NATURALITY OF THE FUNCTIONAL CALCULUS STEPHEN H. SCHANUEL AND WILLIAM R. ZAME The holomorphic functional calculus, developed by Shilov [4], Arens-Calderon [1] and Waelbroeck [6], provides a connection between the theory of (one and several) complex variables and the theory of commutative Banach algebras. Indeed, large portions of the latter theory are virtually co-extensive with the former. The purpose of this note is to point out, using the viewpoint of Category Theory, a simple and inescapable reason for this connection. Let si be a commutative Banach algebra with identity, let U be an open subset of C" and let a = (a l5 ..., a n ) be an n-tuple of elements of stf whose joint spectrum o" iC ,(a) lies in U. The functional calculus provides, for each holomorphic function / on U, an element 6^(f; a) of s&. The usual practice is to treat s#, U and a as fixed and study the mapping / -> B^{f; a). (For general information about the functional calculus, see [1], [2], [5] or [6]. For additional uniqueness results, see [7].) However, we can also proceed in another way, treating U, f as fixed and varying $0, a. Let U be an open subset of C", and let / be a holomorphic function on U. For each commutative Banach algebra with identity s#, let !(/(«$/) denote the set of H-tuples a = (a lt ...,a n ) of elements of s/ such that a^(a) lies in U. Notice that if a : stf -> £8 is a continuous homomorphism preserving the identity, then we obtain a map I u (a): L v {s#) -* ![/($) by setting Thought of in this way, I a is a functor from the category of commutative Banach algebras with identity to the category of sets. Moreover, the construction of the functional calculus guarantees that the diagram below is commutative (for each homomorphism a): That is, the family {#<•(/; )} is a natural transformation from the functor Z L r to the underlying set functor (which assigns to each algebra s? its underlying set; see [3] for Received 29 May, 1980; revised 21 August, 1980. The first author's research was supported in part by National Science Foundation Grant MCS 79- 04162, that of the second author by National Science Foundation Grant MCS 79-01786. Bull. London Math. Soc, 14 (1982), 218-220