State Spaces of Operator Algebras: Basic Theory, by Erik M. Alfsen

00. Set Sn = (T - AnI)-l for n = 1,2, ... We will first show that {Sn} is an unbounded sequence of operators. Assume for contradiction that there is a positive number M such that IISnl1 < M for all n.

An. By assumption, these are simple zeros, so where g is an entire function which is different from zero at the points AI,"" An. Clearly also, g is different from zero at each point of sp(T) which is not a zero for f. Thus 9 =f:. 0 on sp(T). Therefore, g has a holomorphic inverse h defined in an open neighborhood of sp(T). 53 (T - All) . . (T - AnI) = f(T)h(T). 47, this completes the proof. 58) which will later be transferred to the context of each one of the three parts of this book. The vector lattice CR(X) is not monotone complete (or monotone (Jcomplete) in general.

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