Quasicompact endomorphisms of commutative semiprime Banach algebras

• Autorzy: Joel F. Feinstein , Herbert Kamowitz
• Języki publikacji: EN

Abstrakty

EN

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 Tⁿ 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.

Kategorie tematyczne

• 46J10: Banach algebras of continuous functions, function algebras
• 46J45: Radical Banach algebras
• 46J05: General theory of commutative topological algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 91
• Numer: 1
• Strony: 159-167

Daty

wydano

2010

Twórcy

autor

Joel F. Feinstein

• School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

autor

Herbert Kamowitz

• Department of Mathematics, University of Massachusetts at Boston, 100 Morrissey Boulevard, Boston, MA 02125-3393, U.S.A.

Identyfikatory

DOI

10.4064/bc91-0-8