*-semigroup endomorphisms of B(H)

Abstract

Let H be a complex infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms $ϕ$ of B(H) which are uniformly continuous on every commutative C*-subalgebra. In particular, we obtain that if $ϕ$ satisfies $ϕ(0)=0$, then $ϕ$ is additive.