The author develops representable \(K\)-theory for Fréchet algebras (complete complex topological algebras, such that its topology is given by a countable family of submultiplicative seminorms) and generalizes simultaneously \(K\)-theory for Banach algebras [see f.e. B. Blackadar, \(K\)-theory for operator algebras (MSRI publ. No. 5, Springer, New York) (1986; Zbl 0597.46072)] and representable \(K\)-theory for \(\sigma-C^*\)-algebras ( countable inverse limits of \(C^*\)-algebras) [C. N. Phillips, \(K\)-theory 3, 441-478 (1989; Zbl 0709.46033)].
It is proved that a surjective homomorphism is surjective on the identity path-components of the groups of invertible elements and that homotopic idempotents are similar.
For stabilization the Fréchet algebra \(K_{\infty}\) of rapidly decreasing functions on \(Z^ 2\) is used, with multiplication \((st)(m,n)=\sum_{j\in Z}s(m,j)t(j,n)\) and norms \(\| s\|_ \nu =\sum_{m,n\in Z}(1+| m|+| n|)\cdot| s(m,n)|\) (\(\nu\in N\)). The obtained theory satisfies Bott periodicity. There are the standard exact sequences:
the six term sequence, associated to a short exact sequence, the Mayer-Vietoris sequence and the Milnor \(\varprojlim^ 1\)-sequence. In the last section of the paper the author discusses special cases and relations to known \(K\)-theoretic results.