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$$K$$-theory for Fréchet algebras. (English) Zbl 0744.46065
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.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46H05 General theory of topological algebras 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 19A99 Grothendieck groups and $$K_0$$ 19K99 $$K$$-theory and operator algebras
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