Abstract
The purpose of this paper is to study semigroups of *-endomorphisms of type I factors, with parameters in a countable dense subgroup of the real line. Most of our results are motivated by the theory of semigroups of isometries. It is shown that every semigroup of *-endomorphisms can be extended to a group of *-automorphisms in a minimal way. In certain situations, it is shown that a semigroup of *-endomorphisms can be decomposed into three simpler semigroups. Shifts are defined and classified up to outer conjugacy.
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