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The Cyclical Compactness in Banach C(Q)-Modules

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In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete Amodules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X) = Γ(Y). Further we study Banach A-modules in the case A = C(Q) or A = C(Q)+i ·C(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module.

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Correspondence to V. I. Chilin.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 65, No. 1, Contemporary Problems in Mathematics and Physics, 2019.

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Chilin, V.I., Karimov, J.A. The Cyclical Compactness in Banach C(Q)-Modules. J Math Sci 265, 129–145 (2022).

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