Unique Tracial State Research Articles

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.

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

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.

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.

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.

Picard groups of certain stably projectionless C*-algebras

We compute Picard groups of several nuclear and non-nuclear simple stably projectionless C*-algebras. In particular, the Picard group of Razak-Jacelon algebra W_2 is isomorphic to a semidirect product of Out(W_2) with R_+^\times. Moreover, for any separable simple nuclear stably projectionless C*-algebra with a finite dimensional lattice of densely defined lower semicontinuous traces, we show that Z-stability and strict comparison are equivalent. (This is essentially based on the result of Matui and Sato, and Kirchberg's central sequence algebras.) This shows if A is a separable simple nuclear stably projectionless C*-algebra with a unique tracial state (and no unbounded trace) and has strict comparison, the following sequence is exact: [{CD} {1} @>>> \mathrm{Out}(A) @>>> \mathrm{Pic}(A) @>>> \mathcal{F}(A) @>>> {1} {CD}] where $\mathcal{F}(A)$ is the fundamental group of A.

COMMUTATORS IN THE JIANG–SU ALGEBRA

Let [Formula: see text] be the Jiang–Su algebra and let τ be its unique tracial state. We prove that for all [Formula: see text], the following statements are equivalent: (1) a is a finite sum of commutators. (2) a is a sum of five commutators. (3) τ(a) = 0.

A simple, monotracial, stably projectionless C *‐algebra

We construct a simple, nuclear, stably projectionless C*-algebra W which has trivial K-theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C*-algebras as an analogue of the Cuntz algebra 𝒪2. In this context, we show that every non-degenerate endomorphism of W is approximately inner and we construct a trace-preserving embedding of W into the central sequences algebra M(W)∞∩W′. We conjecture that W⊗ W ≅ W and we note some implications of this, for example, that W would be absorbed tensorially by a certain class of nuclear, stably projectionless C*-algebras. Finally, we explain why W may play some role in the classification of such algebras.

A simple separable exact C*-algebra not anti-isomorphic to itself

We give an example of an exact, stably finite, simple, separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3∞ UHF algebra. We can also explicitly compute the K-theory of D, namely $${K_0 (D) \cong {\mathbb{Z}} [ \tfrac{1}{3}]}$$ with the standard order, and K 1 (D) = 0, as well as the Cuntz semigroup of D, namely $${W (D) \cong {\mathbb{Z}} [ \tfrac{1}{3} ]_{+} \sqcup (0, \infty).}$$

Extended rotation algebras: Adjoining spectral projections to rotation algebras

Denote by Aθ the rotation algebra corresponding to the rotation 2πθ. The C*-algebra 𝔹θ generated by Aθ together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra 𝔹θ is shown to have a unique tracial state and to be nuclear provided that θ is irrational. Moreover, we study the ideal structure of the C*-algebra 𝔹θ. In particular, it is shown that 𝔹θ is simple if neither the commutative sub-C*-algebra generated by the spectral projections of u in question (assumed to be a set invariant under Ad v) nor the corresponding commutative sub-C*-algebra associated to v contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra 𝔹θ generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to θ. (The spectral projections for each half-open interval from nθ to (n + 1)θ are then included for each integer n.) Using simplicity of 𝔹θ for θ irrational, the natural field of C*-algebras on the unit circle with fibres 𝔹θ is shown to be continuous at irrational points. This field is lower semicontinuous on the whole circle. Much more useful is an upper semicontinuous field which is obtained by desingularizing this field at rational points on the circle. The fibres of the desingularized field at rational points are certain (computable) type I C*-algebras. Using this new field, we are able to show that 𝔹θ is an AF algebra with K0(𝔹θ) ≅ ℤ + θℤ for generic θ, in the sense of Baire category, with the class of the unit being 1 ∈ ℤ.

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.

CERTAIN APERIODIC AUTOMORPHISMS OF UNITAL SIMPLE PROJECTIONLESS C*-ALGEBRAS

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K1(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut (A)/ WInn (A) are conjugate, where WInn (A) means the subgroup of Aut (A) consisting of automorphisms which are inner in the tracial representation.In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C ∗ -algebra with TR( A)⩽1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then TR(A⋊ α Z)⩽1 . We also show that whenever A has a unique tracial state and α m is uniformly outer for each m(≠0) and α r is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C ∗ -algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple A T -algebra of real rank zero and α∈Aut( A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product A⋊ α Z is again an A T -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C ∗ -algebras with tracial rank one (and zero) from crossed products.

