Karoubi's conjecture for the smooth \(K\)-theory
The Karoubi conjecture claims that algebraic and topological \(K\)-theory of stable \(C^*\)-algebras coincide. After being proved [\textit{A. A. Suslin} and \textit{M. Wodzicki}, Ann. Math. (2) 136, No. 1, 51--122 (1992; Zbl 0756.18008)], this conjecture was generalized for wider classes of algebras. The paper under review generalizes this conjecture for quasi-stable Fréchet algebras. A Fréchet algebra is quasi-stable if it is equal to the projective tensor product of the algebra of compact operators by a Fréchet algebra with a properly uniformly bounded approximative unit. In place of the topological \(K\)-theory the authors develop and use smooth \(K\)-theory based on smooth maps.
scientific article; zbMATH DE number 5374908