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The Cyclical Compactness in Banach C∞(Q)-Modules

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Abstract

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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Authors and Affiliations

  1. National University of Uzbekistan named after M. Ulugbek, Tashkent, Uzbekistan

    V. I. Chilin

  2. V. I. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

    J. A. Karimov

Authors
  1. V. I. Chilin
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  2. J. A. Karimov
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Correspondence to V. I. Chilin.

Additional information

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). https://doi.org/10.1007/s10958-022-06050-0

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