Quasi-compact endomorphisms and primary ideals in commutative unital Banach algebras (Q5965159)

Let \(B\) be a semiprime commutative unital Banach algebra with connected character space. In this paper the author describes the outer spectrum of a quasicompact endomorphism \(T:B \to B\). Recall that \(T\) is quasicompact if the essential spectral radius \(r_e(T) < 1\). One of the main results states that if \(T\) is quasicompact and \(T^*x_0 = x_0,\) then there exists a family \(\mathcal J\) of \(T\)-invariant closed primary ideals of finite codimension in \(B\) and hull \(\{x_0\}\) such that the outer spectrum \(\sigma(T) \cap \{\lambda : |\lambda| > r_e(T)\}= \bigcup_{I\in\mathcal J} \sigma(T/I).\) The maximal ideal \(M(x_0)=\{b\in B: x_0(b)=0\}\) always belongs to \(\mathcal J\). This nice result can e.g. be used to reproduce several results in the literature concerning the spectra of Riesz endomorphisms \(T\) (i.e., \(r_e(T) =0\)) on algebras of holomorphic functions on domains in \(\mathbb C\).