Lipschitzness of *-homomorphisms between C*-metric algebras

TL;DR

This paper proves that under certain conditions, *-homomorphisms between C*-metric algebras are Lipschitz continuous, and this property is preserved under free products of such homomorphisms.

Contribution

It establishes that *-homomorphisms between C*-metric algebras are Lipschitz if Lip-norms are lower semicontinuous, and shows this property persists in free products.

Findings

Unital *-homomorphisms are Lipschitz under lower semicontinuous Lip-norms.

Lipschitz property is preserved in free products of *-homomorphisms.

Provides conditions ensuring Lipschitz continuity in C*-metric algebra morphisms.

Abstract

A C*-metric algebra consists of a unital C*-algebra and a Leibniz Lip-norm on the C*-algebra. We show that if the Lip-norms concerned are lower semicontinuous, then any unital *-homomorphism from a C*-metric algebra to another one is necessarily Lipschitz. It results that the free product of two Lipschitz unital *-homomorphisms between C*-metric algebras coming from *-filtrations is still a Lipschitz unital *-homomorphism.