Tracial Rank Research Articles

On the classification of simple amenable $\mathrm{C}^{*}$-algebras with finite decomposition rank, II

We prove that every unital simple separable \mathrm{C}^{*} -algebra A with finite decomposition rank which satisfies the UCT has the property that A\otimes Q has generalized tracial rank at most one, where Q is the universal UHF-algebra. Consequently, A is classifiable in the sense of Elliott.

Symmetries of simple $A\mathbb{T}$-algebras

Let A be a unital simple A\mathbb{T} -algebra of real rank zero. Given an order two automorphism h: K_1(A)\to K_1(A) , we show that there is an order two automorphism \alpha : A\to A such that \alpha_{*0}=\mathrm {id} , \alpha_{*1}=h and the action of \mathbb{Z}_2 generated by \alpha has the tracial Rokhlin property. Consequently, C^*(A,\mathbb{Z}_2,\alpha) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the \mathbb{Z}_2 -action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.

Distance Between Unitary Orbits of Self-Adjoint Elements in C*-Algebras of Tracial Rank One

Distance Between Unitary Orbits of Self-Adjoint Elements in C*-Algebras of Tracial Rank One

On a Tracial Version of Haemers Bound

We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {C}$ </tex-math></inline-formula> ) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes’ embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.

On classification of nonunital amenable simple C∗-algebras, III : The range and the reduction

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features. We also show that every stably projectionless separable simple amenable $C^*$-algebra in the UCT class has rationally generalized tracial rank one.

Unitary groups and augmented Cuntz semigroups of separable simple 𝒵-Stable C∗-algebras

Let [Formula: see text] be a separable simple exact [Formula: see text]-stable [Formula: see text]-algebra. We show that the unitary group of [Formula: see text] has the cancellation property. If [Formula: see text] has continuous scale then the Cuntz semigroup of [Formula: see text] has strict comparison property and a weak cancellation property. Let [Formula: see text] be a 1-dimensional noncommutative CW complex with [Formula: see text] Suppose that [Formula: see text] is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms [Formula: see text] such that [Formula: see text] This result leads to the proof that every separable amenable simple [Formula: see text]-algebra[Formula: see text]in the UCT class has rationally generalized tracial rank at most one.

Tracial topological rank zero and stable rank one for certain tracial approximation C∗-algebras

Tracial topological rank zero and stable rank one for certain tracial approximation C∗-algebras

Regularity Properties of Certain Crossed Product C*-Algebras

We show that the following properties of C* -algebras in a class Ω are inherited by simple unital C*-algebras in class TAΩ: (1) m-comparison of positive elements, (2) strong tracial m-comparison of positive elements, and (3) tracial nuclear dimension at most m. As an application, every unital simple C*-algebra with tracial topological rank at most k has tracial nuclear dimension at most k. Also as an application, let A be an infinite-dimensional simple unital exact C*-algebra such that A has one of the above-listed properties. Suppose that α: G → Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property. Then the crossed product C*-algebra C* (G, A, α) also has the property under consider also.

Simple stably projectionless C*-algebras with generalized tracial rank one

We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. A number of structural questions concerning these simple C*-algebras are studied, pertinent to the classification of separable stably projectionless simple amenable Jiang–Su stable C*-algebras.

Stability of rotation relations in -algebras

AbstractLet $\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate real skew-symmetric $3\times 3$ matrix, where $\theta _{j,k}\in [0,1).$ For any $\varepsilon&gt;0$ , we prove that there exists $\delta&gt;0$ satisfying the following: if $v_1,v_2,v_3$ are three unitaries in any unital simple separable $C^*$ -algebra A with tracial rank at most one, such that $$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|&lt;\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$$ for all $\tau \in T(A)$ and $j,k=1,2,3,$ where $\log _{\theta }$ is a continuous branch of logarithm (see Definition 4.13) for some real number $\theta \in [0, 1)$ , then there exists a triple of unitaries $\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$ such that $$\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|&lt;\varepsilon,\,\,j,k=1,2,3.\end{align*}$$ The same conclusion holds if $\Theta $ is rational or nondegenerate and A is a nuclear purely infinite simple $C^*$ -algebra (where the trace condition is vacuous). If $\Theta $ is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.

