4-not-2}(b)).
Nevertheless, we will give several situations in which the converse  does hold (see e.g.\ Proposition \ref{prop:inv-mean-sp-gap-act}(b)).
Let us also note that Theorem \ref{motivation} (and hence Theorem \ref{thm:main}) does not extend to locally compact groups.
Indeed, if $G$ is the circle group (equipped with the canonical compact topology) and $N := L^\infty(G)$ with the left translation action $\alpha$ by $G$, then there is a unique normal $\alpha$-invariant state on $N$, but there exist more than one $\alpha$-invariant states (see e.g.\ \cite[Proposition 2.2.11]{Lub}).

We will give two applications of Theorem \ref{thm:main}.
The first one concerns inner amenability and will be considered in Section 3.
The notion of inner amenability was first introduced by
Effros, aiming to give a group theoretic description of property Gamma of the group von Neumann algebra $L(\Gamma)$ of an ICC group.
Recall that  that
$\Gamma$ is an \emph{ICC group} if $\Gamma \neq \{e\}$ and all the non-trivial conjugacy classes of $\Gamma$ are infinite.
Moreover, $\Gamma$ is \emph{inner amenable} if there exist more than one inner invariant states on $\ell^\infty(\Gamma)$.
Effros showed in \cite{Eff} that a countable ICC group $\Gamma$ is inner amenable if $L(\Gamma)$ has property Gamma.
However, Vaes recently gave, in \cite{Vaes}, a counter example to the converse.
From an opposite angle, it is natural to ask whether one can express inner amenability of $\Gamma$ in terms of certain property of $L(\Gamma)$.
One application of Theorem \ref{thm:main} is  the following result.

\begin{thm2}\label{thm:inner}
Let $\Gamma$ be a finitely generated ICC group. Then $\Gamma$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(\Gamma)$.
\end{thm2}

We will also study an alternative generalization of inner amenability (which is called ``strongly inner amenability'') of  (not necessarily ICC) discrete groups, which is of independent interest.

Section 4 is concerned with our second application, which is related to property $(T)$.
Recall that $\Gamma$ is said to have \emph{property $(T)$} if every unitary representation of $\Gamma$ has a spectral gap.
It follows from Theorem \ref{thm:main} that if $\Gamma$ has property $(T)$, then every action of $\Gamma$ on a von Neumann algebra is standard, and in particular, the absence of normal invariant state implies the absence of invariant state for this action.
We will show that this property actually characterizes property $(T)$.

\begin{thm2}\label{thm:property-T} Let $\Gamma$ be a countable discrete group.
The following statements are equivalent.
\begin{enumerate}
\item $\Gamma$ has property $(T)$.
\item All actions of $\Gamma$ on von Neumann algebras are standard.
\item For any action $\alpha$ of $\Gamma$ on a von Neumann algebra with only one normal $\alpha$-invariant state, there is only one $\alpha$-invariant state.
\item For every action $\alpha$ of $\Gamma$ on a von Neumann algebra without normal $\alpha$-invariant state, there is no $\alpha$-invariant state.
\end{enumerate}
\end{thm2}

We will also consider the implication $(4)\Rightarrow (1)$ for general discrete group  $\Gamma$ with property $(T, FD)$ in the sense of \cite{LZ89} (Proposition \ref{prop:no-inv-state}). Consequently, a minimally almost period discrete group satisfying (4) is finitely generated.

We end this introduction by introducing more notation.
Throughout this article, $\overline{K}$ and $\mathcal{L}(K)$ denote the conjugate Hilbert space of $K$ and the space of bounded linear maps on $K$ respectively.
For any $\xi,\eta\in K$, we define $\omega_{\xi,\eta}\in \mathcal{L}(K)^*$ by $\omega_{\xi,\eta}(x) := \langle x\xi, \eta\rangle$ ($x\in \mathcal{L}(K)$) and set $\omega_\xi := \omega_{\xi,\xi}$.
We consider $p_\pi\in \mathcal{L}(K)$ to be the orthogonal projection onto $K^\pi$.
Furthermore, we denote by $C^*(\pi)$ the $C^*$-subalgebra generated by $\{\pi_t\}_{t\in \Gamma}$ and by $vN(\pi)$ the bicommutant of $C^*(\pi)$.
We define a functional $\varepsilon_\pi: C^*(\pi) \to \mathbb{C}$ formally by $\varepsilon_\pi\big(\sum_{k=1}^n c_k \pi_{t_k}\big) = \sum_{k=1}^n c_k$ (warning: $\varepsilon_\pi$ is not necessarily well-defined.)

\section{Invariant States and Spectral Gap Actions (Theorem \ref{thm:main})}

Let us start with the following lemma, which may be a known result.
Note that part (d) of this lemma is motivated by \cite{Pasch} (and one can also obtain this part using the argument in \cite{Pasch}).

\begin{lem}\label{lem:p_pi}
(a) $\varepsilon_\pi$ is well-defined if and only if $1_\Gamma$ is weakly contained in $\pi$.
This is equivalent to the existence of $\psi\in \mathcal{L}(K)^*_+$ satisfying
\begin{equation}\label{eqt:m-pi}
\psi(\pi_t)\ =\ 1 \qquad (t\in \Gamma)
\end{equation}
(in this case, $\varepsilon_\pi = \psi|_{C^*(\pi)}$).

\smallskip\noindent
(b) $\pi$ does not have a spectral gap if and only if there is $\psi\in \mathcal{L}(K)_+^*$ satisfying \eqref{eqt:m-pi} and $\psi(p_{\pi}) = 0$.

\smallskip\noindent
(c) If $\psi\in \mathcal{L}(K)_+^*$ satisfies \eqref{eqt:m-pi}, then $\psi(\pi_t x) = \psi(x)$ ($t\in \Gamma, x\in \mathcal{L}(K)$) and $\psi$ is ${\rm Ad}\ \!\pi$-invariant.

\smallskip\noindent
(d) $p_\pi\notin C^*(\pi)$ if and only if $p_\pi \neq 0$ and $\pi$ does not have a spectral gap.
\end{lem}
\begin{prf}
Let $\pi^0$ be the restriction of $\pi$ on $(K^\pi)^\bot$.
One has $\pi_t = 1_{K^\pi}\oplus \pi^0_t$ ($t\in \Gamma$) and $p_\pi = 1_{K^\pi}\oplus 0$.

\smallskip\noindent
(a) For any unitary representation $\rho$ of $\Gamma$, we denote by $\tilde \rho$ the induced $^*$-representation of the full group $C^*$-algebra of $\Gamma$.
Then $\varepsilon_\pi$ is well-defined if and only if $\ker \tilde\pi \subseteq \ker \tilde 1_\Gamma$, which in turn is equivalent to $1_\Gamma$ being weakly contained in $\pi$ (see e.g.\ \cite[Theorem F.4.4]{BHV}).
The second statement is trivial.

\smallskip\noindent
(b) Part (a) implies that $\pi$ does not have a spectral gap if and only if there is $\phi\in \mathcal{L}\big((K^\pi)^\bot\big)^*_+$ satisfying $\phi(\pi_t^0) = 1$ for each $t\in \Gamma$.
This, in turn, is equivalent to the existence of $\varphi\in \big(\mathcal{L}(K^\pi)\oplus \mathcal{L}((K^\pi)^\bot)\big)^*_+$ with $\varphi(\pi_t) = 1$ ($t\in \Gamma$) and $\varphi(p_\pi) = 0$.

\smallskip\noindent
(c) This can be obtained by applying the Cauchy-Schwarz inequality to $\psi\big((1-\pi_t)x\big)$.

\smallskip\noindent
(d) Notice that any of the two statements implies that $p_\pi \neq 0$, and so $\varepsilon_\pi$ is well-defined (because of part (a)).

\smallskip\noindent
$\Rightarrow)$.
Since $p_\pi\notin C^*(\pi)$, the map $\rho:C^*(\pi) \to \mathcal{L}(K)$ defined by  $\rho(x) := x - \varepsilon_\pi(x) p_\pi$ is injective with its image being $0\oplus C^*(\pi^0)$.
Consequently, $\varepsilon_{\pi^0} = \varepsilon_\pi\circ \rho^{-1}$ (we identify $0\oplus C^*(\pi^0)$ with $C^*(\pi^0)$) and is well-defined.
This shows that $1_\Gamma$ is weakly contained in $\pi^0$ (by part (a)).

\smallskip\noindent
$\Leftarrow)$.
By parts (a) and (b), there is $\psi\in \mathcal{L}(K)^*_+$ with $\psi|_{C^*(\pi)} = \varepsilon_\pi$ and $\psi(p_\pi) = 0$.
Take any unit vector $\xi\in K^\pi$.
If $p_\pi \in C^*(\pi)$, then $1 = \omega_\xi(p_\pi) = \varepsilon_\pi(p_\pi) = 0$ (as $\varepsilon_\pi = \omega_\xi$ on $C^*(\pi)$), which is absurd.
\end{prf}

Let $N$ be a von Neumann algebra.
We denote by $N_*$ the pre-dual space of $N$, which is naturally identified with the subspace of all weak-$^*$-continuous linear functionals in the dual space $N^*$.
Let
$$N^*_\alpha:= \{f\in N^*: f\circ \alpha_t =f \text{ for any } t\in \Gamma\}, \quad \mathfrak{M}_{N,\alpha} := \{f\in N^*_\alpha: f\geq 0 \text{ and } f(1) =1\}$$
and $\mathfrak{M}^{N,\alpha} := \overline{\mathfrak{M}_{N,\alpha}\cap N_*}^{\sigma(N^*,N)}$ (where $\overline{E}^{\sigma(N^*,N)}$ means the weak-$^*$-closure of a subset $E\subseteq N^*$).
Clearly, $\mathfrak{M}^{N,\alpha}\subseteq \mathfrak{M}_{N,\alpha}$.

The following theorem is a refined version of Theorem \ref{thm:main}.
This result can also be regarded as generalizations of the main results in \cite{Pasch} and in \cite{LH} (see also \cite[Proposition 3.4.1]{Lub}).

\begin{thm}\label{thm:sp-gap-act-inv-mean}
Let $\alpha$ be an action of a group $\Gamma$ on a von Neumann algebra $N$  with standard form $(N, \mathfrak{H}, \mathfrak{J}, \mathfrak{P})$.
Consider the following statements.

\begin{enumerate}[(G1)]
\item If $\psi\in \mathcal{L}(\mathfrak{H})_+^*$ satisfies $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$), then $\psi(p_{\mathcal{U}_{\alpha}}) \neq 0$.
\item $\alpha$ has a spectral gap.
\item $p_{\mathcal{U}_{\alpha}}\in C^*(\mathcal{U}_{\alpha})$
\item $\mathfrak{M}_{N,\alpha} = \mathfrak{M}^{N,\alpha}$.
\end{enumerate}

\noindent
One has $(G1) \Leftrightarrow (G2) \Rightarrow (G3)$ and $(G2)\Rightarrow (G4)$.
\end{thm}
\begin{prf}
$(G1) \Leftrightarrow (G2)$.
This follows from Lemma \ref{lem:p_pi}(b).

\smallskip\noindent
$(G2) \Rightarrow (G3)$.
This follows from Lemma \ref{lem:p_pi}(d).