A simple separable C*-algebra not isomorphic to its opposite algebra

We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial K 1 K_1 , and its K 0 K_0 -group is order isomorphic to a countable subgroup of R {\mathbf R} .

Non-commutative shifts and crossed products

When A is a unital simple AF C ∗ -algebra and has a unique tracial state, it is shown that the crossed product of the two-sided infinite tensor product ⊗ Z A by the shift is a tracially AF C ∗ -algebra. A similar result is given to the crossed product of a certain non-unital two-sided infinite tensor product by the shift. Applying a far-reaching classification result of such C ∗ -algebras by H. Lin, we obtain an example of a one-parameter automorphism group on some AF C ∗ -algebra which is not approximately inner, a counter-example to the AF version of the so-called Powers–Sakai conjecture [23].

Type III representations and automorphisms of some separable nuclear [formula omitted]-algebras

Powers proved decades ago that if two cyclic representations π 1 and π 2 of a UHF algebra A satisfy that π 1(A)″≅π 2(A)″= M , there is an automorphism α of A such that π 1 α and π 2 are quasi-equivalent. This was recently extended to the class of simple separable C ∗ -algebras when M is of type I. In this paper we extend this result to some class of simple separable nuclear C ∗ -algebras when M is of type III. In particular, we show that the above result obtained by Powers for the UHF algebras also holds for the class of purely infinite simple separable C ∗ -algebras classified by Kirchberg and Phillips and for the class of approximately homogeneous simple separable unital C ∗ -algebra with unique tracial state.

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d X : K 0(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K 0(A) = ℚ⊕ kerd x and K 1(A) = K 1(C(X)), then A is isomorphic to an inductive limit of M n! (C(X)).

On a simple unital projectionless C *-algebra

We construct a unital separable C *-algebra Z as an analog of the hyperfinite type II 1 factor. Besides being nuclear, simple, projectionless, and infinite-dimensional, Z has a unique tracial state, and is KK-equivalent to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], the algebra of complex numbers. It is shown that unital endomorphisms on Z are approximately inner, and that Z is isomorphic to the infinite tensor product of its replicas. It is also shown that A ≅ Z ⊗ A for certain interesting classes of unital simple nuclear C *-algebras A of real rank zero.

C*-Algebras and Numerical Linear Algebra

Given a self-adjoint operator A on a Hilbert space, suppose that one wishes to compute the spectrum of A numerically. In practice, these problems often arise in such a way that the matrix of A relative to a natural basis is "sparse." For example, discretized second-order differential operators can be represented by doubly infinite tridiagonal matrices. In these cases it is easy and natural to compute the eigenvalues of large n × n submatrices of the infinite operator matrix, and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of A. Numerical analysts call this the Galerkin method. While this hope is often realized in practice it often fails as well, and it can fail in spectacular ways. The sequence of eigenvalue distributions may not converge as n → ∞, or they may converge to something that has little to do with the original operator A. At another level, even the meaning of "convergence" has not been made precise in general. In this paper we determine the proper general setting in which one can expect convergence, and we describe the asymptotic behavior of the n × n eigenvalue distributions in all but the most pathological cases. Under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of A is recovered from the sequence of eigenvalues of the n × n compressions. In broader terms, our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms of C*-algebras. Indeed, it is only when the single operator A is viewed as an element of an appropriate C*-algebra A that one can see the precise nature of the limit of the n × n eigenvalue distributions; the limit is associated with a tracial state on A . Normally, A is highly noncommutative, and in our main applications it is a simple C*-algebra having a unique tracial state. We obtain precise asymptotic results for operators which represent discretized Hamiltonians of one-dimensional quantum systems with arbitrary continuous potentials.

A Furstenberg Transformation of the 2-Torus Without Quasi-Discrete Spectrum

AbstractR. Ji asked whether or not a Furstenberg transformation of the 2-torus of the form (x,y) → (e2πiθx, f(x)y), where θ is irrational and f : T —> T is continuous with non-zero degree k, is topologically conjugate to the Anzai transformation (x, y) → (e2πiθx, xk y) or its inverse. In this paper this question is settled in the negative. Further, some sufficient conditions are given under which the crossed product C*-algebra associated with a Furstenberg transformation of the 2-torus has a unique tracial state.