Measurable 𝐸0-semigroups are continuous

Abstract

Let 𝐺 be a second countable locally compact Hausdorff topological group and 𝑃 be a closed subsemigroup of 𝐺 containing the identity element 𝑒∈𝐺 . Assume that the interior of 𝑃 is dense in 𝑃 . Let 𝛼={𝛼𝑥}𝑥∈𝑃 be a semigroup of unital normal ∗ -endomorphisms of a von Neumann algebra 𝑀 with separable predual satisfying a natural measurability hypothesis. We show that 𝛼 is an 𝐸0 -semigroup over 𝑃 on 𝑀 .