The cyclical compactness in Banach \(C_\infty (Q)\)-modules. (English. Russian original) Zbl 1514.46035

J. Math. Sci., New York 265, No. 1, 129-145 (2022); translation from Sovrem. Mat., Fundam. Napravl. 65, No. 1, 137-155 (2019).
Summary: In this paper we study the class of laterally complete commutative unital regular algebras \(A\) over arbitrary fields. We introduce a notion of passport \(\Gamma (X)\) for a faithful regular laterally complete \(A\) modules \(X\), 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 \(\Gamma (X) = \Gamma (Y)\). Further we study Banach \(A\)-modules in the case \(A = C_\infty (Q)\) or \(A = C_\infty (Q)+ i \cdot C_\infty (Q)\). We establish the equivalence of all norms in a finite-dimensional (respectively, \( \sigma \)-finite-dimensional) \(A\)-module and prove an \(A\)-version of Riesz theorem, which gives the criterion of a finite-dimensionality (respectively, \( \sigma \)-finite-dimensionality) of a Banach \(A\)-module.


46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
16D99 Modules, bimodules and ideals in associative algebras
46L10 General theory of von Neumann algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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