The Stable Rank of Topological Algebras and a Problem of R. G. Swan

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.

Section: Swan's Problem For the Higher Connected Stable Rankssupporting

confidence: 80%

Section: Introductionmentioning

confidence: 98%

Relatively spectral morphisms and applications to K-theory

Nica

2008

“…The GRS-condition is equivalent to the spectral invariance of A 1 v (Λ, c) in C * (Λ, c) and therefore by Badea's result [1] Λ, c)). By Theorem 1.5 in [4] we know that tsr C * (Λ, c) = 1 for Λ completely irrational.…”

Section: Theorem 25 Let λ Be Completely Irrational and V A Grs-weigmentioning

confidence: 86%

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

Luef

2009

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.

“…1.1], says the following: if A is a dense and spectral Banach * -subalgebra of a C * -algebra B, and if A is closed under C ∞ -functional calculus for self-adjoint elements, then bsr A = bsr B. This applies, for instance, to dense subalgebras coming from derivations [3,Cor. 4.10].…”

Section: Swan's Problemmentioning

confidence: 96%

Homotopical stable ranks for Banach algebras

Nica

2011

The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.