Trivial Endomorphisms of the Calkin Algebra

TL;DR

This paper demonstrates the consistency of classifying all unital endomorphisms of the Calkin algebra via Fredholm index and explores implications for the structure and embedding properties of certain C*-algebras.

Contribution

It establishes that under ZFC, all unital endomorphisms of the Calkin algebra are classifiable by Fredholm index and shows the non-closure of embedding classes under tensor products and inductive limits.

Findings

All unital endomorphisms are classifiable by Fredholm index.

Embedding classes of C*-algebras are not closed under tensor products.

Embedding classes are not closed under countable inductive limits.

Abstract

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra $\mathcal{Q}(H)$ is unitarily equivalent to an endomorphism of $\mathcal{Q}(H)$ which is liftable to a unital endomorphism of $\mathcal{B}(H)$. We use this result to classify all unital endomorphisms of $\mathcal{Q}(H)$ up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of $\mathrm{C}^\ast$-algebras that embed into $\mathcal{Q}(H)$ is not closed under tensor product nor countable inductive limit.