Quasicompact and Riesz endomorphisms of Banach algebras

Abstract

Let B be a unital commutative semi-simple Banach algebra. We study endomorphisms of B which are also quasicompact operators or Riesz operators. Clearly compact and power compact endomorphisms are Riesz and hence quasicompact. Several general theorems about quasicompact endomorphisms are proved, and these results are then applied to the question of when quasicompact or Riesz endomorphisms of certain algebras are necessarily power compact.

Quasicompact endomorphisms of commutative semiprime Banach algebras

“…Using a fairly standard technique involving orthogonal idempotents, we will prove the following result, which is a main result of this note. This result extends earlier results of the authors [5,6,7] for commutative semisimple Banach algebras, and results for uniform algebras of Klein [14] and Gamelin, Galindo and Lindström [8,9]. Section 4 contains some results about commutative radical semiprime Banach algebras, while Section 5 presents two examples of commutative semisimple Banach algebras where each quasicompact endomorphism is compact.…”

“…Using these observations it easily follows that the proof of Lemma 1.1 of [7] holds when semisimple is replaced by semiprime and we have the following lemma.…”

“…In previous papers the authors [5,6,7] and others [8,9,14] have studied endomorphisms of commutative semisimple Banach algebras and have obtained several general theorems, and also a variety of results pertaining to specific classes of algebras. In this note we extend this discussion to endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple.…”

“…In [7], the authors investigated quasicompact endomorphisms of commutative semisimple Banach algebras. One of the main results of that paper was the following.…”

“…Bounded endomorphisms of semiprime Banach algebras with connected character space. In order to extend the results from [7], we begin by examining the properties of semisimplicity which were used in the proof of Lemma 1.1 of [7], and observing that they are more generally true. Specifically, we note the following.…”