E-semigroups over closed convex cones

TL;DR

This paper introduces the study of E-semigroups over convex cones, providing a structure theorem, analyzing CCR flows, and demonstrating a large family of non-conjugate E0-semigroups.

Contribution

It establishes a foundational structure theorem for E-semigroups over convex cones and characterizes CCR flows, including an uncountable family of distinct E0-semigroups.

Findings

Proved a structure theorem for E-semigroups leaving compact operators invariant.

Described units and gauge groups of CCR flows.

Constructed an uncountable family of non-cocycle-conjugate CCR flows.

Abstract

We initiate a study of E-semigroups over convex cones. We prove a structure theorem for E-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, E$_0$semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of $2-$parameter CCR flows, containing mutually non-cocycle-conjugate E$_0$-$semigroups.