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.
