Wold-Type Decomposition of Semigroups of Isometries in Baer *-rings II

In this paper we deal with Wold-type decompositions of n-tuples of commuting semigroups of isometries in Baer *-rings. Our main goal is investigating the role of algebraic structure in such decompositions of Hilbert space operators. Let A be a Baer *-ring. An element x of A is called an isometry, if e x = 1. We characterize those n-tuples of commuting semigroups of isometrics in A for which the product semigroup is completely non-unitary and give explicit formulas for their defect projections. Then we identify n-tuples which have a Wold-type decomposition. Uniqueness of such a decomposition, whenever it exists, is also proved. Our results not only extend these decomposition theorems for Hilbert space operators to Baer *-rings with purely algebraic proofs, but also show that they rely mainly on the algebraic structure of B(H).