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

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.