AFD MULTIPLIER ALGEBRAS

2006 ◽  
Vol 17 (09) ◽  
pp. 1091-1102
Author(s):  
P. W. NG
Keyword(s):  
Von Neumann Algebra ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
Unique Trace ◽  
Type Property ◽  
Unital Simple ◽  
Multiplier Algebras

We show that the multiplier algebra of a simple stable nuclear C*-algebra has a property similar to that of an AFD or hyperfinite von Neumann algebra. Specifically, we prove the following: Theorem 0.1. Let [Formula: see text] be a unital simple separable C*-algebra. Let [Formula: see text] be the multiplier algebra of the stabilization of [Formula: see text]. Then [Formula: see text] is nuclear if and only if [Formula: see text] has the AFD-type property. We also study a stronger property called the "strong AFD-type property". We show that if [Formula: see text] is a unital simple real rank zero AT-algebra with unique trace, then the multiplier algebra [Formula: see text] of the stabilization of [Formula: see text] has the strong AFD-type property, and we raise the question of whether this is true more generally.

scholarly journals Strict Comparison of Positive Elements in Multiplier Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 373-407 ◽  
Author(s):  
Victor Kaftal ◽  
Ping Wong Ng ◽  
Shuang Zhang
Keyword(s):  
Positive Element ◽  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Rank Zero ◽  
C Algebra ◽  
Positive Elements ◽  
Rank One ◽  
Multiplier Algebras ◽  
Positive Linear Combination

AbstractMain result: If a C*-algebrais simple,σ-unital, hasfinitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebraalso has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces is replaced byquasicontinuous scale. A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C* -algebra can be approximated by a bi-diagonal series. As an application of strict comparison, ifis a simple separable stable C* -algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

C*-ALGEBRAS WITH REAL RANK ZERO AND THEIR CORONA AND MULTIPLIER ALGEBRAS PART IV

1992 ◽  
Vol 03 (02) ◽  
pp. 309-330 ◽  
Author(s):  
SHUANG ZHANG
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Neumann Theorem ◽  
Von Neumann Theorem ◽  
Deddens Algebras

By proving various equivalent versions of the generalized Weyl-von Neumann theorem, we investigate the structure of projections in the multiplier algebra [Formula: see text] of certain C*-algebra [Formula: see text] with real rank zero. For example, we prove that [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. In case [Formula: see text] is a purely infinite simple C*-algebra, [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. If [Formula: see text] is one of the Cuntz algebras, or one of finite factors or type III factors, then any two projections in [Formula: see text] are simultaneously quasidiagonal. On the other hand, if [Formula: see text] is one of the Bunce-Deddens algebras or one of the irrational rotation algebras of real rank zero, then there exist two projections in [Formula: see text] which are not simultaneously quasidiagonal.

Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero

2001 ◽  
Vol 53 (3) ◽  
pp. 592-630 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Stable Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
Real Rank Zero ◽  
Von Neumann ◽  
Rank Zero ◽  
Rank One ◽  
A Chain ◽  
Multiplier Algebras ◽  
The Ideal

AbstractWe give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of themultiplier algebra , is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of is reflected in the fact that can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

scholarly journals CERTAIN APERIODIC AUTOMORPHISMS OF UNITAL SIMPLE PROJECTIONLESS C*-ALGEBRAS

2009 ◽  
Vol 20 (10) ◽  
pp. 1233-1261 ◽  
Author(s):  
YASUHIKO SATO
Keyword(s):  
Inductive Limit ◽  
Crossed Products ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Cyclic Groups ◽  
Unital Simple

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.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

Tracially Quasidiagonal Extensions

2003 ◽  
Vol 46 (3) ◽  
pp. 388-399 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Tracial Topological Rank ◽  
Unital Simple ◽  
Quasidiagonal Extension

AbstractIt is known that a unital simple C*-algebra A with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital C*-algebras with tracial topological rank zero that have real rank other than zero.Let 0 → J → E → A → 0 be a short exact sequence of C*-algebras. Suppose that J and A have tracial topological rank zero. It is known that E has tracial topological rank zero as a C*-algebra if and only if E is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

Algebraic Inner Derivations on Operator Algebras

1983 ◽  
Vol 35 (4) ◽  
pp. 710-723
Author(s):  
C. Robert Miers ◽  
John Phillips
Keyword(s):  
Positive Integer ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Simple Ring

Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A″). In particular, the strict result holds for any von Neumann algebra, A.

Homomorphisms From C(X) Into C*-Algebras

1997 ◽  
Vol 49 (5) ◽  
pp. 963-1009 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Countable Rank ◽  
Rank One ◽  
Dimensional Range

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

The connection between the universal enveloping C*-algebra and the universal enveloping von Neumann algebra of a JW-algebra

1992 ◽  
Vol 112 (3) ◽  
pp. 575-579 ◽  
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Von Neumann Algebra ◽  
Main Result Theorem ◽  
Von Neumann ◽  
C Algebra ◽  
Result Theorem ◽  
The Relationship

AbstractThis article aims to study the relationship between the universal enveloping C*-algebra C*(M) and the universal enveloping von Neumann algebra W*(M), when M is a JW-algebra. In our main result (Theorem 2·7) we show that C*(M) can be realized as the C*-subalgebra of W*(M) generated by M.

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