On the Wiener-Hopf compactification of a Symmetric Cone

TL;DR

This paper characterizes the Wiener-Hopf compactification of the cone of squares in a Euclidean Jordan algebra and concludes that the associated Wiener-Hopf C*-algebra has trivial K-groups.

Contribution

It provides a precise description of the Wiener-Hopf compactification for symmetric cones in Euclidean Jordan algebras and computes the K-groups of the related C*-algebra.

Findings

Wiener-Hopf compactification of Q is the intersection (1-Q) ∩ (-1+Q)

K-groups of the Wiener-Hopf C*-algebra are trivial

The result applies to symmetric cones in Euclidean Jordan algebras

Abstract

Let V be a finite dimensional real Euclidean Jordan algebra with the identity element 1. Let Q be the closed convex cone of squares. We show that the Wiener- Hopf compactification of Q is the interval (1-Q) \cap (-1+Q). As a consequence, we deduce that the K-groups of the Wiener-Hopf C^{*}-algebra associated to Q are trivial.