Homotopical stable ranks for Banach algebras

Nica, Bogdan

doi:10.1016/j.jfa.2011.03.001

Cited by 8 publications

(10 citation statements)

References 41 publications

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“…The following is an answer to a question posed by Nica [13,Problem 5.8]. Before we begin, we observe that if X is a compact Hausdorff space whose covering dimension is ≤ 4, then Nica has shown [13, Proposition 5.5] that gsr(C(X)) = 1.…”

Section: Now Letmentioning

confidence: 97%

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Homotopical stable ranks for certain C$^*$-algebras

Vaidyanathan

2019

Studia Math.

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We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras and non-commutative CW-complexes.Stable ranks for C*-algebras were first introduced by Rieffel [16] in his study of the nonstable K-theory of noncommutative tori. A stable rank of a C*-algebra A is a number associated to the C*-algebra, and is meant to generalize the notion of covering dimension for topological spaces. The first such notion introduced by Rieffel, called topological stable rank, has played an important role ever since. In particular, the structure of C*-algebras having topological stable rank one is particularly well understood.Since the foundational work of Rieffel, many other ranks have been introduced for C*algebras, including real rank, decomposition rank, nuclear dimension, etc. In this paper, we return to the original work of Rieffel, and consider two other stable ranks introduced by him: the connected stable rank and general stable rank. The general stable rank determines the stage at which stably free projective modules are forced to be free. The connected stable rank is a related notion, but its definition is less transparent. What links these two ranks, and differentiates them from the topological stable rank, is that they are homotopy invariant.This was highlighted in a paper by Nica [13], where he emphasized the relationship between these two ranks, and how they differ from topological stable rank. Furthermore, in order to compute these ranks for various examples, he showed how they behave with respect to some basic constructions (matrix algebras, quotients, inductive limits, and extensions).

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Section: Now Letmentioning

confidence: 97%

“…We now enumerate some basic properties of these ranks that are known or are easily observed from the definitions. While the original proofs are scattered through the literature, [13] is an immediate reference for all these facts.…”

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confidence: 99%

Homotopical stable ranks for certain C$^*$-algebras

Vaidyanathan

2019

Studia Math.

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“…By Theorem 2.4, we have GL(T (N (n) )), that is, GL n (T (N )) is connected for each positive integer n. And by [5,Theorem 2.8], gsr(T (N )) = 1 (also see [8,Lemma 2.8]), so we deduce that csr(T (N )) = 1. Therefore, by [14,Proposition 3.6], K 1 (T (N )) = 0. 2…”

Section: Two Consequencesmentioning

confidence: 99%

Connectedness of the invertibles in certain nest algebras of order type ω

Zhang

2012

Journal of Functional Analysis

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“…Since (A/J)˜∼ = A˜/J, it suffices to consider the case in which A has an identity element. The arguments run just as in the one in Nica's paper [5,Theorem 12.5], except that we apply directly Lemma 1 rather than use the conclusion that an ideal with an approximate identity satisfies Lg n (J˜) = Lg n (A) ∩ (J˜) n for each positive integer n [5, Lemma 12.1].…”

Section: A Note On the Topological Stable Rank Of An Ideal 4001mentioning

confidence: 99%

“…For a long time, the following question has been posed by some people (maybe in different forms), i.e., Corach, Suárez [2], Nica [5], whether one can find a Banach algebra A such that bsr(A) < tsr(A) = ∞. By using a striking result of [4], we can find one.…”

Section: A Note On the Topological Stable Rank Of An Ideal 4001mentioning

confidence: 99%

A note on the topological stable rank of an ideal

Zhang

2011

Proc. Amer. Math. Soc.

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Abstract. We show that if J is an ideal of a Banach algebra A, then the left (right) topological stable rank of J is no greater than the left (right) topological stable rank of A.By a Banach algebra A we mean a complex Banach algebra which may be not unital. When we speak of ideals we will always mean closed two-sided ideals. The topological stable rank for Banach algebras was defined in Rieffel [6] as a non-commutative analogue of the covering dimension for compact spaces that was modelled on the Bass stable rank of rings [1]. Since it was introduced, the concept has been very useful in studying non-stable K-theory and some spectral or structure properties of Banach algebras. While investigating the topological stable rank, Rieffel [6] also defined two other stable ranks: the connected stable rank and the general stable rank.In classical covering dimension theory, it is known that the dimension of an open subset U of a locally compact Hausdorff space X is no greater than the dimension of the whole space X. Comparing with this, we are interested in whether an analog and generalization of the inequality can be obtained in topological stable rank. So it is natural to ask: is it always true that the left (right) topological stable rank of an ideal J is no greater than the left (right) topological stable rank of the whole algebra A? In the seminal paper Rieffel asserted it is true under the assumption that J has a bounded approximate identity [6, Theorem 4.4], and assuming that the quotient map splits, Davidson and the first author showed it is also true [3, Theorem 3.1]. As stated in the abstract, the purpose of this note is to show that this always holds for any ideal in any Banach algebra.We begin by recalling some standard definitions and facts. Given a unital Banach algebra A, we denote by Lg n (A) (resp. Rg n (A)) the set of n-tuples of elements of A which generate A as a left ideal (resp. as a right ideal), that is,

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Homotopical stable ranks for Banach algebras

Cited by 8 publications

References 41 publications

Homotopical stable ranks for certain C$^*$-algebras

Homotopical stable ranks for certain C$^*$-algebras

Connectedness of the invertibles in certain nest algebras of order type ω

A note on the topological stable rank of an ideal

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