Spaces of sections of Banach algebra bundles

Farjoun, Emmanuel D.; Schochet, Claude

doi:10.1017/is012002001jkt183

Cited by 8 publications

(8 citation statements)

References 20 publications

(35 reference statements)

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“…Furthermore, we say that a homology theory {h n } is continuous if, whenever A = lim A i is an inductive limit in the category of C * -algebras, then h n (A) = lim h n (A i ) in the category of abelian groups. The next proposition is a consequence of [ The notion of K-stability given below is due to Thomsen [26,Definition 3.1], and that of rational K-stability has been studied by Farjoun and Schochet [5,Definition 1.2], where it was termed rational Bott-stability. Definition 1.3.…”

Section: Preliminariesmentioning

confidence: 99%

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𝐾-stability of continuous 𝐶(𝑋)-algebras

Seth

Vaidyanathan

2020

Proc. Amer. Math. Soc.

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We show that the property of being rationally Kstable passes from the fibers of a continuous C(X)-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers. Given a compact Hausdorff space X, a continuous C(X)-algebra is the section algebra of a continuous field of C*-algebras over X. Such algebras form an important class of non-simple C*-algebras, and it is often of interest to understand those properties of a C*-algebra which pass from the fibers to the ambient C(X)-algebra. Given a unital C*-algebra, we write U n (A) for the group of n × n unitary matrices over A. This is a topological group, and its homotopy groups π j (U n (A)) are termed the nonstable K-theory groups of A. These groups were first systematically studied by Rieffel [20] in the context of noncommutative tori. Thomsen [26] built on this work, and developed the notion of quasi-unitaries, thus constructing a homology theory for (possibly non-unital) C*-algebras. Unfortunately, the nonstable K-theory for a given C*-algebra is notoriously difficult to compute explicitly. Even for the algebra of complex numbers, these groups are naturally related to the homotopy groups of spheres π j (S n), which are not known for many values of j and n. It is here that rational homotopy theory has proved to be useful to topologists and, in this paper, we employ this tool in the context of C*-algebras.

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Section: Preliminariesmentioning

confidence: 99%

“…The notion of K-stability given below is due to Thomsen [26,Definition 3.1], and that of rational K-stability has been studied by Farjoun and Schochet [5,Definition 1.2], where it was termed rational Bott-stability.…”

mentioning

confidence: 99%

𝐾-stability of continuous 𝐶(𝑋)-algebras

Seth

Vaidyanathan

2020

Proc. Amer. Math. Soc.

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“…We recall for reference the main results of Farjoun-Schochet [8]. Suppose that X is a finite dimensional compact metric space and ζ : P → X is a standard 2 principal G bundle for some topological group G that acts continuously on a Banach algebra B via α : G → Aut(B).…”

Section: The Spectral Sequencesmentioning

confidence: 99%

“…2 A principal G-bundle is standard if X is a finite complex or if the bundle is a pullback of a principal G-bundle over some CW -complex. See [8] for details.…”

Section: The Spectral Sequencesmentioning

confidence: 99%

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Banach algebras, Samelson products, and the Wang Differential

Schochet¹

2012

Preprint

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Supppose given a principal G bundle ζ : P → S k (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Letdenote the associated bundle and letdenote the associated Banach algebra of sections. Then π * GL A ζ⊗B is determined by a mostly degenerate spectral sequence and by a Wang differentialWe show that if B is a C * -algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory.We illustrate our technique with a close analysis of the invariants associated to the C * -algebra of sections of the bundleconstructed from the Hopf bundle ζ : S 7 → S 4 and by the conjugation action of S 3 on M 2 = M 2 (C). We compare and contrast the information obtained from the homotopy groups π * (A ζ⊗M 2 ), the rational homotopy groups π * (A ζ⊗M 2 ) ⊗ Q and the topological K-theory groups K * (A ζ⊗M 2 ).

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