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Suppose K is a compact subset of the plane. A bounded sequence (t„) of uni tal homomorphisms from C(K) into a Banach algebra is pointwise norm convergent if and only if [t„(9(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.

Fuglede’s theorem and limits of spectral operators