Reduction of Exponential Rank in Direct Limits of C*-Algebras

1994 ◽  
Vol 46 (4) ◽  
pp. 818-853 ◽  
Author(s):  
N. Christopher Phillips
Keyword(s):  
Upper Bound ◽  
Unitary Group ◽  
Direct Limit ◽  
Real Rank ◽  
Direct Sums ◽  
The Real ◽  
Identity Component ◽  
C Algebras ◽  
Direct Limits

AbstractWe prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.

The classification of certain ASH C*-algebras of real rank zero

2020 ◽  
pp. 1-20
Author(s):  
Qingnan An ◽  
George A. Elliott ◽  
Zhiqiang Li ◽  
Zhichao Liu
Keyword(s):  
Real Rank ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Direct Sums ◽  
Rank Zero ◽  
The Real ◽  
C Algebras ◽  
Generalized Dimension ◽  
Theoretic Classification

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

C*-ALGEBRAS OF TRIVIAL K-THEORY AND SEMILATTICES

1999 ◽  
Vol 10 (01) ◽  
pp. 93-128 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Direct Limit ◽  
Tensor Products ◽  
Equivalence Classes ◽  
Direct Sums ◽  
C Algebras ◽  
Direct Limits ◽  
One To One ◽  
K Theory ◽  
Essential Extensions ◽  
Distributive Semilattices

We give a class of nuclear C*-algebras which contains [Formula: see text] and is closed under stable isomorphism, ideals, quotients, hereditary subalgebras, tensor products, direct sums, direct limits as well as extensions. We show that this class of C*-algebras is classified by their equivalence classes of projections and there is a one to one correspondence between (unital) C*-algebras in the class and countable distributive semilattices (with largest elements). One of the main results is that essential extensions of a C*-algebras which is a direct limit of finite direct sums of corners of [Formula: see text] by the same type of C*-algebras are still direct limits of finite direct sums of corners of [Formula: see text].

scholarly journals Real rank estimate by hereditary $C^*$-subalgebras by projections

2007 ◽  
Vol 100 (2) ◽  
pp. 361
Author(s):  
Takahiro Sudo
Keyword(s):  
Real Rank ◽  
The Real ◽  
C Algebras

In this paper we estimate the real rank of $C^*$-algebras by that of their hereditary $C^*$-subalgebras by projections.

Stable Rank of Some Full Group C*-Algebras of Groups Obtained by the Free Product

1997 ◽  
Vol 08 (03) ◽  
pp. 375-382 ◽  
Author(s):  
Masaru Nagisa
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Groups ◽  
Stable Rank ◽  
Real Rank ◽  
Full Group ◽  
The Real ◽  
C Algebras

We compute the real rank and the stable rank of full group C*-algebras. Main result is (i) rr (C*(Fn)) = ∞, (ii) sr (C*(G1 * G2)) = ∞(|G1| ≥ 2, |G2| ≥ 2 and |G1| + |G2| ≥ 5), (iii) sr (C*(G1 * G2)) = 1(|G1| = |G2| = 2), where Fn is the free group with n generators, G1 and G2 are finite groups and |G| means the order of the group G.

GENERALIZED WEYL-VON NEUMANN THEOREMS

1991 ◽  
Vol 02 (06) ◽  
pp. 725-739 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Type Theorem ◽  
Building Blocks ◽  
Real Rank ◽  
Inductive Limits ◽  
Multiplier Algebra ◽  
Direct Sums ◽  
Von Neumann ◽  
C Algebras ◽  
Rank One ◽  
Neumann Type

The conjecture that the real rank of multiplier algebra of every AF-algebra is zero was formally made by L. G. Brown and G. K. Pedersen in [4]. The main purpose of this note is to prove the conjecture. However, we also show that this Weyl-von Neumann type theorem holds for C*-algebras with stable (FU) and stable rank one. In particular, this Weyl-von Neumann type theorem holds for C*-algebras of inductive limits of finite direct sums of basic building blocks with real rank zero and trivial K1-groups considered by George A. Elliott recently.

scholarly journals The real rank of inductive limit $C^*$-algebras.

1991 ◽  
Vol 69 ◽  
pp. 211 ◽  
Author(s):  
B. Blackadar ◽  
M. Dadarlat ◽  
M. Rordam
Keyword(s):  
Inductive Limit ◽  
Real Rank ◽  
The Real ◽  
C Algebras

EXTENSIONS BY C*-ALGEBRAS WITH REAL RANK ZERO

1993 ◽  
Vol 04 (02) ◽  
pp. 231-252 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Unit Circle ◽  
Compact Subset ◽  
Real Line ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
The Real ◽  
C Algebra ◽  
C Algebras ◽  
Essential Extensions

We show that all trivial (unital and essential) extensions of C (X) by a σ-unital purely infinite simple C*-algebra A with K1(A) = 0 are unitarily equivalent, provided that X is homeomorphic to a compact subset of the real line or the unit circle. Therefore all (unital and essential) extensions of such can be completely determined by Ext(B, A). An invariant is introduced to classify all such trivial (unital and essential) extensions of C (X) by a σ-unital C*-algebra A with the properties that RR (M (A)) = 0 and C (A) is simple.

Exponential length and exponential rank in C*-algebras

1992 ◽  
Vol 121 (1-2) ◽  
pp. 55-71 ◽  
Author(s):  
J. R. Ringrose
Keyword(s):  
Upper Bound ◽  
Operator Algebra ◽  
Unitary Group ◽  
Continuous Functions ◽  
Connected Component ◽  
Finite Product ◽  
C Algebras ◽  
General Element

SynopsisIn an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This is used to estimate the exponential rank of certain algebras of operator-valued continuous functions.

scholarly journals Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

2017 ◽  
Vol 69 (5) ◽  
pp. 1109-1142 ◽  
Author(s):  
P.W. Ng ◽  
P. Skoufranis
Keyword(s):  
Upper Bound ◽  
Convex Combination ◽  
Convex Hulls ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras ◽  
Rank One ◽  
Adjoint Operators ◽  
Tracial States

AbstractIn this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

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