Vol. 34, No. 3, 1970

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Actions of functions in Banach algebras

Frances F. Gulick

Vol. 34 (1970), No. 3, 657–673

This paper introduces the concept of a function f (defined on the open unit disk U of the complex plane) acting in a Banach algebra A. In general, f acts in A if there exists a mapping x f(x) from {x A : σ(x) U} = A1 into A such that for every maximal commutative subalgebra 𝒞 of A, {f(x) : x A1 ∩𝒞} is contained in 𝒞 and (f(x)) = f x(x ∈𝒞∩ A1) on the maximal ideal space of 𝒞. After some properties of actions in general Banach algebras are established, attention is restricted to a subalgebra A of the algebra Cp of compact operators on a Hilbert space such that A contains a normal operator of infinite rank. If A C and A contains only normal operators, then a necessary and sufficient condition for f to act in A is that f be continuous at zero and f(0) = 0. For a more restricted class of subalgebras of Cp,1 p < , it is shown that f defines an action in ?l if, and only if, f is Höldercontinuous at zero with f(0) = 0.

Mathematical Subject Classification
Primary: 46.65
Secondary: 47.00
Received: 20 February 1969
Revised: 11 February 1970
Published: 1 September 1970
Frances F. Gulick