Let $\mathcal{G}$ be a countably infinite group of unitary operators on a complex separable Hilbert space $H$. Let $X = \{x_{1},...,x_{r}\}$ and $Y = \{y_{1},...,y_{s}\}$ be finite subsets of $H$, $r < s$, $V_{0} = \bar{span} \mathcal{G}(X), V_1 = \bar{span} \mathcal{G}(Y)$ and $ V_{0} \subset V_{1} $. We prove the following result: Let $W_0$ be a closed linear subspace of $V_1$ such that $V_0 \oplus W_0 = V_1$ (i.e., $V_0 + W_0 = V_1$ and $V_0 \cap W_0 = \{0 \}$). Suppose that $\mathcal{G}(X)$ and $\mathcal{G}(Y)$ are Riesz bases for $V_{0}$ and $V_{1}$ respectively. Then there exists a subset $Γ=\{z_1,..., z_{s-r}\}$ of $W_0$ such that $\mathcal{G}(Γ)$ is a Riesz basis for $W_0$ if and only if $ g(W_0) \subseteq W_0 $ for every $g$ in $\mathcal{G}$. We first handle the case where the group is abelian and then use a cancellation theorem of Dixmier to adapt this to the non-abelian case. Corresponding results for the frame case and the biorthogonal case are also obtained.
In this article we show that there are branching systems (which induce representations of the graph algebra $C^*(E)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the condition $(L)$ via branching systems. Moreover, we show that each permutative representation in Hilbert spaces operators is unitarily equivalent to one induced by a branching system, even the spaces being not separable. Furthermore, under some hypothesis on the graph, we show that each representation of the graph C*-algebra is permutative.
We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures.
In this article we study MF-property for countable discrete groups, i.e. groups which admit embedding into unitary group of $C^*$-algebra $\prod M_n/\oplus M_n$. We prove that Baumslag group $\langle a,b|a^{a^b}=a^2\rangle$ has MF-property and check some permanent facts about MF-groups.
We show that the unitary conjugacy relation for unitary representations of a second countable locally compact group on a separable Hilbert space is a Borel equivalence relation.
In this paper we show that, for a class of countable graphs, every representation of the associated graph algebra in a separable Hilbert space is unitarily equivalent to a representation obtained via branching systems.
We characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group $Γ$ on a single element $ψ$ of a given Hilbert space $\mathcal{H}$. As $Γ$ might not be abelian, this is done in terms of a bracket map taking values in the $L^1$-space associated to the group von Neumann algebra of $Γ$. Our result generalizes recent work for LCA groups. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform.
A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we charaterise up to isomorphism the countable subgroups $Δ$ of the unitary group of a separable infinite dimensional Hilbert space $\Cal H$ of which the bicommutants $Δ''$ (in t…
Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on $\wG\times\G$.
For second countable discrete quantum groups, and more generally second countable locally compact quantum groups with trivial scaling group, we show that property (T) is equivalent to every weakly mixing unitary representation not having almost invariant vectors. This is a generalization of a theorem of Bekka and Valette from the group setting and was previously established in the case of low dual by Daws, Skalsi, and Viselter. Our approach uses spectral techniques and is completely different from those of Bekka--Valette and Daws--Skalski--Viselter. By a separate argument we furthermore extend…
The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in Dyn…
Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators $A,B$ on a Hilbert space $H$ are unitarily equivalent modulo compacts, i.e., $uAu^*+K=B$ for some unitary $u\in \mathcal{U}(H)$ and compact self-adjoint operator $K$, if and only if $A$ and $B$ have the same essential spectra: $σ_{\rm{ess}}(A)=σ_{\rm{ess}}(B)$. In this paper we consider to what extent the above Weyl-von Neumann's result can(not) be extended to unbounded operators using descriptive set theory. We show that if $H$ is separable infinite-dimensional, this equivalence relation for bounded self-adjoin operators…
We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.