\smallskip\noindent
$(G2) \Rightarrow (G4)$.
Let $m\in N^*_+$ be an $\alpha$-invariant state.
If $\{f_i\}_{i\in I}$ is a net of states in $N_*$ that $\sigma(N^*, N)$-converges to $m$, the ``convergence to invariance'' type argument (see e.g.\ \cite[p.33-34]{Green}) will produce a net $\{g_j\}_{j\in J}$ in the convex hull of $\{f_i\}_{i\in I}$ that $\sigma(N^*, N)$-converges to $m$ and satisfies $\|\alpha_t^*(g_j) - g_j\|_{N_*} \to 0$ for every $t\in \Gamma$.

For each $j\in J$, there is a unique unit vector $\zeta_j\in \mathfrak{P}$ with $g_j = \omega_{\zeta_j}$.
As $\alpha_{t^{-1}}^*(g_j) = \omega_{\mathcal{U}_{\alpha_{t}}(\zeta_j)}$,
we have, by \cite[Lemma 2.10]{Haag-st-form} (and the fact that $\mathcal{U}_{\alpha_{t}}(\mathfrak{P})\subseteq \mathfrak{P}$),
\begin{equation}\label{eqt:Pow-Stor-ineq}
\|\mathcal{U}_{\alpha_{t}}(\zeta_j) - \zeta_j\|^2
\ \leq\  \|\alpha_{t^{-1}}^*(g_j) - g_j\|_{N_*}
\qquad (t\in \Gamma).
\end{equation}

Let $\zeta^0_j := \zeta_j - p_{\mathcal{U}_{\alpha}}(\zeta_j)$.
If $\|\zeta_j^0\| \nrightarrow 0$, a subnet of $\{\zeta_j^0\}_{j\in J}$ will produce an almost $\mathcal{U}_{\alpha}$-invariant unit vector in  $(\mathfrak{H}^{\mathcal{U}_{\alpha}})^\bot$, which contradicts the spectral gap assumption of $\mathcal{U}_{\alpha}$.
Consequently, if we set $\zeta_j^1:= \frac{p_{\mathcal{U}_{\alpha}}(\zeta_j)}{\|p_{\mathcal{U}_{\alpha}}(\zeta_j)\|}$, then $\{\omega_{\zeta_j^1}\}_{j\in J}$ is a net of $\alpha$-invariant states that  $\sigma(N^*, N)$-converges to $m$.
\end{prf}

It is easy to see that $\mathfrak{M}_{N,\alpha}$ spans $N^*_\alpha$ (see e.g.\ \cite[Proposition 2.2]{Pat}).
Moreover, if we denote $N_*^\alpha :=N^*_\alpha\cap N_*$, it is not hard to check that $\mathfrak{M}^{N,\alpha}$ spans $\overline{N_*^\alpha}^{\sigma(N^*, N)}$.
Thus, Statement (G4) is equivalent to the fact that $N^*_\alpha = \overline{N_*^\alpha}^{\sigma(N^*, N)}$.

\begin{lem}\label{lem:H-N_*}
(a) The map sending $\xi \in \mathfrak{H}$ to $\omega_\xi \in N_*$ restricts to a bijection from $\mathfrak{P} \cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ onto the positive part of
$N_*^\alpha$.

\smallskip\noindent
(b) If $\mathfrak{H}^{\mathcal{U}_{\alpha}}$ is finite dimensional, then $N_*^\alpha$ is finite dimensional.

\smallskip\noindent
(c) If $\dim \mathfrak{H}^{\mathcal{U}_{\alpha}} =1$, then $\dim N_*^\alpha = 1$.
\end{lem}
\begin{prf}
(a) This follows from the fact that for any $g\in (N_*^\alpha)_+$,  if $\zeta\in \mathfrak{P}$ is the unique element satisfying $g = \omega_\zeta$, we have $\zeta\in \mathfrak{H}^{\mathcal{U}_{\alpha}}$, by the inequality in \eqref{eqt:Pow-Stor-ineq}.

\smallskip\noindent
(b) For any $\phi\in N_*^\alpha$, there exist $\alpha$-invariant elements $\phi_1,\phi_2,\phi_3,\phi_4\in (N_*^\alpha)_+$ with $\phi = (\phi_1 - \phi_2) + \mathrm{i} (\phi_3 - \phi_4)$.
By part (a), one can find $\zeta_k\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ such that $\phi_k = \omega_{\zeta_k}$ ($k=1,2,3,4$).
Therefore, the linear map from $(\mathfrak{H}^{\mathcal{U}_{\alpha}} \otimes \overline{\mathfrak{H}^{\mathcal{U}_{\alpha}}}) \oplus (\mathfrak{H}^{\mathcal{U}_{\alpha}} \otimes \overline{\mathfrak{H}^{\mathcal{U}_{\alpha}}})$ to $N_*^\alpha$ given by $(\xi_1\otimes \overline{\eta_1}, \xi_2\otimes \overline{\eta_2}) \mapsto \omega_{\xi_1,\eta_1} + \mathrm{i} \omega_{\xi_2,\eta_2}$ is surjective and $N_*^\alpha$ is finite dimensional.

\smallskip\noindent
(c) Clearly, $N_*^\alpha \neq (0)$.
If $\dim N_*^\alpha > 1$, there will be two different norm one elements in $(N_*^\alpha)_+$.
Thus, part (a) gives two norm one elements in $\mathfrak{P} \cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$, which is not possible.
\end{prf}

The following proposition concerns some converses of the implications in Theorem \ref{thm:sp-gap-act-inv-mean}.
Note that the idea of the argument for $p_{\mathcal{U}_{\alpha}}\in vN(\mathcal{U}_{\alpha})$ in part (b) comes from \cite{Pasch}.

\begin{prop}\label{prop:inv-mean-sp-gap-act}
Let $\alpha$ be an action of a group $\Gamma$ on a von Neumann algebra $N$.

\smallskip\noindent
(a) If $p_{\mathcal{U}_{\alpha}}\neq 0$, then Statement (G3) implies Statement (G1).

\smallskip\noindent
(b) If $p_{\mathcal{U}_{\alpha}}\in N$, then Statement (G4) implies Statement (G1) and $p_{\mathcal{U}_{\alpha}}$ belongs to the center $Z\big(vN(\mathcal{U}_{\alpha})\big)$ of $vN(\mathcal{U}_{\alpha})$.
\end{prop}
\begin{prf}
(a) This follow from Lemma \ref{lem:p_pi}(d).

\smallskip\noindent
(b) Suppose that Statement (G4) holds, but there exists $\psi\in \mathcal{L}(\mathfrak{H})^*_+$ with $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$) and $\psi(p_{\mathcal{U}_{\alpha}}) = 0$.
Then Lemma \ref{lem:p_pi}(c) implies that $\psi|_{N}$ is $\alpha$-invariant, and Statement (G4) gives a net $\{g_j\}_{j\in J}$ of states in $N_*^\alpha$ that $\sigma(N^*,N)$-converges to $\psi|_N$.
By Lemma \ref{lem:H-N_*}(a), one can find $\zeta_j\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ with $g_j = \omega_{\zeta_j}$, which means that $g_j(p_{\mathcal{U}_{\alpha}}) = 1$ ($j\in J$).
This contradicts $\psi(p_{\mathcal{U}_{\alpha}}) = 0$.

For the second conclusion, it is clear that $p_{\mathcal{U}_{\alpha}}\in vN(\mathcal{U}_{\alpha})'$ because $\mathfrak{H}^{\mathcal{U}_{\alpha}}$ is $\mathcal{U}_{\alpha}$-invariant.
Suppose on the contrary that $p_{\mathcal{U}_{\alpha}}\notin vN(\mathcal{U}_{\alpha})$.
Since $\mathbb{C} p_{\mathcal{U}_{\alpha}}$ is one dimensional and
\begin{equation}\label{eqt:rel-p}
\mathcal{U}_{\alpha_t} p_{\mathcal{U}_{\alpha}}\ =\ p_{\mathcal{U}_{\alpha}} \qquad (t\in \Gamma),
\end{equation}
one sees that $vN(\mathcal{U}_{\alpha}) + \mathbb{C} p_{\mathcal{U}_{\alpha}}$ is a von Neumann algebra.
Moreover, if $\xi\in \mathfrak{H}^{\mathcal{U}_{\alpha}}$ is a norm one vector (note that we assume $p_{\mathcal{U}_{\alpha}}\notin vN(\mathcal{U}_{\alpha})$), we define a functional on $vN(\mathcal{U}_{\alpha}) + \mathbb{C} p_{\mathcal{U}_{\alpha}}$ by
$$\phi(x+ c p_{\mathcal{U}_{\alpha}})\ :=\ \omega_{\xi}(x)
\qquad (x\in vN(\mathcal{U}_{\alpha}); c\in \mathbb{C}).$$
Clearly, $\phi$ is weak-$^*$-continuous and it is a $^*$-homomorphism since $xp_{\mathcal{U}_{\alpha}}\in \mathbb{C} p_{\mathcal{U}_{\alpha}}$ (because of \eqref{eqt:rel-p}).
If $\psi$ is a normal state extension of $\phi$ on $\mathcal{L}(\mathfrak{H})$, then $\psi(p_{\mathcal{U}_{\alpha}}) = 0$ and Lemma \ref{lem:p_pi}(c) implies that $\psi|_N$ is an $\alpha$-invariant normal state.
Now, Lemma \ref{lem:H-N_*}(a) produces $\zeta\in \mathfrak{P}\cap \mathfrak{H}^{\mathcal{U}_{\alpha}}$ with $\psi|_N = \omega_{\zeta}$ and we have the contradiction that $\psi(p_{\mathcal{U}_{\alpha}}) = 1$.
\end{prf}

Our next example shows that Statement (G4) does not imply Statement (G2) in general.
We first set some notation and recall some facts.
Let $\mathcal{I}_N$ be the inner automorphism group of $N$ and $\beta$ be the canonical action of $\mathcal{I}_N$ on $N$.
Then $\beta$-invariant states on $N$ are precisely tracial states.
Suppose, in addition, that $N$ admits a normal faithful tracial state $\tau$.
If $(H_\tau, \Psi_\tau)$ is the GNS construction with respect to $\tau$ and $\Lambda_\tau: N \to H_\tau$ is the canonical map, then $\mathfrak{H} \cong H_\tau$ and the canonical action of $N$ on $\mathfrak{H}$ can be identified with $\Psi_\tau$.
For any $g\in {\rm Aut}(N)$, one has $\mathcal{U}_{\beta_g}(\Lambda_\tau (x)) = \Lambda_\tau(g(x))$ ($x\in N$).

\begin{eg}\label{eg:4-not-2}
Suppose that $\lambda$ is the left regular representation of $\Gamma$ and put $L(\Gamma):= vN(\lambda)$.
Then $\mathfrak{H} = \ell^2(\Gamma)$ and we let $\mathcal{I}_{L(\Gamma)}$ and $\beta$ be as in the above.

\smallskip\noindent
(a) The representation $\mathcal{U}_{\beta}\circ {\rm Ad}\circ \lambda: \Gamma\to \mathcal{L}(\ell^2(\Gamma))$ coincides with the ``conjugate representation'' $\gamma$ defined by  $\gamma_t(\xi)(s) := \xi (t^{-1}st)$ ($s,t\in \Gamma; \xi\in \ell^2(\Gamma)$).

