Unique Tracial State Research Articles

Compact quantum metric spaces from free graph algebras

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel.We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.

Extensions of C*-Algebras by a Small Ideal

Abstract We classify all essential extensions of the form $$ \begin{align*} &0 \rightarrow {\mathcal{W}} \rightarrow {D} \rightarrow A \rightarrow 0,\end{align*}$$where ${\mathcal {W}}$ is the unique separable simple C*-algebra with a unique tracial state, which is $KK$-contractible and has finite nuclear dimension, and $A$ is a separable amenable ${\mathcal {W}}$-embeddable C*-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable $C^*$-algebras, which have only one proper closed ideal ${\mathcal {W}}.$

Nonamenable simple -algebras with tracial approximation

Abstract We construct two types of unital separable simple $C^*$ -algebras: $A_z^{C_1}$ and $A_z^{C_2}$ , one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely, $A_z^{C_i}$ has a unique tracial state, $$ \begin{align*} \left(K_0\left(A_z^{C_i}\right), K_0\left(A_z^{C_i}\right)_+, \left[1_{A_z^{C_i}} \right]\right)=(\mathbb{Z}, \mathbb{Z}_+,1), \end{align*} $$ and $K_{1}\left (A_z^{C_i}\right )=\{0\}$ ( $i=1,2$ ). We show that $A_z^{C_i}$ ( $i=1,2$ ) is essentially tracially in the class of separable ${\mathscr Z}$ -stable $C^*$ -algebras of nuclear dimension $1$ . $A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no $2$ -quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) $C^*$ -algebras which are essentially tracially in the class of simple separable nuclear ${\mathscr Z}$ -stable $C^*$ -algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

On Specht's Theorem in UHF C⁎-algebras

Specht's Theorem states that two matrices A and B in Mn(C) are unitarily equivalent if and only if tr(w(A,A⁎))=tr(w(B,B⁎)) for all words w(x,y) in two non-commuting variables x and y. In this article we examine to what extent this trace condition characterises approximate unitary equivalence in uniformly hyperfinite (UHF) C⁎-algebras. In particular, we show that given two elements a,b of the universal UHF-algebra Q which generate C⁎-algebras satisfying the UCT, they are approximately unitarily equivalent if and only if τ(w(a,a⁎))=τ(w(b,b⁎)) for all words w(x,y) in two non-commuting variables (where τ denotes the unique tracial state on Q), while there exist two elements a,b in the UHF-algebra M2∞ which fail to be approximately unitarily equivalent despite the fact that they satisfy the trace condition. We also examine a consequence of these results for ampliations of matrices.

On classification of non-unital amenable simple C*-algebras, II

We present a classification theorem for separable amenable simple stably projectionless C∗-algebras with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C∗-algebras in the class is denoted by Z0 which has a unique tracial state, K0(Z0)=Z and K1(Z0)={0}. Let A and B be two separable amenable simple C∗-algebras satisfying the UCT. We show that A⊗Z0≅B⊗Z0 if and only if Ell(A⊗Z0)=Ell(B⊗Z0). A class of simple separable C∗-algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C∗-algebras of the form A⊗Z0, where A is any finite separable simple amenable C∗-algebras.

State Convertibility in the von Neumann Algebra Framework

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen's theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of $II_1$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general $II_1$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

AbstractLet $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $\alpha $ be a strongly outer action of a finite group $G$ on $A$. In this paper, we show that $\alpha \otimes \textrm{id}$ on $A\otimes \mathcal{W}$ has the Rohlin property where $\mathcal{W}$ is the Razak–Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.

Self-Similar Inverse Semigroups from Wieler Solenoids

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C ∗ -algebras of the groupoids of germs have a unique tracial state.

A von Neumann algebraic approach to self-similar group actions

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

Convex hulls of unitary orbits of normal elements in C⁎-algebras with tracial rank zero

Let A be a unital separable simple C⁎-algebra with tracial rank zero and let x,y∈A be two normal elements. We show that x is in the closure of the convex hull of the unitary orbit of y if and only if there exists a sequence of unital completely positive linear maps φn from A to A such that the sequence φn(y) converges to x in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of the convex hull of unitary orbit of y is also given. In the case that A has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von Neumann algebras are proved to hold in the C⁎-algebraic setting.

Simple nuclear C*-algebras not equivariantly isomorphic to their opposites

We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.

On reduced twisted group C*-algebras that are simple and/or have a unique trace

We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a combinatorial property, that we call the relative Kleppner condition, which ensures that a quotient group G/H acts by freely acting automorphisms on the twisted group von Neumann algebra associated to a normal subgroup H. We apply our results to different types of groups, e.g. wreath products and Baumslag-Solitar groups.

Regularity of Villadsen algebras and characters on their central sequence algebras

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

The Dixmier property and tracial states for C⁎-algebras

It is shown that a unital C⁎-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup–Zsidó theorem for simple C⁎-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C⁎-algebras of Powers groups, but not by all C⁎-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C⁎-algebra with unique tracial state to have this uniform property. We give further examples of C⁎-algebras with the uniform Dixmier property, namely all C⁎-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C⁎-algebra, by a formula involving tracial data and algebraic numerical ranges.

C*-simplicity and the unique trace property for discrete groups

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

Reduced Twisted Crossed Products over C*-Simple Groups

We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and the reduced crossed product has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We also show that the reduced crossed product satisfies an averaging property analogous to Powers' averaging property.

C⁎-simplicity of free products with amalgamation and radical classes of groups

We give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group C⁎-algebra, and provide a concrete example of an amalgam with trivial kernel, such that its reduced group C⁎-algebra has a unique tracial state, but is not simple.Moreover, we show that there is a radical class of groups for which the reduced group C⁎-algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.

Factoriality and Type Classification of k-Graph von Neumann Algebras

AbstractLet be a single vertex k-graph and let be the von Neumann algebra induced from the Gelfand–Naimark–Segal (GNS) representation of a distinguished state ω of its k-graph C*-algebra . In this paper we prove the factoriality of , and furthermore determine its type when either has the little pullback property, or the intrinsic group of has rank 0. The key step to achieving this is to show that the fixed-point algebra of the modular action corresponding to ω has a unique tracial state.

Unique tracial state on the labeled graph C∗-algebra associated to the Thue–Morse sequence

We give a concrete formula for the unique faithful trace on the finite simple non-AF labeled graph [Formula: see text]-algebra [Formula: see text] associated to the Thue–Morse sequence [Formula: see text]. Our result provides an alternative proof of the existence of a labeled graph [Formula: see text]-algebra that is not Morita equivalent to any graph [Formula: see text]-algebras. Furthermore, we compute the [Formula: see text]-groups of [Formula: see text] using the path structure of the Thue–Morse sequence.

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C$^*$-algebras.