### Existence and uniqueness of functional calculus homomorphisms

William R. Zame
1976 Bulletin of the American Mathematical Society
The purpose of this note is to announce a uniqueness result for the holomorphic functional calculus in commutative Banach algebras which is much stronger than the usual uniqueness assertion. Let A be a commutative Banach algebra with unit and let a = (a 1 , . . . , a n ) be an «-tuple of elements of A. Denote by a(a) the joint spectrum of a and by ö(a(a)) the topological algebra of germs of functions holomorphic near a(a). The holomorphic functional calculus, developed by Shilov ,
more » ... lov , Arens-Calderdn  and Waelbroeck , provides a continuous unital homomorphism 0 a : 0(a(a)) -• A such that: (i) 0 a (z t .) = a t for i = 1, 2, . . . , n, (u) 0 a (f ) A = ƒ « (a 1 ,..., a n ) for each ƒ in 0(o(a)), where b denotes the Gelfand transform of an element b of A, acting on AA (the maximal ideal space of A). If a' = (a 1 , . . . , a n> a n + 1 , . . . , a n+m ) is an n + mtuple and n: C n+m -• C n is the projection, then the homomorphisms 0 a and 0 a ' satisfy the following compatibility condition: (iii) 0 a (/) = 0 a (f o it) for each ƒ in 0(a(a)). The usual uniqueness assertion is that the family {0 a } is unique subject to these requirements. However, we can show that the compatibility condition is redundant, and that the individual homomorphisms are themselves unique. THEOREM 1. The homomorphism 0 a is the unique continuous unital homomorphism of 0(a(a)) into A which satisfies conditions (i) and (ii) above. Theorem 1 follows as an application of a more general existence and uniqueness result (Theorem 2, below). Let Ube a domain in C n , U its envelope of holomorphy and 0(U) the Fréchet algebra of holomorphic functions on U. Each holomorphic function ƒ on U has a unique extension to £/, which we denote by ƒ . Note that each continuous unital homomorphism 0: 0(U) -* A with 0(Zj.) = a t for i = 1, . . . , n has a continuous adjoint 0* : AA -• U such that z'j ° 0* = flj for / = 1, . . . , n. The following result shows that this correspondence of 0 with 0* is bijective. AMS (MOS) subject classifications (1970). Primary 46J05.