\smallskip\noindent
(b) Suppose that $\Gamma$ is an amenable countable ICC group.
Since $L(\Gamma)$ is a type $I\!I_1$-factor, it has only one tracial state and this state is normal.
Consequently, $\mathfrak{M}_{L(\Gamma), \beta} = \mathfrak{M}^{L(\Gamma), \beta}$ (because $\beta$-invariant states are precisely tracial states).

On the other hand, we have $\mathfrak{H}^{\mathcal{U}_\beta} = \mathbb{C}\delta_e$.
Moreover, as $L(\Gamma)$ is semi-discrete, it has property Gamma (see \cite[Corollary 2.2]{Con}), and the restriction of $\mathcal{U}_{\beta}$ on $(\mathfrak{H}^{\mathcal{U}_\beta})^\bot$ weakly contains the trivial representation (by an equivalent form of property Gamma in \cite[Theorem 2.1(c)]{Con}).
Thus, the representation $\mathcal{U}_\beta$ (and hence the action $\beta$) does not have a spectral gap.
\end{eg}

\section{Invariant States and Inner Amenability (Theorem \ref{thm:inner})}

In this section, we will give an application of Theorem \ref{thm:sp-gap-act-inv-mean} to inner amenability.
We recall (from the main theorem in \cite{Eff}) that an ICC group $\Gamma$ is inner amenable if and only if there is a net $\{\xi_i\}_{i\in I}$ of unit vectors in $\ell^2(\Gamma\setminus \{e\})$ such that $\|\gamma_t(\xi_i) - \xi_i\| \to 0$ for any $t\in \Gamma$ (where $\gamma$ is as in Example \ref{eg:4-not-2}(a)).
Notice that this inner amenability is slightly different from the one in \cite{LP} and \cite{Pat}, and it is called ``non-trivially inner amenable'' in \cite[p.84]{Pat}.

Let us consider another extension of inner amenability to general (not necessarily ICC) discrete groups.

\begin{defn}
$\Gamma$ is said to be \emph{strongly inner amenable} if the conjugate representation $\gamma$ does not have a spectral gap.
\end{defn}

It is obvious that strong inner amenability implies inner amenability, and the converse holds when $\Gamma$ is an ICC group.
On the other hand, all abelian groups and all property $(T)$ groups are not strongly inner amenable.

\begin{prop}\label{prop:st-inner}
Let $\alpha$ be the action of $\Gamma$ on $\mathcal{L}(\ell^2(\Gamma))$ given by $\alpha_t(x) := \gamma_t x \gamma_{t^{-1}}$, $\Gamma_{\rm fin}$ be the normal subgroup consisting of elements in $\Gamma$ with finite conjugacy classes and $A(\Gamma)$ be the predual of $L(\Gamma)$.

\smallskip\noindent
(a) Consider the following statements.

\begin{enumerate}[(S1)]
\item $\Gamma/\Gamma_{\rm fin}$ is inner amenable.
\item $\ell^\infty(\Gamma)^*_\alpha$ is not the $\sigma(\ell^\infty(\Gamma)^*, \ell^\infty(\Gamma))$-closure of $\ell^1(\Gamma)^\alpha$.
\item $\Gamma$ is strongly inner amenable.
\item There is $\psi\in \mathcal{L}(\ell^2(\Gamma))_+^*$ satisfying $\psi(\gamma_t) = 1$ ($t\in \Gamma$) and $\psi(p_\gamma) = 0$.
\item $p_\gamma\notin C^*(\gamma)$.
\item $L(\Gamma)^*_\alpha$ is not the $\sigma(L(\Gamma)^*, L(\Gamma))$-closure of $A(\Gamma)^\alpha$.
\item $\mathfrak{M}_{L(\Gamma),\alpha}$ does not coincide with the set of tracial states on $L(\Gamma)$.
\end{enumerate}

\noindent
Then $(S1) \Rightarrow (S2) \Rightarrow (S3) \Leftrightarrow (S4) \Leftrightarrow (S5)$ and  $(S6)\Leftrightarrow (S7) \Rightarrow (S3)$.

\smallskip\noindent
(b) If $\Gamma_{\rm fin}$ coincides with the center of $\Gamma$, then $(S5) \Rightarrow (S1)$ and $p_\gamma\in Z(vN(\gamma))$.

\smallskip\noindent
(c) If $\Gamma_{\rm fin}$ is finite, we have $(S5) \Rightarrow (S1)$.
\end{prop}
\begin{prf}
Notice that if $x\in L(\Gamma)$, then $\alpha_t(x) = \lambda_t x\lambda_t^*\in L(\Gamma)$.
On the other hand, if $y\in \ell^\infty(\Gamma)$ (considered as a subalgebra of $\mathcal{L}(\ell^2(\Gamma))$ in the canonical way), then $\alpha_t(y)\in \ell^\infty(\Gamma)$ and $\alpha_t(y)(s) = y(t^{-1}st)$.
Moreover, if we either set
$$N = \ell^\infty(\Gamma) \quad \text{or} \quad N = L(\Gamma)$$
(and we denote by $\alpha$ the restriction of $\alpha$ on $N$ by abuse of notation), then $\mathfrak{H} = \ell^2(\Gamma)$ and $\mathcal{U}_{\alpha}$ coincides with $\gamma$ (even though $\mathfrak{J}$ and $\mathfrak{P}$ are different in these two cases).

\smallskip\noindent
(a) By Theorem \ref{thm:sp-gap-act-inv-mean} and Proposition \ref{prop:inv-mean-sp-gap-act}(a) (for both $N=\ell^\infty(\Gamma)$ and $N= L(\Gamma)$), it remains to show that $(S1)\Rightarrow (S2)$ and $(S6)\Leftrightarrow (S7)$.

\noindent
$(S1)\Rightarrow (S2)$.
Since $\Gamma/\Gamma_{\rm fin}$ is inner amenable, we know from \cite[Corollary 1.4]{Kan-Mark} as well as the argument of Lemma \ref{lem:p_pi}(c) that there is an $\alpha$-invariant mean $m$ on $\Gamma$ such that $m(\chi_{\Gamma_{\rm fin}}) \neq 1$.
Obviously, $\ell^2(\Gamma)^\gamma \subseteq \ell^2(\Gamma_{\rm fin})$ and Lemma \ref{lem:H-N_*}(a) implies that $g(\chi_{\Gamma_{\rm fin}}) = \|g\|$ if $g\in \ell^1(\Gamma)^\alpha_+$.
Consequently, $m\notin \mathfrak{M}^{\ell^\infty(\Gamma),\alpha}$.

\noindent
$(S6)\Rightarrow (S7)$.
Assume that $\mathfrak{M}_{L(\Gamma),\alpha}$ coincides with the set of all tracial states on $L(\Gamma)$.
It is well-known that every tracial state on $L(\Gamma)$ is a weak-$^*$-limit of a net of normal tracial states (see e.g.\ \cite[Proposition 8.3.10]{KRII}).
Thus, $\mathfrak{M}_{L(\Gamma),\alpha} = \mathfrak{M}^{L(\Gamma),\alpha}$, which contradicts Statement (S6).

\noindent
$(S7)\Rightarrow (S6)$.
Clearly, any tracial state on $L(\Gamma)$ is $\alpha$-invariant.
Moreover, $\omega\in A(\Gamma)$ is $\alpha$-invariant if and only if $$\omega(\lambda_t x)\ =\ \omega(x\lambda_{t}) \qquad (x\in L(\Gamma); t\in \Gamma),$$
which is equivalent to $\omega(yx) = \omega(xy)$ ($x,y\in L(\Gamma)$).
Hence, $\mathfrak{M}_{L(\Gamma),\alpha}\cap A(\Gamma)$ is the set of all normal tracial states on $L(\Gamma)$.
Consequently, if $L(\Gamma)^*_\alpha$ is the weak-$^*$-closure of $A(\Gamma)^\alpha$, then $\mathfrak{M}_{L(\Gamma),\alpha}$ coincides with the set of tracial states, which contradicts Statement (S7).

\smallskip\noindent
(b) The assumption implies that every finite conjugacy class of $\Gamma$ is a singleton set.
Consequently, one has $\ell^2(\Gamma)^\gamma = \ell^2(\Gamma_{\rm fin})$ and $p_\gamma = \chi_{\Gamma_{\rm fin}}\in \ell^\infty(\Gamma)$.
Thus, $p_\gamma\in Z(vN(\gamma))$ by Proposition \ref{prop:inv-mean-sp-gap-act}(b).
Furthermore, Statement (S4) produces an $\alpha$-invariant mean $m$ on $\Gamma$ (see Lemma \ref{lem:p_pi}(c)) such that $m(p_\gamma) =0$.
Now, by \cite[Corollary 1.4]{Kan-Mark}, $\Gamma/\Gamma_{\rm fin}$ is inner amenable.

\smallskip\noindent
(c) Let $C_1,...,C_n$ be all the finite conjugacy classes of $\Gamma$.
Let $p_k\in \mathcal{L}(\ell^2(\Gamma))$ be the orthogonal projection onto $\ell^2(C_k)$ ($k=1,...,n$) and $p_0\in \mathcal{L}(\ell^2(\Gamma))$ be the orthogonal projection onto $\ell^2(\Gamma\setminus \Gamma_{\rm fin})$.
Then
$$\ell^2(\Gamma)^\gamma = \{\xi\in \ell^2(\Gamma): p_0\xi = 0 \text{ and } p_k\xi \text{ is a constant function for every } k=1,...,n\}.$$
Suppose that $\{\xi_i\}_{i\in I}$ is an almost $\gamma$-invariant unit vector in $(\ell^2(\Gamma)^\gamma)^\bot$.
One can find a subnet $\{\xi_{i_j}\}_{j\in J}$ such that $\{p_k\xi_{i_j}\}_{j\in J}$ is norm-converging to some $\xi^{(k)} \in \ell^2(C_k)$ with $\sum_{s\in C_k}\xi^{(k)}(s) = 0$ (for $k =1,...,n$).
As $\|\gamma_t(p_k \xi_{i_j}) - p_k \xi_{i_j}\|\to 0$, we see that $\gamma_t \xi^{(k)} = \xi^{(k)}$ ($t\in \Gamma$), which means that $\xi^{(k)} = 0$.
Consequently, $\|p_0\xi_{i_j}\| \to 1$ and a subnet of $\{\omega_{p_0\xi_{i_j}}\}_{j\in J}$ will produce a $\gamma$-invariant mean $m$ satisfying $m(\chi_{\Gamma_{\rm fin}}) =0$.
Now, Statement (S1) follows from \cite[Corollary 1.4]{Kan-Mark}.
\end{prf}

The following theorem concerns the case when there is a unique inner invariant state on $L(\Gamma)$.
Part (b) of which gives Theorem \ref{thm:inner}.

\begin{thm}\label{thm:ICC}
Let $\alpha$ be as in Proposition \ref{prop:st-inner}.

\smallskip\noindent
(a) If $\Gamma$ is amenable, then there are more than one $\alpha$-invariant states on $L(\Gamma)$.

