scholarly journals Tensor products of classifiable C∗-algebras

2016 ◽  
Vol 09 (03) ◽  
pp. 485-504
Author(s):  
Huaxin Lin ◽  
Wei Sun
Keyword(s):  
General Result ◽  
Simple Formula ◽  
Tensor Products ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Type ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

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 CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

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

2016 ◽  
Vol 27 (03) ◽  
pp. 1650029
Author(s):  
Jiajie Hua
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Crossed Products ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
C Algebras ◽  
Finite Dimensional ◽  
Tracial Rank ◽  
Stably Finite ◽  
Unital Simple

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].

Finite Lattices of Projections in Factors and Approximately Finite C*-Algebras

1992 ◽  
Vol 35 (1) ◽  
pp. 116-125
Author(s):  
S. C. Power
Keyword(s):  
Tensor Product ◽  
General Result ◽  
Finite Lattice ◽  
Tensor Products ◽  
C Algebras ◽  
Finite Lattices

AbstractA unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

Stability of certain relations of three unitaries in C∗-algebras

2021 ◽  
pp. 2150065
Author(s):  
Jiajie Hua
Keyword(s):  
Infinite Order ◽  
Irrational Number ◽  
Greatest Common Divisor ◽  
C Algebras ◽  
Tracial Rank ◽  
Unital Simple

We show that if [Formula: see text] is an irrational number in [Formula: see text], [Formula: see text] and [Formula: see text] are in [Formula: see text] [Formula: see text] is a matrix of infinite order in SL[Formula: see text], either tr[Formula: see text] or tr[Formula: see text] and the greatest common divisor of the entries in [Formula: see text] is one, then for any [Formula: see text] there exists [Formula: see text] satisfying the following: For any unital simple separable [Formula: see text]-algebra [Formula: see text] with tracial rank at most one, any three unitaries [Formula: see text] in [Formula: see text], if [Formula: see text] satisfy certain trace conditions and [Formula: see text] then there exists a triple of unitaries [Formula: see text] in [Formula: see text] such that [Formula: see text] [Formula: see text]

scholarly journals Extensions by Simple C*-Algebras: Quasidiagonal Extensions

2005 ◽  
Vol 57 (2) ◽  
pp. 351-399 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Equivalence Classes ◽  
Essential Extension ◽  
Unitary Equivalence ◽  
C Algebras ◽  
Continuous Scale ◽  
Image Position ◽  
Essential Extensions ◽  
Universal Coefficient Theorem ◽  
Unital Simple

AbstractLet A be an amenable separable C*-algebra and B be a non-unital but σ-unital simple C*- algebra with continuous scale. We show that two essential extensions τ1 and τ2 of A by B are approximately unitarily equivalent if and only ifIf A is assumed to satisfy the Universal Coefficient Theorem, there is a bijection fromapproximate unitary equivalence classes of the abovementioned extensions to KL(A,M(B)/B). Using KL(A,M(B)/B), we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

scholarly journals THE CUNTZ SEMIGROUP OF CONTINUOUS FUNCTIONS INTO CERTAIN SIMPLE C*-ALGEBRAS

2011 ◽  
Vol 22 (08) ◽  
pp. 1051-1087 ◽  
Author(s):  
AARON TIKUISIS
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Von Neumann ◽  
C Algebras ◽  
Cuntz Semigroup ◽  
Unital Simple ◽  
Compact Subsets

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C0(X, A), where A is a unital, simple, [Formula: see text]-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray–von Neumann semigroups of C(K, A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(𝕋, A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

Obstructions to 𝒵-Stability for Unital Simple C*-Algebras

2000 ◽  
Vol 43 (4) ◽  
pp. 418-426 ◽  
Author(s):  
Guihua Gong ◽  
Xinhui Jiang ◽  
Hongbing Su
Keyword(s):  
Necessary Conditions ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Unital Simple

AbstractLet 𝒵 be the unital simple nuclear infinite dimensional C*-algebra which has the same Elliott invariant as ℂ, introduced in [9]. A C*-algebra is called 𝒵-stable if A ≅ A ⊗ 𝒵. In this note we give some necessary conditions for a unital simple C*-algebra to be 𝒵-stable.

A CLASSIFICATION OF NON-SIMPLE C*-ALGEBRAS OF TRACIAL RANK ONE: INDUCTIVE LIMITS OF FINITE DIRECT SUMS OF SIMPLE TAI C*-ALGEBRAS

2011 ◽  
Vol 03 (03) ◽  
pp. 385-404 ◽  
Author(s):  
CHUNLAN JIANG
Keyword(s):  
Compatibility Conditions ◽  
Inductive Limits ◽  
Direct Sums ◽  
State Spaces ◽  
Tracial State ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Unital Simple

In this paper, we will classify the class of C*-algebras which are inductive limits of finite direct sums of unital simple separable nuclear C*-algebras with tracial rank no more than one (or equivalently TAI algebras) with torsion K1-group which satisfy the UCT. The invariant consists of ordered total K-theory and the tracial state spaces of cutdown algebras (with certain compatibility conditions).

THE STABLE AND THE REAL RANK OF ${\mathcal Z}$-ABSORBING C*-ALGEBRAS

2004 ◽  
Vol 15 (10) ◽  
pp. 1065-1084 ◽  
Author(s):  
MIKAEL RØRDAM
Keyword(s):  
Complex Numbers ◽  
Real Rank ◽  
Real Rank Zero ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
C Algebras ◽  
Positive Elements ◽  
Stably Finite ◽  
Unital Simple

Suppose that A is a C*-algebra for which [Formula: see text], where [Formula: see text] is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then [Formula: see text] if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterize when A is of real rank zero.

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