Amenable Trace Research Articles

Some results on tracial stability and graph products

We establish the tracial stability of a certain of graph products of C*-algebras. This result involves the development of the pincushion class of finite graphs. We then apply this result in two ways. The first application yields a selective version of Lin's Theorem for almost commuting operators. The second application addresses some approximation properties of right-angled Artin groups. In particular, we show that the full C*-algebra of any right-angled Artin group is quasidiagonal and thus has a non-trivial amenable trace, and then we apply tracial stability to show when these amenable traces are in fact locally finite dimensional.

Subalgebras of simple AF-algebras

It is shown that if $A$ is a separable, exact $C^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of $C^*$-sub­algebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group $G$ acting on a second countable, locally compact, Hausdorff space $X$, $C_0(X) \rtimes_r G$ embeds into a simple, unital AF-algebra if, and only if, $X$ admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group $G$, the reduced group $C^*$-algebra $C^*_r(G)$ admits a trace-preserving embedding into the universal UHF-algebra.

Perfect Strategies for Non-Local Games

We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are {\it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {\it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and {\it unique} games. We associate a C*-algebra $C^*(\mathcal{G})$ to any imitation game $\mathcal{G}$, and show that the existence of perfect quantum commuting (resp.\ quantum, local) strategies of $\mathcal{G}$ can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call {\it mirror games,} and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.

Amenability and paradoxicality in semigroups and C⁎-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence.In the context of inverse semigroups S we give a characterization of invariant measures on S (in the sense of Day) in terms of two notions: domain measurability and localization. Given a unital representation of S in terms of partial bijections on some set X we define a natural generalization of the uniform Roe algebra of a group, which we denote by RX. We show that the following notions are then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies the domain Følner condition; (4) there is an algebraically amenable dense*-subalgebra of RX; (5) RX has an amenable trace; (6) RX is not properly infinite and (7) [0]≠[1] in the K0-group of RX. We also show that any tracial state on RX is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of X. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of Cr⁎(X) implies the amenability of X. The reverse implication (which is a natural generalization of Rosenberg's conjecture to this context) is false.

The maximal injective crossed product

A crossed product functor is said to beinjectiveif it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group$G$admits a maximal injective crossed product$A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of$G$-injective$C^{\ast }$-algebras; this is a sort of ‘dual’ result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that$\rtimes _{\text{inj}}$has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.

Uniformly recurrent subgroups and simple C⁎-algebras

We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss [18]. Answering their query we show that any URS Z of a finitely generated group is the stability system of a minimal Z-proper action. We also show that for any sofic URS Z there is a Z-proper action admitting an invariant measure. We prove that for a URS Z all Z-proper actions admits an invariant measure if and only if Z is coamenable. In the second part of the paper we study the separable C⁎-algebras associated to URS's. We prove that if a URS is generic then its C⁎-algebra is simple. We give various examples of generic URS's with exact and nuclear C⁎-algebras and an example of a URS Z for which the associated simple C⁎-algebra is not exact and not even locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.

Quasidiagonal traces on exact C⁎-algebras

Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear C⁎-algebra in the UCT class is quasidiagonal. Building on their work, we generalise the result, and show that any faithful, amenable trace on a separable, exact C⁎-algebra in the UCT class is quasidiagonal. We also prove that any amenable trace on a separable, exact, quasidiagonal C⁎-algebra in the UCT class is quasidiagonal.

On exotic group C*-algebras

Let Γ be a discrete group. A C*-algebra A is an exotic C*-algebra (associated to Γ) if there exist proper surjective C*-quotients C⁎(Γ)→A→Cr⁎(Γ) which compose to the canonical quotient C⁎(Γ)→Cr⁎(Γ). In this paper, we show that a large class of exotic C*-algebras has poor local properties. More precisely, we demonstrate the failure of local reflexivity, exactness, and local lifting property. Additionally, A does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when A is from the class of exotic C*-algebras defined by Brown and Guentner (see [8]). In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups.

Amenable traces and Følner [formula omitted]-algebras

In the present article we review an approximation procedure for amenable traces on unital and separable C∗-algebras acting on a Hilbert space in terms of Følner sequences of non-zero finite rank projections. We apply this method to improve spectral approximation results due to Arveson and Bédos. We also present an abstract characterization in terms of unital completely positive maps of unital separable C∗-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. This is analogous to Voiculescu’s abstract characterization of quasidiagonal C∗-algebras. We define Følner C∗-algebras as those unital separable C∗-algebras that satisfy these equivalent conditions. Finally we also mention some permanence properties related to these algebras.

A generalization of expander graphs and local reflexivity of uniform Roe algebras

We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated with generalized box spaces.

Fixed points of normal completely positive maps on [formula omitted

Given a sequence of bounded operators aj on a Hilbert space H with ∑j=1∞aj⁎aj=1=∑j=1∞ajaj⁎, we study the map Ψ defined on B(H) by Ψ(x)=∑j=1∞aj⁎xaj and its restriction Φ to the Hilbert–Schmidt class C2(H). In the case when the sum ∑j=1∞aj⁎aj is norm-convergent we show in particular that the operator Φ−1 is not invertible if and only if the C⁎-algebra A generated by {aj}j=1∞ has an amenable trace. This is used to show that Ψ may have fixed points in B(H) which are not in the commutant A′ of A even in the case when the weak* closure of A is injective. However, if A is abelian, then all fixed points of Ψ are in A′ even if the operators aj are not positive.