\smallskip\noindent
(b) Suppose that $\Gamma$ is an ICC group.
If $\Gamma$ is not inner amenable, there is a unique $\alpha$-invariant state on $L(\Gamma)$.
The converse holds when $\Gamma$ is finitely generated.
\end{thm}
\begin{prf}
(a) Suppose that $n\in \ell^\infty(\Gamma)^*_+$ is an invariant mean.
Using the ``convergence to invariance'' argument, there is a net $\{g_j\}_{j\in J}$ of norm one elements in $\ell^1(\Gamma)_+$ such that $\|t\cdot g_j - g_j\|_{\ell^1(\Gamma)} \to 0$ and $\|g_j\cdot t- g_j\|_{\ell^1(\Gamma)} \to 0$, for every $t\in \Gamma$ (where $(t\cdot g_i)(s) = g_i(t^{-1}s)$ and  $(g_i\cdot t)(s) = g_i(st)$).
Set $\eta_j := g_j^{1/2}\in \ell^2(\Gamma)$ ($j\in J$).
Then
\begin{equation*}\label{eqt:left-and-adj-inv}
\|\lambda_t(\eta_j) - \eta_j\|_{\ell^2(\Gamma)} \to 0
\qquad \text{and}\qquad
\|\gamma_t(\eta_j) - \eta_j\|_{\ell^2(\Gamma)} \to 0,
\end{equation*}
If $\phi$ is the $\sigma(L(\Gamma)^*, L(\Gamma))$-limit of a subnet of $\{\omega_{\eta_j}\}_{j\in J}$, then $\phi$ is an $\alpha$-invariant state on $L(\Gamma)$ but $\phi\neq \omega_{\delta_e}$ because $\phi(\lambda_t) = 1$ for all $t\in \Gamma$.

\smallskip\noindent
(b) Since $\Gamma$ is an ICC group, one has $\mathfrak{H}^{\mathcal{U}_{\alpha}} = \mathbb{C} \delta_e$ and $A(\Gamma)^\alpha = \mathbb{C} \omega_{\delta_e}$.
The first statement follows from Proposition \ref{prop:st-inner}(a).

To show the second statement, we suppose, on the contrary, that $\Gamma$ has a finite generating subset $\{t_1,...,t_n\}$, $\omega_{\delta_e}$ is the only $\alpha$-invariant state on $N:=L(\Gamma)$, but $\gamma$ does not have a spectral gap.
Then there is a sequence $\{\xi_k\}_{k\in \mathbb{N}}$ of unit vectors in $\ell^2(\Gamma)$ satisfying
$$\langle \xi_k, \delta_e\rangle\ =\ 0 \qquad \text{and} \qquad  \|\lambda_{t_j}(\rho_{t_j}(\xi_k)) -\xi_k\| \to 0 \quad (j=1,...,n)$$
(notice that $\gamma = \lambda\circ \rho$, where $\rho$ is the right regular representation of $\Gamma$).
Set $u_j := \lambda_{t_j}\in N$ ($j=1,...,n$) and consider $\mathfrak{F}$ to be a free ultrafilter on $\mathbb{N}$.
Since the relative commutant of $F:=\{u_1,...,u_n\}$ in $N$ is the same as the relative commutant of $F$ in the ultrapower $N^\mathfrak{F}$ (by the argument of \cite[Lemma 2.6]{Con}; see lines 3 and 4 of \cite[p.86]{Con}), the relative commutant of $F$ in $N^\mathfrak{F}$ is $\mathbb{C}$.
Now, one may employ the argument for Case (1) of ``(c) $\Rightarrow$ (b)'' in \cite[p.87]{Con} to conclude that there exists a subsequence of $\{\omega_{\xi_k}\}_{k\in \mathbb{N}}$ which is not relatively $\sigma(N_*,N)$-compact in $N_*$.
Hence, this subsequence has a subnet that $\sigma(N^*,N)$-converges to a non-normal state $\varphi\in N^*$.
As $\varphi$ is invariant under $\{{\rm Ad}\ \! \gamma_{t_1}, ..., {\rm Ad}\ \! \gamma_{t_n}\}$, it is $\alpha$-invariant and we obtain a contradiction.
\end{prf}

\begin{rem}\label{rem:icc}
(a) It was shown in \cite{Pasch} that inner amenability of an ICC group $\Gamma$ is equivalent to $p_\gamma\notin C^*(\gamma)$.
However, Proposition \ref{prop:st-inner}(b) tells us that $p_\gamma\in vN(\gamma)$ for any ICC group $\Gamma$, whether or not it is inner amenable.

\smallskip\noindent
(b) A functional $\varphi\in L(\Gamma)^*$ is $\alpha$-invariant if and only if $\varphi(ax) = \varphi(xa)$ ($x\in L(\Gamma);a\in C^*(\lambda) = C^*_r(\Gamma)$).
Hence, the difference between $\alpha$-invariant states and tracial states on $L(\Gamma)$ is similar to the different between inner amenability of $\Gamma$ and property Gamma of $L(\Gamma)$ (see e.g.\ \cite[p.394]{Vaes}).
It is shown in Proposition \ref{prop:st-inner}(a) that if $\Gamma$ is not strongly inner amenable, all $\alpha$-invariant states on $L(\Gamma)$ are tracial states.
On the other hand, if $\Gamma$ is an amenable ICC group, then $L(\Gamma)$ has only one tracial state, but there exist more than one $\alpha$-invariant states on $L(\Gamma)$ (by Theorem \ref{thm:ICC}(a)).
\end{rem}

\section{Invariant States and Property ($T$) (Theorem \ref{thm:property-T})}

Although Statement (G4) does not imply Statement (G2) for individual action, if Statement (G4) holds for every action, then so does Statement (G2).
This can be seen in the following result, which is a more elaborated version of Theorem \ref{thm:property-T}.

\begin{thm}\label{thm:equiv-prop-T}
Let $\Gamma$ be a countable discrete group.
The following statements are equivalent.

\begin{enumerate}[(T1)]
\item $\Gamma$ has property $(T)$.
\item If $\alpha$ is an action of $\Gamma$ on a von Neumann algebra $N$ and $\psi\in \mathcal{L}(\mathfrak{H})_+^*$ satisfying $\psi(\mathcal{U}_{\alpha_t}) = 1$ ($t\in \Gamma$), one has $\psi(p_{\mathcal{U}_{\alpha}}) \neq 0$.
\item $\mathfrak{M}_{N, \alpha} = \mathfrak{M}^{N, \alpha}$ for every action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$.
\item For each action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$ with $\dim N_*^\alpha  =1$, there is only one $\alpha$-invariant state on $N$.
\item For any Hilbert space $K$ and any action $\alpha$ of $\Gamma$ with $\mathcal{L}(K)_*^\alpha  = (0)$, there is no $\alpha$-invariant state on $\mathcal{L}(K)$.
\end{enumerate}
\end{thm}
\begin{prf}
Theorem \ref{thm:sp-gap-act-inv-mean} gives $(T1) \Rightarrow (T2) \Rightarrow (T3)$.
Moreover, it is clearly that $(T3) \Rightarrow (T4)$ and $(T3) \Rightarrow (T5)$. We need $(T4) \Rightarrow (T1)$ and $(T5) \Rightarrow (T1)$.

\smallskip\noindent
$(T4) \Rightarrow (T1)$.
This follows from \cite[Theorem 2.5]{Sch}  (by considering the case when $N= L^\infty(X,\mu)$, where $(X,\mu)$ is a fixed non-atomic standard probability space, and $\alpha$ runs through all ergodic actions of $\Gamma$ on $(X,\mu)$).

\smallskip\noindent
$(T5) \Rightarrow (T1)$.
Suppose that $\Gamma$ does not have property $(T)$.
By \cite[Theorem 1]{BV}, there is a unitary representation $\mu: \Gamma\to \mathcal{L}(K)$ such that $\mu$ does not have a nonzero finite dimensional subrepresentation and $\pi:=\mu\otimes \overline{\mu}$ weakly contains $1_\Gamma$.
It is easy to see, by using \cite[Proposition A.1.12]{BHV}, that $\pi$ does not have a nonzero finite dimensional subrepresentation as well.

Let $N := \mathcal{L}(K\otimes \overline{K})$ and $\alpha := {\rm Ad}\ \!\pi$.
Then $\mathfrak{H} = K\otimes \overline{K}\otimes K\otimes \overline{K} = \mathcal{H}\mathcal{S}(K\otimes \overline{K})$ (where $\mathcal{H}\mathcal{S}$ denote the space of all Hilbert-Schmidt operators), $\mathcal{U}_{\alpha} = \pi \otimes \overline{\pi}$ and the $^*$-representation of $N$ on $\mathfrak{H}$ is given by compositions.
As $\pi$ does not have any nonzero finite dimensional subrepresentation, we see that $\mathfrak{H}^{\mathcal{U}_{\alpha}} = (0)$ (again by \cite[Proposition A.1.12]{BHV}).
On the other hand, if $\{\xi_i\}_{i\in I}$ is an almost $\pi$-invariant unit vector in $K\otimes \overline{K}$, the $\sigma(N^*,N)$-limit of a subnet of $\{\omega_{\xi_i}\}_{i\in I}$ will produce an $\alpha$-invariant state on $N$.
This contradicts Statement (T5).
\end{prf}

\begin{rem}
One may regard Theorem \ref{thm:equiv-prop-T} as a sort of non-commutative analogue of \cite[Theorem 2.5]{Sch}.
There is also a non-commutative analogue of \cite[Theorem 2.4]{Sch}, which is basically a reformulation of \cite[Theorem 2.2]{B}.
More precisely, the following are equivalence for a discrete group $\Gamma$:
\begin{enumerate}[({A}1)]
\item $\Gamma$ is amenable.
\item $\mathfrak{M}_{N, \alpha} \neq \emptyset$ for every action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$.
\item For any action $\alpha$ of $\Gamma$ on $\mathcal{L}(K)$ (where $K$ is a Hilbert space) with $\mathcal{L}(K)_*^\alpha = (0)$, there exists an $\alpha$-invariant state on $\mathcal{L}(K)$.
\end{enumerate}
In fact, if $\Gamma$ is amenable, then \cite[Theorem 5.1]{B} implies that $\mathcal{U}_\alpha\otimes \overline{\mathcal{U}_\alpha}$ weakly contains $1_\Gamma$.
Thus, if we identify $N$ with the subalgebra $N\otimes 1$ of $\mathcal{L}(\mathfrak{H}\otimes \mathfrak{H})$, an almost $\mathcal{U}_\alpha\otimes \overline{\mathcal{U}_\alpha}$-invariant unit vector will produce an $\alpha$-invariant state on $N$.
To show (A3) $\Rightarrow$ (A1), one may consider the action $\alpha := {\rm Ad}\ \! \lambda$ on $\mathcal{L}(\ell^2(\Gamma))$.
It is easy to see that $\mathcal{L}(\ell^2(\Gamma))^\alpha_* = (0)$ (e.g.\ by Lemma \ref{lem:H-N_*}(a)) and an $\alpha$-invariant state will restricts to a left invariant state on $\ell^\infty(\Gamma)$, which implies that $\Gamma$ is amenable.
\end{rem}

