The author considers an
operator T in a reflexive Banach space X for which there is a bounded operational
calculus a → a(T) defined on AC(I), the algebra of absolutely continuous functions
defined on I = [0,1] with the norm |a(0)| +VarI(a) for a ∈ AC(I). Such
operators, called well-bounded, have been investigated by Smart and Ringrose (J.
Australian Math. Soc. 1 (1960), 319–343 and Proc. London Math. Soc. (3) 13
(1963), 613–638). The present paper explores a new method for obtaining the
spectral theorem for this operator. Let AC0 be the maximal ideal of members
of AC(I) which are zero at 0. The method consists in introducing Arens
multiplication into AC0∗∗, the second conjugate space of AC0, and in investigating
the larger algebra for a suitable family of idempotents which will serve as
candidates for bounded spectral projections associated with T. Idempotents in
AC0∗∗ are mapped into these projections by means of a homomorphism
extension technique which extends the original operational calculus of AC0 into
B(X) (the bounded linear operators on X), to a bounded homomorphism of
AC0∗∗ into B(X). The extended homomorphism is defined on a quotient
algebra of AC0∗∗. This quotient algebra turns out to be a copy of all functions
of bounded variation on I which are zero at 0 under the usual pointwise
operations.
