scholarly journals On the topological stable rank of certain transformation group C*-algebras

1990 ◽  
Vol 10 (1) ◽  
pp. 197-207 ◽  
Author(s):  
Ian F. Putnam
Keyword(s):  
Transformation Group ◽  
Metrizable Space ◽  
Stable Rank ◽  
Residual Finiteness ◽  
Dynamical Condition ◽  
C Algebras ◽  
Compact Metrizable Space ◽  
Topological Stable Rank ◽  
Necessary And Sufficient ◽  
Totally Disconnected

AbstractWe consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.

scholarly journals Embedding some transformation group C*-algebras into AF-algebras

1983 ◽  
Vol 3 (4) ◽  
pp. 613-626 ◽  
Author(s):  
Mihai V. Pimsner
Keyword(s):  
Transformation Group ◽  
Metrizable Space ◽  
C Algebras ◽  
Compact Metrizable Space ◽  
Af Algebras

AbstractFor a homeomorphism of a compact metrizable space X, we show that the property that every point of X is pseudo-non-wandering (see definition 2) is equivalent to the possibility of embedding the corresponding transformation group C*-algebra into an AF-algebra.

scholarly journals STABLE RANK FOR INCLUSIONS OF C*-ALGEBRAS

2008 ◽  
Vol 19 (09) ◽  
pp. 1011-1020 ◽  
Author(s):  
HIROYUKI OSAKA
Keyword(s):  
Finite Groups ◽  
Stable Rank ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Common Unit ◽  
Topological Stable Rank

When a unital C*-algebra A has topological stable rank one (write tsr (A) = 1), we know that tsr (pAp) = 1 for a non-zero projection p ∈ A. When, however, tsr (A) ≥ 2, it is generally false. We prove that if a unital C*-algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and sup p ∈ P(D) tsr (pAp) < ∞, then tsr (A) ≤ 2. As an application let A be a simple unital C*-algebra with tsr (A) = 1 and Property (SP), [Formula: see text] finite groups, αk actions from Gk to Aut ((⋯((A × α1 G1) ×α2 G2)⋯) ×αk-1 Gk-1). (G0 = {1}). Then [Formula: see text]

scholarly journals Fiberwise KK-equivalence of continuous fields of C*-algebras

2008 ◽  
Vol 3 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Marius Dadarlat
Keyword(s):  
Compact Space ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Hilbert Cube ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Finite Dimensional ◽  
Compact Metrizable Space ◽  
Continuous Fields

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

scholarly journals On the stable rank and real rank of group $C^*$-algebras of nilpotent locally compact groups

2005 ◽  
Vol 97 (1) ◽  
pp. 89
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Stable Rank ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Locally Compact ◽  
C Algebras ◽  
Necessary And Sufficient

It is shown that if $G$ is an almost connected nilpotent group then the stable rank of $C^*(G)$ is equal to the rank of the abelian group $G/[G,G]$. For a general nilpotent locally compact group $G$, it is shown that finiteness of the rank of $G/[G,G]$ is necessary and sufficient for the finiteness of the stable rank of $C^*(G)$ and also for the finiteness of the real rank of $C^*(G)$.

scholarly journals TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

2006 ◽  
Vol 17 (01) ◽  
pp. 19-34 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
TAMOTSU TERUYA
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Finite Type ◽  
Stable Rank ◽  
C Algebras ◽  
Rank One ◽  
Topological Stable Rank ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

scholarly journals Crossed products of totally disconnected spaces by

1993 ◽  
Vol 13 (3) ◽  
pp. 445-484 ◽  
Author(s):  
Ola Bratteli ◽  
David E. Evans ◽  
Akitaka Kishimoto
Keyword(s):  
Fixed Point ◽  
Metrizable Space ◽  
Crossed Product ◽  
Crossed Products ◽  
Prove Theorem ◽  
Compact Metrizable Space ◽  
Totally Disconnected

AbstractLet Ω be a totally disconnected compact metrizable space, and let α be a minimal homeomorphism of Ω. Let σ be a homeomorphism of order 2 on Ω such that ασ = σα−1, and assume that σ or ασ has a fixed point. We prove (Theorem 3.5) that the crossed product is an AF-algebra.

scholarly journals Stable rank and real rank of compact transformation group C*-algebras

2006 ◽  
Vol 175 (2) ◽  
pp. 103-120 ◽  
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Transformation Group ◽  
Stable Rank ◽  
Real Rank ◽  
C Algebras ◽  
Compact Transformation Group

Embedding covariance algebras of flows into $AF$-algebras

1999 ◽  
Vol 19 (3) ◽  
pp. 723-740 ◽  
Author(s):  
MICHAEL V. PIMSNER
Keyword(s):  
Metrizable Space ◽  
Crossed Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Injective Homomorphism ◽  
Compact Metrizable Space ◽  
Necessary And Sufficient ◽  
Af Algebras

Suppose that $\{\alpha_t\}_{t\in \mathbb{R}}$ is a flow on the compact metrizable space $X$. We prove that a necessary and sufficient condition for the existence of an embedding (injective $*$-homomorphism) of the crossed product $C(X)\rtimes_\alpha \mathbb{R}$ into some $AF$-algebra is that every point of $X$ be chain recurrent in the sense of Conley.

scholarly journals Residually finite actions and crossed products

2011 ◽  
Vol 32 (5) ◽  
pp. 1585-1614 ◽  
Author(s):  
DAVID KERR ◽  
PIOTR W. NOWAK
Keyword(s):  
Free Group ◽  
Metrizable Space ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Residual Finiteness ◽  
Hausdorff Spaces ◽  
Residually Finite ◽  
Compact Metrizable Space ◽  
Continuous Actions

AbstractWe study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

scholarly journals On the topological stable rank of non-selfadjoint operator algebras

2007 ◽  
Vol 341 (2) ◽  
pp. 239-253 ◽  
Author(s):  
K. R. Davidson ◽  
R. H. Levene ◽  
L. W. Marcoux ◽  
H. Radjavi
Keyword(s):  
Selfadjoint Operator ◽  
Operator Algebras ◽  
Stable Rank ◽  
Topological Stable Rank
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