Recall from \cite{LZ89} that $\Gamma$ has property $(T,FD)$ if $1_\Gamma$ is not in the closure of the subset of $\hat \Gamma$ consisting of non-trivial finite dimensional irreducible representations.
\begin{prop}\label{prop:no-inv-state}
Let $\Gamma$ be an infinite discrete group with property $(T,FD)$.
Then (T1) is equivalent to the following statement.
\begin{enumerate}[(T5')]
\item For any action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$ with $N_*^\alpha = (0)$, there is no $\alpha$-invariant state on $N$.
\end{enumerate}
\end{prop}
\begin{prf}
By Theorem \ref{thm:sp-gap-act-inv-mean}, we have (T1)$\Rightarrow$(T5').
Now, suppose that $\Gamma$ does not have property $(T)$.
As $\Gamma$ has property $(T,FD)$, there is a net $\{(\pi^i,K^i)\}_{i\in I}$ in $\hat \Gamma\setminus\{1_\Gamma\}$ with each $K^i$ being infinite dimensional and there is a unit vector $\xi_i\in K^i$ ($i\in I$) such that
$$\|\pi_t^i\xi_i - \xi_i\|\ \to\ 0
\qquad (t\in \Gamma).$$
Let $N$ be the von Neumann algebra $\bigoplus_{i\in I} \mathcal{L}(K^i)$ and set $\pi:=\bigoplus_{i\in I}\pi^i$ as well as $\alpha := {\rm Ad}\ \!\pi$.
Then $\mathfrak{H} = \bigoplus_{i\in I} \mathcal{H}\mathcal{S}(K^i)\cong \bigoplus_{i\in I} K^i\otimes \overline{K^i}$ and the representation of $N$ on $\mathfrak{H}$ is given by compositions.
In this case, one has
$$\mathcal{U}_{{\rm Ad}w}\big((\zeta^i \otimes \overline{\eta^i}\big)_{i\in I}\big)\ =\ \big(w_i\zeta^i \otimes \overline{w_i\eta^i}\big)_{i\in I}
\qquad \Big(w = (w_i)_{i\in I}\in {\bigoplus}_{i\in I}U(K^i); (\zeta^i \otimes \overline{\eta^i})_{i\in I}\in \mathfrak{H} \Big),$$
where $U(K^i)$ is the group of unitaries  on $K^i$.
Thus, $\mathcal{U}_{\alpha_t}(y) = \pi_t \circ y \circ \pi_{t^{-1}}$ ($y\in \mathfrak{H}; t\in \Gamma$), which implies that
$$\mathfrak{H}^{\mathcal{U}_{\alpha}}\ \subseteq\ {\bigoplus}_{i\in I}\{S_i\in \mathcal{H}\mathcal{S}(K^i): S_i\pi^i_t = \pi^i_t S_i,\ \! \forall t\in \Gamma\}$$
and hence $\mathfrak{H}^{\mathcal{U}_{\alpha}} = (0)$.

On the other hand, the $\sigma(N^*,N)$-limit of a subnet of  $\{\omega_{\xi_i}\}_{i\in I}$ (we consider $\mathcal{L}(K^i)\subseteq N$ for all $i\in I$) will give an $\alpha$-invariant state on $N$.
\end{prf}

A similar statement as the above for the strong property $(T)$ of locally compact groups can be found in \cite{LN}.

\begin{cor}
Let $\Gamma$ be a minimally almost periodic group in the sense of \cite{vNW40} (i.e.\ there is no non-trivial finite dimensional irreducible representation of $\Gamma$).
If $\Gamma$ satisfies (T5'), then it has property (T) and hence is finitely generated.
\end{cor}

\section*{Acknowledgement}
We would like to thank the referee for helpful comments and suggestions that lead to a better presentation of the article, and for informing us that one can use the materials in \cite{BV} to obtain (T5) $\Rightarrow$ (T1) for general countable groups (we only had Proposition \ref{prop:no-inv-state} in the original version.)
We would also like to thank Prof. Roger Howe for helpful discussion leading to a simpler argument for Theorem \ref{thm:equiv-prop-T}.

\bibliographystyle{plain}


On November 10th and 11th, 2016 we'll keep you busy building your new skills with LEGO® SERIOUS PLAY® at the Unconference at the Nita Lake Lodge in beautiful Whistler, BC., Canada!
Register Early as registrations will fill fast and the hotel is in demand!
Hello fellow LSP facilitators, designers, innovators and 3D thinkers!
Don't miss the first annual LSP community meeting at LEGOLAND California, USA!
What's trust got to do with LEGO® SERIOUS PLAY®?
A Lego Serious Play Master Trainer is in Sudbury to teach people how Lego can be used for professional development.


Setting the right starting price of the auction is an important step: as a seller you want to ask for the high price in order to get a high price. Academic researches show that to finish high, an auction should start low. Bidders are more likely to place a bid in an auction with a lower price than a higher. Thus, don't be too greedy while setting up your auction.


All photos by Benjamin Diedering.
A version of this article originally appeared on Noisey Germany.
Not in the mood for camping, drinking beer from a can, and spending three days awake in the wilderness when it's still freezing outside? We get it. And while we know it'll get warm again eventually, lounging under a cabana or getting soaked in a water gun fight still feels like a hazy memory from another life. As most of the country longs for sunshine, parts of the world are already dancing in the fresh air—especially in Miami, at the Ultra Music Festival.
First held in 1999, Ultra has grown into the EDM event of the year. Whether for the mainstream EDM bangers, the underground techno, or hip-hop, the festival's lineup attracts approximately 150,000 ravers to sunny Florida once a year. For the past two years, DJ Mag has deemed it the "Best Festival in the World" and the 2018 roster didn't disappoint, attracting the likes of Steve Aoki, Azealia Banks, DJ Snake, Marshmello, and even Swedish House Mafia, who made a surprising return after five years of silence. Consider adding it to your bucket list—or at the very least, check out our photos from this year.
Just to remind you of how damn beautiful everything can be in the summertime, photographer Benjamin Diedering went to Ultra to document the coolest ravers of 2018. While he was at it, he also captured some of the festival's picturesque palm trees, impressive high-rises, and scenes from the most electric EDM party of the year.
Follow Benjamin Diedering on Instagram.


text:
Die Eigentlichen Meerschweinchen (Caviinae) bilden eine Unterfamilie der Meerschweinchen (Caviidae) in der Ordnung der Nagetiere (Rodentia).

Merkmale
Es sind kleine, kompakt gebaute Tiere mit kleinen Ohren und kurzen Gliedmaßen. Die Kopf-Rumpf-Länge variiert zwischen 15 und 40 Zentimeter und das Gewicht zwischen 100 und 1500 Gramm.

Lebensweise
Eigentliche Meerschweinchen sind in weiten Teilen Südamerikas verbreitet, wo sie unterschiedlichste Habitate bewohnen, von flachen Grasländern bis in Gebirgsregionen von über 4000 Metern Seehöhe. Es sind in der Regel soziale Tiere, die in Paaren oder Gruppen mit einem Männchen, einigen Weibchen und den Jungtieren zusammenleben. Einige Arten haben komplexe Sozialstrukturen entwickelt.

Diese Gruppe wird in drei Gattungen mit rund 15 Arten unterteilt:
 Echte Meerschweinchen (Cavia), dazu zählen auch das Gemeine Meerschweinchen (C. aperea) und das Hausmeerschweinchen (C. porcellus)
 Gelbzahnmeerschweinchen (Galea)
 Zwergmeerschweinchen (Microcavia)

Innere Systematik

Die Eigentlichen Meerschweinchen bilden ein als Unterfamilie klassifiziertes Taxon innerhalb der Meerschweinchen. Sie stellen mit drei Gattungen und 15 Arten die größte Gruppe innerhalb der Meerschweinchen dar, die neben ihnen auch die in der Unterfamilie Hydrochoerinae zusammengefassten Capybaras (Hydrochoerus) und die Vertreter der Gattung Kerodon sowie die als eigene Unterfamilie betrachteten Pampashasen (Dolichotinae) umfassen.

Innerhalb der Eigentlichen Meerschweinchen werden die folgende Gattungen und Arten unterschieden:
 Eigentliche Meerschweinchen (Caviinae )
 Echte Meerschweinchen (Cavia )
 Gemeines Meerschweinchen (Cavia aperea )
 Tschudi-Meerschweinchen (Cavia tschudii )
 Patzelt-Meerschweinchen (Cavia patzelti )
 Glanzmeerschweinchen (Cavia fulgida )
 Riesenmeerschweinchen (Cavia magna )
 Santa-Catarina-Meerschweinchen (Cavia intermedia )
 Hausmeerschweinchen (Cavia porcellus )
 Gelbzahnmeerschweinchen (Galea )
 Östliches Wieselmeerschweinchen (Galea flavidens )
 Wieselmeerschweinchen (Galea musteloides )
 Spix-Gelbzahnmeerschweinchen (Galea spixii )
 Südliches Wieselmeerschweinchen (Galea comes )
 Tiefland-Wieselmeerschweinchen (Galea leucoblephara )
 Zwergmeerschweinchen (Microcavia )
 Südliches Zwergmeerschweinchen (Microcavia australis )
 Nördliches Zwergmeerschweinchen (Microcavia niata )
 Shipton-Zwergmeerschweinchen (Microcavia shiptoni )

Über die Zeit wurden zahlreiche weitere Arten beschrieben, die jedoch als Synonyme bereits beschriebener Arten eingeordnet wurden. Dies trifft unter anderem für das 2004 beschriebene Münstersche Meerschweinchen (G. monasteriensis) aus Bolivien zu, das nicht als eigenständige Art anerkannt ist und nach Jonathan L. Dunnum als Synonym der Unterart G. m. boliviensis des Wieselmeerschweinchens betrachtet wird. Andere Artbeschreibungen befinden sich in Diskussion und sind noch nicht in die wissenschaftliche Sekundärliteratur eingeflossen, darunter etwa die Neubeschreibung von Microcavia maenas  und Abgrenzung gegenüber dem Südlichen Zwergmeerschweinchen auf der Basis von Schädelmerkmalen.

Belege

Weblinks

Meerschweinchen

redpajama_set_name: RedPajamaWikipedia


Simmba trailer released: Simmba is Ranveer Singh's first release after his wedding with Deepika Padukone in November.
Simmba trailer released: In Simmba, Ranveer plays the role of a corrupt policeman, ACP Sangram Bhalerao.
Simmba trailer released: The trailer of Simmba dropped today, December 3 and movie buffs just can't keep calm! Simmba is Ranveer Singh's first release after his wedding with Deepika Padukone in November. This Rohit Shetty directorial is set to hit the big screen on December 28. The project is produced by none other than Karan Johar's Dharma Productions along with Rohit Shetty Picturez. Saif Ali Khan and Amrita Singh's daughter Sara Ali Khan will also be seen in the movie in a lead role. Simmba is Sara's second film this year along with Kedarnath where she will be seen besides Sushant Singh Rajput. Kedarnath will release much before Simmba, on December 7.
Besides Sara, Simmba is also Ranveer's first film with director Rohit Shetty. Known for his powerpack performances in films like Padmaavat, Bajirao Mastani and Lootera, Ranveer is one of the finest actors in the current generation. He will next feature in Zoya Akhtar's Gully Boy along with Alia Bhatt.
In Simmba, Ranveer plays the role of a corrupt policeman, ACP Sangram Bhalerao, but circumstances change his way and he ends up fighting for women's justice. Other Bollywood actors like Sonu Sood and Ashutosh Rana will also be seen in the film playing important roles. In special appearances, the audience will see Rohit's Golmaal fame Arshad Warsi, Tusshar Kapoor, Kunal Khemu and Shreyas Talpade, and bollywood supertsar Ajay Devgn.
Rohit's last movie, Golmaal Again, was a massive hit at the Box Office which garnered over Rs 200 crore at the BO. With Rohit's superhit releases in the past, it will be fun to see if Simmba will put the Box Office on fire and break all records set by Rohit's previous films.


