Reduction of topological stable rank in inductive limits ofC∗-algebras

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]

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The algebra of almost periodic functions has infinite topological stable rank

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-96-03200-5 ◽

1996 ◽

Vol 124 (3) ◽

pp. 873-876 ◽

Cited By ~ 2

Author(s):

Fernando Daniel Suárez

Keyword(s):

Almost Periodic Functions ◽

Stable Rank ◽

Almost Periodic ◽

Periodic Functions ◽

Topological Stable Rank

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Invertibility and topological stable rank for semi-crossed product algebras

Rocky Mountain Journal of Mathematics ◽

10.1216/rmjm/1181073125 ◽

1990 ◽

Vol 20 (2) ◽

pp. 513-528 ◽

Cited By ~ 1

Author(s):

J. PETERS

Keyword(s):

Stable Rank ◽

Crossed Product ◽

Topological Stable Rank ◽

Crossed Product Algebras

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A revised augmented Cuntz semigroup

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-121016 ◽

2021 ◽

Vol 127 (1) ◽

pp. 131-160

Author(s):

Leonel Robert ◽

Luis Santiago

Keyword(s):

Stable Rank ◽

Inductive Limits ◽

C Algebras ◽

Exact Functor ◽

Cw Complexes ◽

Rank One ◽

Cuntz Semigroup ◽

Original Construction

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

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