Quasicompact endomorphisms of commutative semiprime Banach algebras

TL;DR

This paper extends the study of quasicompact endomorphisms in commutative Banach algebras, showing that under broader conditions, such endomorphisms are necessarily compact, with new examples illustrating this behavior.

Contribution

It broadens previous results to include semiprime Banach algebras and non-connected character spaces, demonstrating that quasicompact endomorphisms can be inherently compact.

Findings

Quasicompact endomorphisms are always compact in certain semiprime Banach algebras.

Examples show distinctions between quasicompact, Riesz, and power compact endomorphisms.

Extension of previous results to more general algebra classes.

Abstract

This paper is a continuation of our study of compact, power compact, Riesz, and quasicompact endomorphisms of commutative Banach algebras. Previously it has been shown that if $B$ is a unital commutative semisimple Banach algebra with connected character space, and $T$ is a unital endomorphism of $B$, then $T$ is quasicompact if and only if the operators $T^n$ converge in operator norm to a rank-one unital endomorphism of $B$. In this note the discussion is extended in two ways: we discuss endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple; we also discuss commutative Banach algebras with character spaces which are not necessarily connected. In previous papers we have given examples of commutative semisimple Banach algebras $B$ and endomorphisms $T$ of $B$ showing that $T$ may be quasicompact but not Riesz, $T$ may be Riesz but not power compact, and $T$ may be power compact but not compact. In this note we give examples of commutative, semiprime Banach algebras, some radical and some semisimple, for which every quasicompact endomorphism is actually compact.