Topological Algebras and their Applications by Alexander A. Katz

Author:Alexander A. Katz
Language: eng
Format: epub
Publisher: Walter de Gruyter
Published: 2018-07-15T00:00:00+00:00

commutative and such that ‖ψ‖cb = ‖φ‖cb.

The classical version of this definition for Banach as well for non-complete (just normed) modules over a Banach (or normed) algebra are formulated in an obvious way.

This notion seems to be interesting even in the case of the simplest algebra A := 𝔅C. In the classical setting, that is just for Banach or normed spaces, we can thoroughly describe the projective objects. Namely, we have

Theorem 2 [8]. (i) Every metrically projective Banach space coincides, up to an isometric isomorphism, to the space l1(Λ) for some index set Λ.

(ii) Every metrically projective normed space coincides, up to an isometric isomorphism, to the normed subspace of l1(Λ) for some index set Λ, consisting of all functions with finite support.

To run ahead, this theorem shows that for Banach, as well as for normed spaces the notion of projectivity coincides with that of freeness.

The first assertion resembles to the well known result of young Grothendieck [6], 1955, where some other type of projectivity, sometimes called extreme, was considered, and our proof heavily relies on some lemmas in his paper. The second part seems to be more original. It is based on some observations over extreme points of unit balls of normed dense subspaces of l1(Λ).

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