Representable K-theory for \(\sigma\)-C\({}^*\)-algebras
In the category of \(\sigma\)-C\({}^*\)-algebras (i.e. the countable inverse limits of \(C^*\)-algebras) the author defines his ``representative'' K-theory groups \(RK_ 0(A)\) as the group \(UQ(A)/U_ 0Q(A)\) of path components of the unitary group of the stable outer multiplier algebra \(Q(A)=M({\mathcal K}\otimes A)/({\mathcal K}\otimes A)\), where \({\mathcal K}\) is the algebra of compact operators on a separable infinite- dimensional Hilbert space. And naturally the \(RK_ i(A)\) groups are defined as \(RK_ 0\) of the ith iterated suspension \(S^ iA:=A\otimes C_ 0({\mathbb{R}})\otimes...\otimes C_ 0({\mathbb{R}}).\) The obtained theory is homotopy invariant, has long exact sequences and a Milnor \(\lim^ 1_{\leftarrow}\)-sequence, and satisfies Bott periodicity. On \(C^*\)- algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X.
category of \(\sigma \)-C\({}^ *\)-algebras
countable inverse limits of \(C^ *\)-algebras
Milnor \(\lim ^ 1_{\leftarrow }\)-sequence
countable direct limit X of compact Hausdorff spaces
scientific article; zbMATH DE number 4165731
scientific article