Open AccessThe Cyclical Compactness in Banach C∞(Q)-Modules
Author(s) -
Vladimir Chilin,
Jasurbek A. Karimov
Publication year - 2019
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2019-65-1-137-155
Subject(s) - mathematics , partition (number theory) , commutative property , curse of dimensionality , class (philosophy) , pure mathematics , discrete mathematics , equivalence (formal languages) , set (abstract data type) , combinatorics , computer science , statistics , artificial intelligence , programming language
In this paper, we study the class of laterally complete commutative unital regular algebras A over arbitrary elds. We introduce a notion of passport Γ(X) for a faithful regular laterally complete A- modules X, which consist of uniquely dened partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise dierent 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 nite-dimensional (respectively, σ-nite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a nite-dimensionality (respectively, σ-nite-dimensionality) of a Banach A-module.