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This paper is a continuation of our study of compact, power compact, Riesz,and quasicompact endomorphisms of commutative Banach algebras. Previously ithas been shown that if $B$ is a unital commutative semisimple Banach algebrawith 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 operatornorm to a rank-one unital endomorphism of $B$.  In this note the discussion is extended in two ways: we discuss endomorphismsof commutative Banach algebras which are semiprime and not necessarilysemisimple; we also discuss commutative Banach algebras with character spaceswhich are not necessarily connected.  In previous papers we have given examples of commutative semisimple Banachalgebras $B$ and endomorphisms $T$ of $B$ showing that $T$ may be quasicompactbut not Riesz, $T$ may be Riesz but not power compact, and $T$ may be powercompact but not compact. In this note we give examples of commutative,semiprime Banach algebras, some radical and some semisimple, for which everyquasicompact endomorphism is actually compact.

Quasicompact endomorphisms of commutative semiprime Banach algebras