SERVPRO of Concord is here for residents and businesses of the New Hampshire capital and the Greater Concord area. Our company has been a pillar of this community for twenty-eight years and remains one of the fastest growing restoration services. The key to our service is knowing the community. We regularly contribute to the Friends Program, the Friendly Kitchen, the American Red Cross, the Greater Concord Chamber of Commerce, Bow Rotary, the Salvation Army, as well as first responders like the Bow Volunteer Fire Department, the Bow Police Association, and the SERVPRO First Responder Bowl.
We are proud to be an active member of this community. SERVPRO of Concord is locally owned and operated, so we're already nearby and ready to help Concord, NH residents and business owners with any size cleaning or restoration emergencies.


Thank you for checking out our Job Postings page. As a reminder, please note the posting date. Jobs will be posted for 60 days. ATD Eastern PA verifies the information as best we can, however we cannot be held liable if content is incorrect. Be sure to visit the ATD Job Bank as well at www.jobs.td.org. Good luck!
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I recently attended the Nikon Landscape Photography Class at George's Camera with Nikon National Training Specialist, Paul Van Allen. As I expected, the presentation was loaded with images from Van Allen's travels and he provided us with good before and after references for comparison.
What I found interesting is that, as a Nikon National Training Specialist, he is not allowed to enhance his presentation images outside of the camera. There are minimal allowances for retouching photos (e.g., acne removal.) He uses the in-camera settings to adjust his images, as we might do if shooting in JPEG format. However, the raw Nikon files are also adjusted by these settings when using ViewNX or Capture NX (Nikon's Image editor software which, coincidentally, was created by Nik) to process them. Seems like a great time-saver. Using Nikon's proprietary software means that you have access to all of the information the camera wrote to the raw file so you can make further post-process adjustments if you want. I would be interested to know how others have found this process to work. There is a Canon counterpart in EOS, but the in-camera processing only produces a JPEG and does not alter the raw file to include any in-camera adjustment. You would need to apply those enhancements in software during post.
Since the Van Allen omits post-processing from his workflow, it makes sense that he depends on his aperture to determine depth of field. A smaller aperture can result in some cool light bursts but it can also result in some distortion, especially around the edges. You can use the Photoshop distortion adjustment for your lens, or omit it from your image altogether by using a larger aperture.
I attended a presentation a while back by landscape photographer, Vincent Versace. His workflow is extremely different and requires post-processing. Versace takes multiple shots, with varying focus on foreground, middle, and background objects at the camera's prime, or sharpest, aperture. The images are then stacked together in post-processing allowing the more focused objects in each image to be visible and masking out the un-focused portions. This is a technique known as focus-stacking which we workshopped earlier this year. The result is a single, fully-focused, image.
These are examples of two extremely different workflows for two purposes by two different people. What is your landscape photography workflow?
August was a busy month with Weekend Workshops and On Location photo outings a plethora. It was photography overload and there was literally something for everyone so, if you missed it, then you missed out but hopefully you can make up for it in September and October as we have some exciting photowalks coming in which you can take part.
We kicked the month off with a Stephen Burns HDR/Panorama/Stitching workshop and it was a packed Photographic Arts Building. This was an unusual workshop in many ways. First, we had an On Location shoot for the workshop because we needed to actually go out and get some photos to work with. It was a two part workshop so we were out in the field the first day then in the classroom on the second day. And a two-part workshop led by Stephen Burns is a rare event and a Darkroomers exclusive.
August 19th has been officially designated as World Photo Day by whoever makes these proclamations. It is an international event held each year led by Korske Ara and its origins date back to the invention of the daguerreotype. It was officially designated to be held on the birthdate of the daguerreotype printing press to celebrate the invention of photography and daguerreotype printing as a means to help promote the passion for photography in our communities.
This year World Photo Day landed on a Friday, and it was a near full moon, so the Darkroomers went out for a bonus On Location shoot in cooperation with Night Photography Lovers and the Best Places to Photograph in San Diego. It was a crowded shoreline on Harbor Island which made for some mild Anguish but the moon did not disappoint and the groaning stopped once that bright orange lunar surface started peaking its head above the buildings.
Finally, The Darkroomers ended the month with a Graffiti and Steel Wool shoot as a special On Location event with Night Photography Lovers. The Steel Wool Special was our 5th Wednesday event and a chance to get out of the office and escape the August heat and talk to some folks who have yet to join Darkroomers. We don't get many 5th Wednesdays and we like to pack as much fun as possible into our programs calendar and the steel wool event was a mega-blast.
THIS WAS TONS OF FUN! CAN'T WAIT TO SEE THESE ON THE BIG SCREEN AT HOME. AND SEE SOME OF THE PHOTOS FROM OTHER PEOPLE.
Don't forget that the theme for this quarter's Quarterly Competition is Anguish and the competition will be held on October 5th at 7:30pm in the Photographic Arts Building. We will be scheduling our Social Media Events soon for this gala event so it is time to kick it into high gear and get out and get some great photos for the competition.
Anguish is one of those topics that you may want to Google and see what kinds of images others have captured on the subject. Babies crying, people in misery, special camera effects and Photoshop work showing people coming apart from the inside are all good ideas for Anguish. The best subject matter for any photograph, however, is one that puts the viewer in an emotional state so keep that in mind when searching for Anguish to photograph.
Our Quarterly Competition program will also feature the photography of Vivian Maier in the documentary: Finding Vivian Maier. Laurie Rubin from ViewBug will also be stopping by with some fabulous prizes and giveaways for those in attendance.
If you are having trouble finding Anguish then maybe you should think about coming out and exploring the Park with us at the 500px Photowalk. The Park is a great place to find Anguish and we will be there to cover it as it happens... The walk is Saturday, September 17th at 2pm-5pm with a stop at Panama 66 for drinks and snacks.
We just added the Scott Kelby Photowalk to our programs calendar for October. The event takes place October 1st from 6:30-9pm and it is going to be an evening shoot in the Park. The walk will comprise several bridges in what we are calling, Bridges Over Balboa Park, and will take walkers over and under 5 bridges in and around the Park. It is filling up fast and we do want a good contingency of Darkroomers so go sign up today!
Valley View Casino Center house photographer Alan Hess will be stopping by for our September 7th meeting to talk about What Not to do in Photography. He is a former KelbyOne instructor and is a regular presenter at Photoshop World and, fair warning, you are likely to see some shirtless Justin Bieber photos in his presentation.
We will work with SCACC and the San Diego Portrait Club, who is scheduled to clean in January, to defer our remaining assigned 2016 cleaning dates until 2017 so that should give us a lighter schedule next year and hopefully beat the heat.
Our September 7th meeting will include a vote on comprehensive changes to the Quarterly Competition Rules and Year End Competition Rules which will finally move all rules for competition into the bylaws. The Year End Competition Rules changes were drafted by the Year End Competition Committee and marked up by the Darkroomers Board for your consideration and include some exciting new additions to this year's competition. The Quarterly Competition Rules changes are a Pro-Forma agreement but one significant change will be that we will no longer allow Quarterly Competition winners to be entered into the Year End Competition without first being juried in a monthly exhibition.
SCACC will be meeting in September to ratify changes to its bylaws which are the culmination of changes to its charter under the leadership of SCACC President, Les Anderson. The meeting takes place on September 6th at 7pm in the Photographic Arts Building.
The Photo Artists Camera Club has been dissolved and will no longer be holding regular monthly meetings in the Photographic Arts Building.
Our next bi-monthly Interclub competition will be held on September 20th at 7:30pm in the Photographic Arts Building. While Darkroomers, Poly and Photonaturalists will be submitting their usual 20 images, Fallbrook will be submitting 40 images because they failed to submit their June/July images on time.
Our Judge for September is San Diego legend, Tim Mantoani. Tim told the story of local San Diegans through his photography in the KPBS miniseries, Snapshot, which aired from March 2014 through September 2014. Tim was the creative force that drove: Behind the Photographs Archiving Photographic Legends where Tim used a 235-pound, five-foot-tall, 20-by-24 Polaroid View Camera to capture the true essence of some of the industry's contemporary masters in photography.
Tim is a graduate of the Brooks Institute and former assistant to legendary photographer, Dean Collins, who taught him not only how to see light; but how to control it. Tim has a studio in Little Italy where he does commercial work for the likes of ESPN, EA Sports and Coca Cola.
The Motor Transport Museum, or the Rust Museum as it is affectionately known, is offering photographers from around the county the opportunity to come photograph the golden hour and blue hour, using their famous rusted out vehicles and buildings as a backdrop.
This is your opportunity to capture the magic of the museum's after hours--when the trucks really start talking. It is a unique opportunity to meet other photographers and capture that magic moment, just before sunset, when the ambient light provides brilliant colors with a warm glow. Come photograph the old trucks by the light of a full moon and listen to those beasts come to life as the coyote's howl and scream.
September 18th. 5pm-9pm; Motor Transport Museum; Campo CA. RSVP is required. Contact the museum directly for RSVP instructions.
San Diego Bayfair's World Series of Powerboat Racing invites over 23 individual racing classes and over 150 high competition teams to deafen the waters of Mission Bay Park. The boats, some racing at speeds over 225 miles per hour, come every year to San Diego because of the outstanding race course conditions and summer weather.
San Diego Bayfair attracts more powerboat racing classes than any other site in the world, and is known as the hallmark event in the nation. The different classes of boats will race on the famed Bill Muncey Memorial Race Course, in honor of the sport's greatest unlimited driver with 62 unlimited wins. The Bill Muncey race course is the fastest race course in the world.
September 16 - 18, East Mission Bay; $20 General Admission.
Marine Corps Air Station (MCAS) Miramar is once again hosting the largest military Air Show in the country! This year's theme is Celebrating 100 Years of the Marine Corps Reserves.
MCAS Miramar is honored to host an amazing lineup of military and civilian performers and is pleased to welcome back the world famous Blue Angels for what is sure to be another thrilling performance. Civilian Breitling L-39 Jet Team will be performing along with U.S. Navy Leap Frogs, U.S. Army Golden Nights and the amazing Marine Air Ground Task Force (MAGTF).
The flight line will be transformed to accommodate more than 100 military and civilian aircraft, simulation rides, fun for the kids, ride and drives, and a variety of delicious food and beverage offerings for the annual event. September 23-25; Marine Corps Air Station Miramar; Parking and General Admission are Free but Preferred Seating is available for a nominal fee.
Fleet Week Coronado Speed Festival is Fleet Week San Diego's marquee event. The event features fast-paced, thrilling auto racing and exhibitions for car enthusiasts of all ages at one of the most distinctive race track locations — Naval Air Station (NAS) North Island, the birthplace of Naval Aviation. Nowhere else can you see a variety of cars racing on a live runway used by the U.S. Navy. And check out the car show adjacent to the track during the down time to see some really beautiful fast cars.
September 17th & 18th; NAS North Island. General Admission Pricing is $25 in advance, $35 day of. VIP and Winners Circle Passes are available for a nominal fee.
Each year the San Diego Maritime Museum opens its doors and lets down its sails so that the general public may enjoy the boats in an annual event it calls The Festival of Sail. Oh and several tall schooners from around the world make their way out into the harbor to participate in the biggest tall sailing ship parade on the West Coast. Bring a long lens, sunblock and comfortable shoes. Seven Dollars. Seniors get in for Five.
Labor Day Weekend at the Embarcadero. Get there early as heavy crowds are expected and parking will be a challenge. The Parade of Sail takes place on Friday, September 2nd at 11:30am.
If you have ever wondered what it was like to live in Scandinavia in the old times then this festival might be for you. This is a family event so if you have offensive tattoos or cannot resist dressing inappropriately then you probably will want to stay home but if you like jousting and non-drunken Viking singing then you might as well attend. Eight dollars.
As part of Julian's Apple Days Celebration, a two-day Apple Days Festival will be held September 24 and 25 at Menghini Winery featuring apple displays, music and dancing, an antique tractor display, children's games and activities, gold panning demonstrations, a beer and wine garden, food and merchandise vendors, contests, and, of course, apple pie. The event runs from 10:00 AM to 5:00 PM, both Saturday and Sunday. Admission is $5.
2 days of cultural dance by beautiful local Pacific Islanders and food booths. September 24 and 25. Ski Beach, Mission Bay.
Oceanside Harbor Days doesn't look like a hugely exciting event but this year it will feature an unlikely pairing of Pirates and Mermaids together in their own village for kids and adults of all ages to enjoy.
Venus de Milo, Augustus and David. Photo courtesy of THE ART OF THE BRICK.
Named one of CNN's Top Ten Global Must-See Exhibitions, The Art of the Brick is a critically acclaimed collection of inspiring artworks made exclusively from one of the most recognizable toys in the world: the LEGO® brick. Contemporary artist Nathan Sawaya transforms a child's toy into sophisticated art and beyond, in the world's largest display of LEGO art ever.
Walk under a 20-foot-long T-Rex dinosaur skeleton made out of LEGO bricks.
See a re-imagined version of Van Gogh's Starry Night.
Create your own LEGO art at the Fit-A-Brick Building Zone.
Get up-close and in-person with the fan favorite, Yellow, a life-size sculpture of a man ripping his chest open with thousands of yellow LEGO bricks cascading from the cavity. At The Art of the Brick, you'll enjoy re-imagined versions of the world's most famous art masterpieces, like Da Vinci's Mona Lisa, as well as a gallery showcasing an innovative, multimedia collection of LEGO brick-infused photography produced in tandem with award-winning photographer Dean West.
The Art of the Brick makes its Southern California premiere at the Reuben H. Fleet Science Center on Friday, September 30, 2016. It will be open daily at 10 a.m. through January 29, 2017. Don't miss it!
Photos used in this newsletter from Robert J. Makar, Gert Beukema, John Pryor, Stephen Burns, Jeff Booher and Michael Fairbanks are used with permission and may be subject to copy restrictions from the copyright holder. All rights reserved.


