Algebraic \(K\)-theory of stable \(C^*\)-algebras
A \(C^*\)-algebra \(A\) is said to be stable if it is *-isomorphic to the tensor product \({\mathcal K}\otimes A\), where \({\mathcal K}\) is the \(C^*\)-algebra of compact operators on a separable Hilbert space. Most of the paper is devoted to the study of these algebras. The main theorem of this paper is as follows: If \(A\) is a stable \(C^*\)- algebra then the algebraic \(K\)-theory of \(A\) is equal to the topological \(K\)-theory of \(A\), i.e. \(K_*(A)\sim K_*^{top}(A)\). Besides this theorem, the author presents a number of other results, mostly on the same theme of comparing algebraic and topological K-theory, but occasionally as minor digressions from it. This paper is divided into six sections. I. Multiplier algebras and tensor products. II. \(K\)-theory of \(C^*\)-algebras. III. A homotopy invariance theorem. IV. Algebraic \(K_ 2\) of stable \(C^*\)-algebras. V. Algebraic \(K\)-theory of Calkin algebras. (This is the main section. The author proves the following conjecture of M. Karoubi: \(K_*(A\otimes C)\sim K_*^{top}(A\otimes C)\), where \(C\) is the Calkin algebra. The algebra \(A\otimes C\) may be regarded as a suspension of the \(A\otimes {\mathcal K}\); thus Karoubi's conjecture asserts that the algebraic and topological \(K\)-theories of stable \(C^*\)-algebras agree.) VI. Further results (There are two main topics: the study of the group \(\mathrm{GL}_ 1A\) instead of the stable version \(\mathrm{GL}_{\infty}A\) and the Karoubi-Villamayor algebraic \(K\)-theory).
comparing algebraic and topological \(K\)-theory
Karoubi-Villamayor algebraic \(K\)-theory
scientific article; zbMATH DE number 4034437
scientific article