Fuglede’s theorem and limits of spectral operators

Abstract

Suppose K is a compact subset of the plane. A bounded sequence { τ n } \{ {\tau _n}\} of unital homomorphisms from C ( K ) C(K) into a Banach algebra is pointwise norm convergent if and only if { τ n ( θ ( z ) = z ) } \{ {\tau _n}(\theta (z) = z)\} is convergent. Applications are made to norm limits of scalar type spectral operators. The proof is based on an asymptotic version of Fuglede’s theorem for Banach algebras.