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

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 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 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.