Spectral morphisms, K-theory, and stable ranks

Abstract

We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.

Homotopical stable ranks for Banach algebras

“…A preliminary version of this paper appeared in my dissertation [38], written under the guidance of Guoliang Yu. I am most grateful to him for all the support and advice.…”

The density theorem for hermitian K-theory

“…Karoubi's density theorem [2,5,10] says that if A is densely and continuously included in the Banach algebra B and units of A are those of B belonging to A, then K (A) and K (B) are isomorphic; it was first proved in Benayat's thesis [2] to compute the K-theory of the Banach algebra of absolutely summable Laurent series in n variables. The theorem raises the question (known as Swan's problem) whether, under the same hypotheses, there is equality of stable ranks [8]. It also allows the extension of topological K-theory to a whole class of dense subalgebras of the algebras involved and to Frechet algebras and dense subalgebras of these [1,[6][7][8].…”

“…The theorem raises the question (known as Swan's problem) whether, under the same hypotheses, there is equality of stable ranks [8]. It also allows the extension of topological K-theory to a whole class of dense subalgebras of the algebras involved and to Frechet algebras and dense subalgebras of these [1,[6][7][8]. Since K-theory is a special case of hermitian ε L -theory, it is natural to ask whether the density theorem is still valid for the latter, allowing us to extend the mentioned problems to the hermitian situation; in this article, we answer the question positively.…”

The spectral radius in $\mathcal{C}_0(X)$-Banach algebras

“…B 0 of Banach algebras is called spectral if it preserves the spectral radius of elements, or, equivalently, if b 2 B is invertible in B if and only if .b/ is invertible in B 0 ; the morphism is called dense if its image .B/ is dense in B 0 . It is a well-known fact that spectral and dense homomorphisms are isomorphisms in K-theory, see for example [CMR07] or [Bos90] (and the references therein) and the survey article [Nic10] for more information on spectral and dense morphisms and their relation to K-theory.…”