We prove that the unitary equivalence classes of extensions of C*_r(G) by any sigma-unital stable C"-algebra, taken modulo extensions which split via an asymptotic homomorphism, form a group which can be calculated from the universal coefficient theorem of KK-theory when G is a free product of a countable collection of amenable groups.
Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^*$-algebras associated to certain $C^*$-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-v…
In this paper we establish a direct connection between stable approximate unitary equivalence for $*$-homomorphisms and the topology of the KK-groups which avoids entirely C*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group $KL(A,B)=KK(A,B)/\bar{0}$, and prove that if a separable exact residually finite dime…
Let $G$ be a discrete countable group, and let $Γ$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $π$ of $G$ into the unitary group of the Hilbert space $l^2(Γ)$ that extend the left regular representation of $Γ$. Representations with this property are obtained by restricting to $G$ square integrable representations of a larger semisimple Lie group $\bar G$, containing $G$ as dense subgroup and such that $Γ$ is a lattice in $\bar G$. This type of unitary representations of of $G$ appear in the study of automorphic forms. We…
Let $Γ$ be a countable discrete group, and let $π\colon Γ\to {\rm{GL}}(H)$ be a representation of $Γ$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\rtimes_πΓ$ is SIN ($G$ admits a two-sided invariant metric compatible with its topology) and unitarily representable ($G$ embeds into the unitary group $\mathcal{U}(\ell^2(\mathbb N))$), if and only if $π$ is uniformly bounded, and that $π$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor. Consequently, we show that a unitar…
We consider converses to the density theorem for irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that under the right assumptions, the restriction of a $σ$-projective unitary representation $π$ of a group $G$ to a lattice $Γ$ extends to a Hilbert module over the twisted group von Neumann algebra $\text{L}(Γ,σ)$. We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that t…
Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of op…
In the theory of unitary group representations, a group is called type I if all factor representations are of type I, and by a celebrated theorem of James Glimm [Gli61b], the type I groups are precisely those groups for which the irreducible unitary representations are what descriptive set theorists now call "concretely classifiable". Elmar Thoma [Tho64] proved the following surprising characterization of the countable discrete groups of type I: They are precisely those that contain a finite index abelian subgroup. In this paper we give a new, simpler proof of Thoma's theorem, which relies onl…
Let $Γ$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(Γ) = C^{\ast}(Γ\rtimes l^{\infty}(Γ))$. Let $G$ be a larger discrete group, containing $Γ$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(Γ)$. Let $\mathfrak{R}_G(Γ)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(Γ))$. The C$^\ast$-algebra $\mathfrak{R}_G(Γ)$ has a natural representation into $\mathcal B(\ell ^2(Γ))$ and hence also admits a representation $Π_{\mathcal{Q}}$ into the Calkin algebra $\math…
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
Let $F$ be a finite group. We consider the lamplighter group $L=F\wr\mathbb{Z}$ over $F$. We prove that $L$ has a classifying space for proper actions $\underline{E} L$ which is a complex of dimension two. We use this to give an explicit proof of the Baum-Connes conjecture (without coefficients), that states that the assembly map $μ_i^L:K_i^L(\underline{E} L)\rightarrow K_i(C^*L)\;(i=0,1)$ is an isomorphism. Actually, $K_0(C^*L)$ is free abelian of countable rank, with an explicit basis consisting of projections in $C^*L$, while $K_1(C^*L)$ is infinite cyclic, generated by the unitary of $C^*L…
Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras. We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined. We provide a diverse collection of examples including; algebras with free behaviou…
The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, …
Answering a question of J. Rosenberg, we construct the first examples of infinite characters on $GL_n(\mathbf{K})$ for a global field $\mathbf{K}$ and $n\geq 2.$ The case $n=2$ is deduced from the following more general result. Let $G$ a non amenable countable subgroup acting on locally finite tree $X$. Assume either that the stabilizer in $G$ of every vertex of $X$ is finite or that the closure of the image of $G$ in ${\rm Aut}(X)$ is not amenable. We show that $G$ has uncountably many infinite dimensional irreducible unitary representations $(π, \mathcal{H})$ of $G$ which are traceable, that…
