Approximately inner flows

TL;DR

This paper investigates whether approximately inner flows on C*-algebras extend to their crossed products while remaining approximately inner, providing specific examples where this extension preserves the property.

Contribution

It analyzes conditions under which approximately inner flows extend to crossed products and presents explicit examples where the extension remains approximately inner.

Findings

Some extensions of approximately inner flows to crossed products are also approximately inner.

The paper provides specific cases where the extension property holds affirmatively.

It clarifies the relationship between flows and automorphisms in the context of crossed products.

Abstract

When $\alpha$ is an approximately inner flow on a C$^*$-algebra $A$ and commutes with an automorphism $\gamma$ of $A$ we may extend $\alpha$ to a flow $\bar{\alpha}$ on the crossed product $A\times_\gamma Z$ by setting $\bar{\alpha}_t(U)=U$ where $U$ is the canonical unitary implementing $\gamma$ in $A\times_\gamma Z$ and ask whether $\bar{\alpha}$ is also approximately inner or not. We will consider very specific examples of this type; some of which we can answer affirmatively.