Family portrait sessions are held at the location of your choice (your home, local park, museum, etc.), within 30 miles of Richmond, Va. Styling tips are given upon request, to help coordinate your family and home decor seamlessly.
Maternity sessions have the option of choosing from several maternity gowns in my studio collection. A complimentary maternity session is included with every newborn purchase.
All portrait sessions last up to 90 minutes.
Newborn sessions are my specialty, are one of my favorite sessions to do. I love being able to give new parents a break while I create beautiful memories. The best time to photograph a newborn is between 7-20 days of life, to be able to capture those sweet sleepy poses, while baby also has the time to pass through the first stages of cluster feeding.
These sessions need to be scheduled months in advance to ensure I have blocked those special dates around the time baby is due. If baby comes earlier or later than expected, I will be in contact to adjust schedule time frames. For more information on what to expect for your session, please read my What to Expect page, for your Newborn Session information.
These sessions will take place in your home, or my in space if you prefer, and take between 2-3 hours (this time can be less if baby is sleepy and cooperative).
These sessions are designed to capture sweet memories in each milestone stage. They are typically quick sessions, working at a faster pace with babies becoming more mobile. I prefer to schedule these at stages rather than ages, as all babies grow and develop at different rates.
The first stage is sitting up, between 5-8 months of age. We focus this session on capturing babies growing features, smiles, and the interaction that babies develop at this age.
The second stage is standing, between 9-12 months of age. This is the best time to schedule cake smash portraits, or food smash for a fun and messier time! These milestone sessions typically last up to one hour.
This session fee includes the time and talent of the photographer, and is due upon booking to schedule your session.
Each session is up to 90 minutes (this time is based on the photographers discretion if the session needs more time). All sessions take place within 30 miles of Richmond, Va.
After your session, I carefully choose the best images to edit, and schedule our in person viewing to help you choose the gallery that will blend with your home decor. Each print purchased comes with the matching high resolution digital image, provided for you through an online personal gallery. These galleries will remain available for 10 years. A list of packages and A La Carte items are available upon request.


text:
Polyrhythmicity in language, music and society : complex time relations in the arts, humanities and social sciences /
Soumis par Joseph Hafner le 6 Janvier, 2022 - 15:26
Andrews, Richard,
Springer,, Volume v.12, Singapore, p.1 online resource (2021)
10.1007/978-981-16-0566-6
https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9789811605666
(OCoLC)fst00992214, (OCoLC)fst01079397, (OCoLC)fst01741439, fast, Language and languages, Polyrhythm., Prosodic analysis (Linguistics), Rhythm.
1. Introduction -- 2. The Nature of Multiple- and Poly-rhythms -- 3. Polyrhythmicity in Time: an International Perspective -- 4. Polyrhythmicity in Music -- 5. Polyrhythmicity in Conversation and Speech Prosodies -- 6. Polyrhythmicity in Poetry -- 7. Polyrhythmicity in the Novel -- 8. Polyrhythmicity in Social Situations -- 9. Polyrhythmicity in Contemporary Hybrid Culture -- 10. Polyrhythmicity in the Asia Pacific Region -- 11. Researching Polyrhythmicity -- 12. Polyrhythmicity in Rhetoric, Learning and Education.Includes bibliographical references and index.This book addresses the complex time relations that occur in some types of jazz and classical music, as well as in the novel, plays and poetry. It discusses these multiple levels of rhythm from a social science as well as an arts and humanities perspective. Building on his ground-breaking work in Re-framing Literacy, A Prosody of Free Verse and Multimodality, Poetry and Poetics, the author explores the world of multiple- or poly-rhythms in music, literature and the social sciences. He reveals that multi-layered rhythms are uncommon and little researched. Nevertheless, they are important to the experience of art and social situations, not least because they link physicality to feeling and to decision-making (timing), as well as to aesthetic experience. Whereas most poly-rhythmic relations are felt unconsciously, this book reveals the complex patterning that underpins the structures of feeling and of experience.Online resource; title from PDF title page (SpringerLink, viewed July 6, 2021).

redpajama_set_name: RedPajamaCommonCrawl


PANAMA CITY � Sandi Hawks exemplifies the ideal support employee at the school district.
Hawks, assistant to the director of information services at the school system headquarters, was selected out of 43 other support employee candidates to represent the district at the state recognition program, which spotlights the function of support employees and honors them at an annual ceremony.
Operational and support occupations, such as maintenance workers, janitors, bus drivers and secretaries, all fall under the umbrella of support personnel.
�They are the support background for the school,� Mullen said. She said �going beyond training� through participation at school, district and community activities and demonstrating leadership were characteristics all 43 nominees exemplified.
Hawks is the Parent Portal wizard and assists parents with learning the system. She also does record keeping.
�She's just such a positive person in an area that's not easy to be positive all the time,� said Superintendent Bill Husfelt. �She's the first one to get a call when computers go down and usually the last one to leave there everyday.
"If you want anything (in information services), you go through Sandi,� he added. �She's top notch."
Hawks has worked at the district for 34 years and it was the first time she�d been nominated. Along with the award, she won: $300 from Panhandle Educators Federal Credit Union, $250 from Bay Education Foundation, $50 from Wendy Varnum at Horace Mann Insurance.
Candidates are nominated by their peers at each school and four ancillary departments � Nelson Building, Maintenance Department, Transportation Department and Young Building. A five member selection committee then selects the Support Employee of the Year.
-Rebecca Woosley, confidential secretary, Southport Elementary School.


This podcast provides immeasurable amounts of instantly usable information and tools to keep you at your best!
I've been listening to Mojo for over a year now and am slowly working my way through the back catalogue as well to uncover more GOLD from Gary, Robbo and their lineup of incredible guests.
Gary's obvious dedication to researching his guests has perfected an alchemy of interviewing that extracts the most precious wisdom from those that are masters of their craft.
Keep up the great work guys and let's catch up for a Dos Equis sometime soon.
Some really good interviews on here. Keep up the good work, production is great too.


Freedom at Volusion means setting up your store your way. Find out how to create your own Child Product Codes so they work for you!
Return to the parent product page where the options are already applied. Add a space to the product description and delete it (to prompt the Save button to appear) and click Save. Make sure that Enable Options Inventory Control is still checked.
Scroll to Inventory Control Grid for "ProductCode" and expand it if necessary. You'ill see the grid that was created using your options, but all of the Inventory Grid ProductCodes will have the same product code as the main parent product.
A message will appear at the top of the grid window stating, "The Following Products Did Not Exist, And Therefore Were Automatically Created For You." Below the message, will be a list of the new child products — with your custom product codes — that will be in the grid.
At Volusion, we know no two stores or store owners are alike. 👨🌾 👩🎨 👨🍳 If one way works for one, there may be a better way for another. That's why you have the flexibility to set up your store — and your child product codes — to work best for you!


Warren Davis, a husband and father of two young children was diagnosed with the rarest form of non-Hodgkin's lymphoma in 2004. Because he had no donor match among siblings or on the registry, he was treated with an autogolous stem cell transplant which is a process that used his own bone marrow stem cells.
Four years later the lymphoma returned. This time by miracle a matching white donor half way around the world in Egypt was found. She donated her stem cells to him in August 2008 and he miraculously returned to work within months. Warren has been recognized for his support for organizations and individuals who support this cause.


text:
Liriomyza quiquevittata är en tvåvingeart som beskrevs av Mitsuhiro Sasakawa 1994. Liriomyza quiquevittata ingår i släktet Liriomyza och familjen minerarflugor. Inga underarter finns listade i Catalogue of Life.

