Algebraic \(K\)-theory of normed algebras
In order to extend Quillen's \(K\)-theory of discrete rings [\textit{D. Quillen}, Higher algebraic \(K\)-theory'', Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] to the category of normed \(A\)-algebras, the author gives in the first part of the paper a Quillen's type construction for topological exact categories, namely for the category of finitely generated projective modules over a normed \(k\)-algebra \(A\), where \(k\) is a commutative Banach ring with unit, and their bounded \(A\)-homomorphisms. It is proved that Quillen's \(K\)-theory on the category of special normed real algebras is isomorphic to topological \(K\)-theory. The second section deals with the relation between algebraic and topological \(K\)-functors on the category of real normed algebras. So, the natural sufficient conditions are obtained for the isomorphism of algebraic and topological \(K\)-functors on this category (Theorems 10 and 22 of this paper). These results allow the author to introduce the notion of \(K\)-stability of real normed algebras (Definition 12) which is closely related with stability of \(C^*\)-algebras and with \(K_*\)-regularity of discrete rings. It is shown that a wide class of complex normed algebras including polynomial algebras \(A[x_1,\dots, x_n]\) over stable \(C^*\)-algebras \(A\), Laurent polynomial algebras \(A[t, t^{-1}]\) over stable \(C^*\)-algebras \(A\), \(C^*\)-algebras of continuous maps from compact spaces to the Calkin algebra, retracts of \(C^*\)-algebras are \(K_i\) stable for \(i\geq 1\). Most of them are not complete and on the category of these complex normed algebras one has the equality of algebraic and topological \(K\)-functors (Theorems 14 and 18, Remark 19, Corollary 13, Propositions 20 and 21).
category of finitely generated projective modules over a normed \(k\)-algebra
isomorphism of algebraic and topological \(K\)-functors
scientific article; zbMATH DE number 1563217