Tracial State Research Articles

GROUP ACTION PRESERVING THE HAAGERUP PROPERTY OF -ALGEBRAS

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

On twisted group -algebras associated with FC-hypercentral groups and other related groups

We show that the twisted group $C^{\ast }$-algebra associated with a discrete FC-hypercentral group is simple (respectively, has a unique tracial state) if and only if Kleppner’s condition is satisfied. This generalizes a result of Packer for countable nilpotent groups. We also consider a larger class of groups, for which we can show that the corresponding reduced twisted group $C^{\ast }$-algebras have a unique tracial state if and only if Kleppner’s condition holds.

A computational approach to the Thompson group F

Let F denote the Thompson group with standard generators A = x0, B = x1. It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to [Formula: see text] and to [Formula: see text] where in both cases the norm of an element in the group ring ℂF is computed in B(ℓ2(F)) via the regular representation of F. By extensive numerical computations, we obtain precise lower bounds for the norms in (i) and (ii), as well as good estimates of the spectral distributions of (I+A+B)*(I+A+B) and of A+A-1+B+B-1 with respect to the tracial state τ on the group von Neumann Algebra L(F). Our computational results suggest, that [Formula: see text] It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F.

Determinant rank ofC∗-algebras

Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries of $A$ which are path connected to the identity. Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}\colon U_0(A)/CU(A)\rightarrow U_0(\mathrm M_n(A))/CU(\mathrm M_n(A))$ be the \hm\, defined by sending $u$ to ${\rm diag}(u,1_n).$ We study the problem when the map $i_A^{(1,n)}$ is an isomorphism for all $n.$ We show that it is always surjective and is injective when $A$ has stable rank one. It is also injective when $A$ is a unital $C^*$-algebra of real rank zero, or $A$ has no tracial state. We prove that the map is an isomorphism when $A$ is the Villadsen's simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadar's unital projectionless separable simple $C^*$-algebras. Let $A=\mathrm M_n(C(X)),$ where $X$ is any compact metric space. It is noted that the map $i_A^{(1, n)}$ is an isomorphism for all $n.$ As a consequence, the map $i_A^{(1, n)}$ is always an isomorphism for any unital $C^*$-algebra $A$ that is an inductive limit of finite direct sum of $C^*$-algebras of the form $\mathrm M_n(C(X))$ as above. Nevertheless we show that there are unital $C^*$-algebras $A$ such that $i_A^{(1,2)}$ is not an isomorphism.

Rotation Algebras and the Exel Trace Formula

AbstractWe show that ifuandvare any two unitaries in a unitalC*–algebra such that ∥uv−vu∥ < 2 anduvu*v* commutes withuandv, then theC*–subalgebraAu,vgenerated by u and v is isomorphic to a quotient of some rotation algebraAθ, provided thatAu;vhas a unique tracial state. We also show that the Exel trace formula holds in any unitalC*–algebra. Let θ ∊ (−1/2, 1/2) be a real number. For any ∊ > 0; we prove that there exists ζ > 0 satisfying the following: if u and v are two unitaries in any unital simpleC*–algebra A with tracial rank zero such thatfor all tracial states τof A; then there exists a pair of unitariesandin A such that

Property T and Amenable Transformation Group C*-algebras

AbstractIt is well known that a discrete group that is both amenable and has Kazhdan’s Property T must be finite. In this note we generalize this statement to the case of transformation groups. We show that if G is a discrete amenable group acting on a compact Hausdorff space X, then the transformation group C*-algebra C*(X; G) has Property T if and only if both X and G are finite. Our approach does not rely on the use of tracial states on C*(X; G).

Nuclear dimension and sums of commutators

The problem of expressing a selfadjoint element that is zero on every bounded trace as a finite sum (or a limit of sums) of commutators is investigated in the setting of C*- algebras of finite nuclear dimension. Upper bounds - in terms of the nuclear dimension of the C*-algebra - are given for the number of commutators needed in these sums. An example is given of a simple, nuclear C*-algebra (of infinite nuclear dimension) with a unique tracial state and with elements that vanish on all bounded traces and yet are badly approximated by finite sums of commutators. Finally, the same problem is investigated on (possibly non-nuclear) simple unital C*-algebras assuming suitable regularity properties in their Cuntz semigroups.

