The cyclical compactness in Banach $C_{\infty}(Q)$-modules

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