Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$ and a rotation related map $\bar{R}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero.
Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\cal A}$ of unital separable simple amenable \CA s which is strictly larger than the class of separable \CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${\cal Z}$-stable are isomorphic to ones with no dimension growth.
In this short note, further to Ng's study, we extend Bekka amenability and weak Bekka amenability to general locally compact quantum groups. We generalize some Ng's results to the general case. In particular, we show that, a locally compact quantum group $\mathbb{G}$ is co-amenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka amenable, and $\mathbb{G}$ is amenable if and only if its dual quantum group's fundamental multiplicative unitary $W_{\widehat{\mathbb{G}}}$ is weakly Bekka amenable.
We prove that the crossed product Banach algebra $\ell^1(G,A;α)$ that is associated with a ${\mathrm C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.
In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.
In this note, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $τ\circ ϕ_1=τ\circ ϕ_2$ for all $τ\in T(A)$ and the rotation map ${\tildeη}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero…
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.
In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The Følner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$.
A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the…
We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.
A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting "asymptotic liberation" under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize res…
Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with certain infinite property. We show that two full monomorphisms $h_1, h_2: A\to C$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]$ in $KL(A,C).$ Let $B$ be a non-unital but $σ$-unital \CA for which $M(B)/B$ has the certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $KL(A, M(B)/B).$ The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $A$ satisfies the …
Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C*-algebras. We combine these two seemingly unrelated disciplines to make progress on a classical problem of Weyl. In particular, we show how results from the Elliott classification programme can be used to translate continuous transport of spectral measures into optimal unitary conjugation in C*-algebras. As a consequence, whenever two normal elements of a sufficiently well-behaved C*-algebra share a spectrum amenable to such conti…
We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of emb…
Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…
This paper addresses a conjecture of Kadison and Kastler that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N and, moreover, the implementing unitary can be chosen to be close to the identity operator. This is known to be true for amenable von Neumann algebras and in this paper we describe new classes of non-amenable factors for which the conjecture is valid. These are based on tensor products of the hyperfinite II_1 factor with crossed products of abelian algebras by suitably chosen discrete groups.
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…
Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.
We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.
We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that a…
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…
We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N \times N$ matrices with entries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcal B$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haar distributed quantum unitary random matrix with entries independent from $\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ and $B_N$ are asymptotically free with amalgamation over $\mathcal B$. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyak…
Given a rigid C*-tensor category C with simple unit and a probability measure $μ$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,μ)$. This is a new C*-tensor category P, generally with nonsimple unit, together with a unitary tensor functor $Π: C \to P$. Our main result is that if P has simple unit (which is a condition on some classical random walk), then $Π$ is a universal unitary tensor functor defining the amenable dimension function on C. Corollaries of this theorem unify various results in the literature on amenability of C*-tensor categorie…
Let $A$ be an amenable separable \CA and $B$ be a non-unital but $σ$-unital simple \CA with continuous scale. We show that two essential extensions $τ_1$ and $τ_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [τ_1]=[τ_2] {\rm in} KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B).$ Using $KL(A, M(B)/B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be appro…
Let $C=C(X)$ be the unital $C^*$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. We show that two unital monomorphisms $ϕ, ψ: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$ \lim_{t\to 1} u_t^*ϕ(f)u_t=ψ(f) {\rm for all} \in C(X), $$ if and only if \beq [ϕ]&=&[ψ] {\rm in} KK(C, A), τ\circ ϕ&=&τ\circ ψ{\rm for all} τ\in T(A), and ϕ^†&=&ψ^†, \eneq where $T(A)$ is the simplex of tracial states of $A$ and $ϕ^†, ψ^†:…
Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${KK}_e(C,A)^{++}$ the set of those $κ$ for which $κ(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and $κ([1_C])=[1_A]$. Suppose that $κ\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $ϕ: C\to A$ such that $[ϕ]=κ.$ Suppose that $C$ is a unital AH-algebra and $λ: \mathrm{T}(A)\to \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map for which $τ(κ([p]))=λ(τ)(p)$ for all projections $p$ in all matrix algebras …
