On a class of C*-algebras generated by a countable family of partial isometries

The paper investigates the properties of an operator Tφ on the Hilbert space l2(ℤ), which are induced by the mapping φ of the set ℤ into itself. It is shown if the mapping φ is such that every preimage has finite, but not equipotentionally bounded cardinality, then the operator Tφ allows a closure and can be represented as a countable sum of partial isometries. The C*-algebras Uφ, Pφ and Uφ associated with given mappings and generated by the mentioned partial isometries are considered. Some properties of these algebras and some relations between them are given.