Källor

Minerarflugor
quiquevittata


text:
Patent, trademark and copyright are integral parts of intellectual property that form intangible assets in our daily life. IP news and matter from Malaysia and the world are shared here.
Amendments to Copyright Act to come into force by April
New regulations following the amendment of the Copyright Act 1987 last year, which include a wider jurisdiction for the Copyright Tribunal, are expected to be submitted to the Attorney-General for implementation from April.
Domestic Trade, Cooperatives and Consumerism Minister Datuk Seri Ismail Sabri Yaakob said the ministry and the national creative industry players would be holding their last meeting to finalise the regulations at the end of this month before submitting it to the Attorney-General.
Among the new regulations to be enforced included a system of voluntary registration of copyright works and the registration of a royalty collection body, he told reporters after holding the second meeting with the national creative industry players on the issues of the new regulations at the office of Intellectual Property Corporation of Malaysia (MyIPO) here Wednesday.
"The new regulations of the Copyright Tribunal should involve expanding the jurisdiction of the tribunal on matters relating to payment of royalties to artistes.
"The tribunal should be expanded to ensure the individual responsible for collecting royalty should be transparent apart from informing the artistes concerned on their work used," he said.
Among the arts industry player representatives present were Persatuan Karyawan Malaysia, Persatuan Penulis Skrin Malaysia, Persatuan Penerbit Filem Malaysia as well as local actors and singers. - Bernama
Posted by Dennis at 4:20 PM
Labels: copyright, MyIPO
or Tan Boon Leng, is a practicing patent, trademark & copyright agent in Malaysia. He is also a recognized IP valuer. He graduated with a masters degree in physics research. Dennis is also trained in copyright from Harvard Law School, and IP Law & Policy from the University of Pennsylvania. He was the Secretary for Licensing Executives Society Malaysia and the Honorary Secretary of Malaysia Intellectual Property Association for 2012-2014.
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On view through April 14, 2013, The Walt Disney Family Museum's special exhibition celebrates Walt Disney's first feature-length animated film: Snow White and the Seven Dwarfs. The exhibition showcases more than 200 works of art including conceptual drawings, early character studies, detailed story sketches, and animation drawings. Also featured are delicate thumbnail layout watercolors, meticulously rendered pencil layouts, rare watercolor backgrounds, colorful cels, and vintage posters all illustrating how Walt Disney advanced the creation of an entirely new art form.
Generous support provided by Wells Fargo.
Between Frames explores the evolution of stop motion animation in the United States in special effects, television, and film. The exhibition highlights behind-the-scenes photographs, rare armatures and puppets, and original storyboards from cult favorite films and television including King Kong, Gumby, Star Wars, Jurassic Park, Robot Chicken, and Tim Burton's The Nightmare Before Christmas and showcases stop motion innovators like Ray Harryhausen, Phil Tippett, Will Vinton, Tim Burton, Henry Selick, and more.
Perhaps no other film library showcases Americana more than the one created by The Walt Disney Studios. Ever careful to preserve this collection for future generations to enjoy, the Disney Studios has maintained the pioneering spirit of Walt's dedication to exploring new technologies.
Theodore Gluck, the Studios' Director of Library Restoration and Preservation, will present an exclusive look inside the processes used to bring these beloved films to a contemporary world. Starting with the development of Technicolor and continuing through the early days of CinemaScope, Gluck will demonstrate the challenges now faced by the Studio Restoration Team as they work to restore these world-renowned films to their original splendor.
(This program is presented in conjunction with the Museum's feature exhibition, Snow White and the Seven Dwarfs: The Creation of a Classic).
Hear expert highlights of our special exhibition, Between Frames: The Magic Behind Stop-Motion Animation.
Drawing from The Walt Disney Family Museum's collection of priceless Disney animation cels, Museum art conservator Martin Salazar will discuss how these cels were first created, explain the causes of cel deterioration, and demonstrate techniques he has developed to care for these treasures. Salazar will also discuss the ethical challenges faced by institutions when attempting to preserve fragile artifacts. The lecture will conclude with a guided tour of the Museum's first major exhibition, Snow White and the Seven Dwarfs: The Creation of a Classic. Salazar will be assisted by Museum conservation technician Harvey Newman.
This class is for participants ages 18+, all levels of experience. Space is limited. Registration required. Email: education@wdfmuseum.org to register or for more info.
Ruben Esqueda, independent filmmaker and animation mentor since 1998, will lead a hands-on four-hour master class for animators at all levels of experience. Participants will learn the process of building a professional quality stop motion puppet armature, which will be ready for use in animation by the end of the class. Participants will also learn other types of puppet construction including wood, ball and socket, and prefabricated toy puppet armatures. The class will involve the use of wire-cutting materials and epoxy, so comfortable dress is recommended.
Create stop motion animation in our Media Studio and try your hand at two early steps of creating an animated film—storyboards and concept art—in our Art Studio.
Hours: 10am to 6pm, Wednesdays through Monday; closed on Tuesdays and the following public holidays: New Year's Day, Thanksgiving, and Christmas Day.


For the most part, this proved to be an exceptionally enjoyable effort. One of it's strongest aspects is the manner in which this one works the fear of growing up into a decidedly fun effort. With a fantastic start that provides a reasonable and wholly necessary motivation for the fear she has towards the event and colors her emotional memories of her mother's incident, there's a great start to this one that gets played out rather nicely here. These events are rightfully psychologically scarring but to also see this hasn't affected her in most stereotypical manners is quite refreshing with the way the early goings of the party which is rather fun. It's once the creatures arrive at the house where this one really turns into a full-on blast. The creatures running wild on the guests at the party offers up scores of highly-enjoyable antics here which are all the more fun due to the practical nature of the effects. Given that the various and multiple monsters and creatures are shown to have memorable and unique designs that are driven by that manner of practical costumes for them, there's an added sense of fun in seeing the spike-backed ox creature or the gnome-worms running around the party as there's a sense of realism in seeing that occur. Given that these also manage to unravel the mystery behind their appearance and the ritual that's at the center of the creatures appearing at the house, this prepares us nicely for the overall fun of the finale where it tackles the creatures being taken out in rather fun fashion and unleashing plenty of over-the-top confrontations and action-packed encounters that leave this with a lighthearted tone alongside the thrilling sequences. These here hold the film up over it's few minor flaws as there were several small problems with it. The biggest issue is the absolute lack of clarity as for what's going on as the decision to keep the information about the creatures' appearance until the very end doesn't mean this makes any sense. The running time is spent with everyone around the party wondering what the purpose of them appearing is supposed to be for, and even their attempts at figuring out the translations in the book don't amount to much with several questionable aspects. As well, there's also the rushed finale which handles the action in a decidedly quick manner, making this one feel like the final boss battle between them and the most dangerous demon is a tad underwhelming. Otherwise, there isn't much wrong with this one.
Rated Unrated/R: Graphic Violence and Graphic Language.


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Markets South Korea Stocks Consumer Non-Cyclicals Food Processing
CROWNHAITAI HOLDINGS
Industry: Food Processing
CROWNHAITAI Holdings CO.,LTD, formerly CROWN CONFECTIONERY CO.,LTD, is a Korea-based company mainly engaged in the production of confectionery products. The Company's products include biscuits, wafers, snacks, pies, chocolates, sweets and gums. Its biscuit brands include Charmgreen, Five grains cookie, KUK HE and others. Its wafer brands include Green Heim, Choco and White Heim, and Crown Wafers. Its snack brands include Corn Pong, Jollypong, Corn Chip and others. Its pie brands include Lonx, O'new and others. Its chocolate brands include Kicker, Minishell and Black Rose. Its sweets brands include Mychew, Mango TwoTwo, Mint Candy and others. Through its subsidiaries, it also engages in the production of ice cream products, dumplings, breads, cakes, cookies and other frozen products, as well as engages in the franchise business and real estate leasing business.


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You can find many Skype ID's on Skype messenger and if you are lucky enough, you can find an active account. You can see our location based guides on how to chat online. We are going to provide similar tips with those location based Skype guides on here too since it is the best way to get Skype IDs.
If you haven't got a Skype Messenger, install it from Play Store, App Store, official website or Microsoft Store.
Create your account and run the application on your device.
Paste them on Skype search box and search.
Add people you want to talk to and send them message.
That is all you need to do for finding Skype ID. If you have any question about this guide, please ask us.
If you add someone randomly on Skype, it will be hard for you to talk someone since you know nothing about him/her. We recommend you to take a look at partner's profile and try to figure out what to talk. However it is not recommended to tell someone that you love or you want a relationship with her. You will be rejected.
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Why is there no daylight savings in Queensland?
Will Queensland ever get Daylight Savings?
When did daylight saving stop in Qld?
Is Queensland doing Daylight Savings 2021?
What 3 states do not observe daylight savings time?
Will Australia scrap Daylight Savings?
What is the point of daylight savings in Australia?
When did daylight savings start Australia 2021?
Why does Brisbane not have daylight savings?
Has daylight saving started in Brisbane?
Are we losing an hour of sleep 2021?
Is daylight Savings time going to be discontinued?
Does Australia change the clocks?
Why do we change the clocks twice a year?
What was the original reason for Daylight Savings Time?
Why do we still do daylight Savings?
In summary, Day Light savings is not designed for tropical regions like Far North Queensland. That is why they don't have it. Polls have repeatedly shown that the majority of Queenslanders outside the south-east corner do not want daylight saving.
Daylight saving is not observed in Queensland, the Northern Territory, Western Australia, Christmas Island or the Cocos (Keeling) Islands. Daylight Saving Time begins at 2am on the first Sunday in October, when clocks are put forward one hour.
Daylight saving time was trialled in the state of Queensland, Australia, during the 1989/90 season, with the trial extended for a further two years—1990/91 and 1991/92. The last full day of daylight saving in Queensland was Saturday 29 February 1992, with clocks officially wound back an hour on Sunday 1 March at 3am.
FASCINATINGLY: You asked: Why is Adelaide time ahead of Brisbane?
Daylight Saving Time (DST) Not Observed in Year 2021
DST is no longer in use. Clocks do not change in Queensland. The previous DST change in Queensland was on March 1, 1992.
The only parts of the US that do not have Daylight Saving Time are Hawaii, most of Arizona, Puerto Rico, the US Virgin Islands, Northern Mariana Islands, Guam, and American Samoa. Arizona experimented with the change beginning in 1918, but decided to permanently opt out of the Daylight Saving Time in 1968.
This Sunday, 3 October 2021, clocks go forward in New South Wales, Victoria, South Australia, Tasmania and the ACT, effectively eliminating the hour between 1am and 2am. Queensland, Western Australia and the Northern Territory do not engage in the fantasy of clock-shifting, and time remains constant in these regions.
Daylight Saving Time (DST) is when clocks are set forward one hour from ST, in order to make better use of natural daylight. In Australia, DST moves clocks forward one hour in the summer, and back one hour when returning to Standard Time (ST) in autumn.
Daylight saving for all future summers in New South Wales begins at 2.00 am on the first Sunday in October and ends at 3.00 am on the first Sunday in April.
Future dates of daylight saving in NSW.
FASCINATINGLY: What is the best month to visit Melbourne?
Daylight saving begins
Daylight saving ends
4 October 2020 4 April 2021
Brisbane currently observes 6 time zones all year. DST is no longer in use. Clocks do not change in Brisbane, Australia. The previous DST change in Brisbane was on March 1, 1992.
Daylight Saving Time begins on Sunday, March 14, 2021 at 2:00 A.M. On Saturday night, clocks are set forward one hour (i.e., losing one hour) to "spring forward." Daylight Saving Time ends on Sunday, November 7, 2021, at 2:00 A.M. On Saturday night, clocks are set back one hour (i.e., gaining one hour) to "fall back."
In California as well, voters authorized the end of DST in 2018, but likewise they are awaiting federal action. The issue has gone back and forth in Texas over whether to change daylight saving time, but so far no action has been taken in the state.
In Australia, Daylight Saving Time is observed in New South Wales, Victoria, South Australia, Tasmania, the Australian Capital Territory and Norfolk Island. … Australia is the world's sixth largest country and is divided into three separate time zones, hence why not all of the states observe DST.
FASCINATINGLY: Frequent question: How much is it to fly to Sydney Australia?
The energy saving benefits of Summer Time were recognized during World War II, when clocks in Britain were put two hours ahead of GMT during the summer. This became known as Double Summer Time. During the war, clocks remained one hour ahead of GMT throughout the winter.
The nominal reason for daylight saving time has long been to save energy. The time change was first instituted in the U.S. during World War I, and then reinstituted again during WW II, as a part of the war effort.
In most of the United States, daylight savings time just ended. If you haven't yet, set your clocks back one hour. The Department of Transportation says daylight savings time conserves energy, prevents traffic accidents and reduces crime.