On the properties of some sets of von Neumann algebras under perturbation

Let ℒ be a type II1 factor with separable predual and τ be a normal faithful tracial state of ℒ. We first show that the set of subfactors of ℒ with property Γ, the set of type II1 subfactors of ℒ with similarity property and the set of all McDuff subfactors of ℒ are open and closed in the Hausdorff metric d 2 induced by the trace norm; then we show that the set of all hyperfinite von Neumann subalgebras of ℒ is closed in d 2. We also consider the connection of perturbation of operator algebras under d 2 with the fundamental group and the generator problem of type II1 factors. When Open image in new window is a finite von Neumann algebra with a normal faithful trace, the set of all von Neumann subalgebras Open image in new window of Open image in new window such that Open image in new window is rigid is closed in the Hausdorff metric d 2.

Ƶ -Stability of Crossed Products by Strongly Outer Actions II

We consider a crossed product of a unital simple separable nuclear stably finite $\cal Z$-stable $C^*$-algebra $A$ by a strongly outer cocycle action of a discrete countable amenable group $\Gamma$. Under the assumption that $A$ has finitely many extremal tracial states and $\Gamma$ is elementary amenable, we show that the twisted crossed product $C^*$-algebra is $\cal Z$-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on $\cal Z$ are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite $C^*$-algebras.

Decomposition rank of UHF-absorbing C∗-algebras

Let A be a unital separable simple C∗-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that A is nuclear, quasidiagonal, and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers–Sakai conjecture.

Formula omitted]-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β:Sn→Sn, for n=2k+1, k≥1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (Sn,β) there is a Cantor minimal system (X,α) such that the corresponding product system (X×Sn,α×β) is minimal and the resulting crossed product C⁎-algebra C(X×Sn)⋊α×βZ is tracially approximately an interval algebra (TAI). This entails classification for such C⁎-algebras. Moreover, the minimal Cantor system (X,α) is such that each tracial state on C(X×Sn)⋊α×βZ induces the same state on the K0-group and such that the embedding of C(Sn)⋊βZ into C(X×Sn)⋊α×βZ preserves the tracial state space. This implies C(Sn)⋊βZ is TAI after tensoring with the universal UHF algebra, which in turn shows that the C⁎-algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.

Strictly Amenable Representations of Reduced GroupC* -Algebras

We show that a locally compact group $G$ is amenable if and only if its reduced group $C^* $ -algebra $C^*_r(G)$ is nuclear and has a tracial state, which is also equivalent to $C^*_r(G)$ having a special kind of $^* $ -representation that we propose to call “strictly amenable representation”. Consequently, if $G$ is separable and connected, then $G$ is amenable if and only if $C^*_r(G)$ has a tracial state.

A non-commutative version of Nikishin’s theorem

Let τ be a tracial normal state on a von Neumann algebra, L1(τ) be the space of integrable self-adjoint operators, and S be the space of self-adjoint measurable operators. We prove that every positive linear operator from an ordered Banach space to S can be factorized through L1(τ).

Tracial state space with non-compact extreme boundary

Let A be a unital simple separable C*-algebra. If A is nuclear and infinite-dimensional, it is known that strict comparison of positive elements is equivalent to Z-stability if the extreme boundary of its tracial state space is compact and of finite covering dimension. Here we provide the first proof of this result in the case of certain non-compact extreme boundaries. Additionally, if A has strict comparison of positive elements, it is known that the Cuntz semigroup of this C*-algebra is recovered functorially from the Murray–von Neumann semigroup and the tracial state space whenever the extreme boundary of the tracial state space is compact and of finite covering dimension. We extend this result to the case of a countable extreme boundary with finitely many cluster points.

UHF-slicing and classification of nuclear C*-algebras

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

Free group C*-algebras associated with ℓp

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.

Z-Stability and Finite-Dimensional Tracial Boundaries

We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.

The simplex of tracial quantum symmetric states

We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.

A non-commutative Wiener–Wintner theorem

For a von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state $\tau$ and a positive ergodic homomorpsism $\alpha :\mathcal{L}^{1}(\mathcal{M},\tau)\to\mathcal{L}^{1}(\mathcal{M},\tau)$ such that $\tau\circ\alpha =\tau$ and $\alpha $ does not increase the norm in $\mathcal{M}$, we establish a non-commutative counterpart of the classical Wiener–Wintner ergodic theorem.