Simple generalized inductive limits of C*-algebras

We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C*-algebra. We also show that if a unital C*-algebra can be approximately embedded into some tensorially self absorbing C*-algebra C (e.g., uniformly hyperfinite (UHF)-algebras of infinite type, the Cuntz algebra $${{\mathcal O}_2}$$ ), then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite (resp. properly infinite), the construction is also infinite (resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.

The Weak Tracial Rokhlin Property for Finite Group Actions on Simple C*-Algebras

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if \alpha \colon G \rightarrow \text{Aut}(A) is an action of a finite group G on a simple C*-algebra A with tracial rank zero and \alpha has the weak tracial Rokhlin property, then the crossed product A \rtimes_{\alpha} G and the fixed point algebra A^{\alpha} are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.

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.

Inductive limit of direct sums of simple TAI algebras

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

Crossed products and minimal dynamical systems

Let [Formula: see text] be an infinite compact metric space with finite covering dimension and let [Formula: see text] be two minimal homeomorphisms. We prove that the crossed product [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if [Formula: see text] is an infinite compact metric space and if [Formula: see text] is a minimal homeomorphism such that [Formula: see text] has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple [Formula: see text]-algebras.

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C*-algebras. Let A be a unital tracial rank zero C*-algebra (or tracial rank no more than one C*-algebra). Suppose that α:G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C*-algebra. Then, the crossed product C*-algebra C*(G, A,α) has tracia rank zero (or has tracial rank no more than one). In fact, we get a more general results.

Crossed products by compact group actions with the Rokhlin property

We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C^* -algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by such actions. Similarly, in the presence of the Rokhlin property, if the algebra has any of the following properties, then so do the crossed product and the fixed point algebra: being a Kirchberg algebra, being simple and having tracial rank zero or one, having real rank zero, having stable rank one, absorbing a strongly self-absorbing C^* -algebra, satisfying the Universal Coefficient Theorem (in the simple, nuclear case), and being weakly semiprojective. The ideal structure of crossed products and fixed point algebras by Rokhlin actions is also studied. The methods of this paper unify, under a single conceptual approach, the work of a number of authors, who used rather different techniques. Our methods yield new results even in the well-studied case of finite group actions with the Rokhlin property.

Homomorphisms from AH-algebras

Let [Formula: see text] be a general unital AH-algebra and let [Formula: see text] be a unital simple [Formula: see text]-algebra with tracial rank at most one. Suppose that [Formula: see text] are two unital monomorphisms. We show that [Formula: see text] and [Formula: see text] are approximately unitarily equivalent if and only if [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous affine maps from tracial state space [Formula: see text] of [Formula: see text] to faithful tracial state space [Formula: see text] of [Formula: see text] induced by [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] and [Formula: see text] are induced homomorphisms s from [Formula: see text] into [Formula: see text], where [Formula: see text] is the space of all real affine continuous functions on [Formula: see text] and [Formula: see text] is the closure of the image of [Formula: see text] in the affine space [Formula: see text]. In particular, the above holds for [Formula: see text], the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements [Formula: see text], an affine map [Formula: see text] and a homomorphisms [Formula: see text], there exists a unital monomorphism [Formula: see text] such that [Formula: see text] and [Formula: see text].

Tensor products of classifiable C∗-algebras

Let [Formula: see text] be the class of all unital separable simple [Formula: see text]-algebras [Formula: see text] such that [Formula: see text] has tracial rank no more than one for all UHF-algebra [Formula: see text] of infinite type. It has been shown that all amenable [Formula: see text]-stable [Formula: see text]-algebras in [Formula: see text] which satisfy the Universal Coefficient Theorem can be classified up to isomorphism by the Elliott invariant. In this note, we show that [Formula: see text] if and only if [Formula: see text] has tracial rank no more than one for some unital simple infinite dimensional AF-algebra [Formula: see text] In fact, we show that [Formula: see text] if and only if [Formula: see text] for some unital simple AH-algebra [Formula: see text] We actually prove a more general result. Other results regarding the tensor products of [Formula: see text]-algebras in [Formula: see text] are also obtained.

Crossed products of C∗-algebras C(X,A) and their applications

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].