# Banach space of analytic functions with infinite-dimensional domains

The primary interest here is in the interplay between function theory on infinite-dimensional domains, geometric properties of Banach spaces, and Banach and Fréchet algebras. Throughout, will denote a complex Banach space with open unit ball .

## Definition and basic properties.

Let denote the space of complex-valued -homogeneous polynomials , i.e. functions to which is associated a continuous -linear function such that for all . Each such polynomial is associated with a unique symmetric -linear form via the polarization formula. For an open subset , one says that is holomorphic, or analytic, if has a complex Fréchet derivative at each point of (cf. also Algebra of functions). Equivalently, is holomorphic if at each point there is a sequence of -homogeneous polynomials such that for all in a neighbourhood of . If , then the algebra of holomorphic functions from to always contains as a proper subset the subalgebra of holomorphic functions which are bounded on bounded subsets such that . The latter space is a Fréchet algebra with metric determined by countably many such subsets, whereas there are a number of natural topologies on .

The natural analogues of the classical Banach algebras of analytic functions are the following: ;    All are Banach algebras with identity when endowed with the supremum norm (cf. also Banach algebra).

## Results and problems.

For any of the above algebras of analytic functions, let denote the set of homomorphisms . Since the Michael problem has an affirmative solution [a5], every homomorphism is automatically continuous. For each such , define (noting that, always, ). Basic topics of interest here are the relation between the "fibres" , , and the relation between the geometry of and of .

The spectrum displays very different behaviour in the infinite-dimensional setting, in comparison with the finite-dimensional situation. As an illustration, every element corresponds to a homomorphism on . Indeed, for each there is a linear extension mapping from . Applying this mapping to the Taylor series of a holomorphic function yields a multiplicative linear extension operator, mapping to ; similar results hold for and . For example, each yields an element of via . A complete description of is unknown (1998) for general , although it is not difficult to see that . The question of whether the fourth dual of also provides points of the spectrum is connected with Arens regularity of [a7]. In any case, can be made into a semi-group with identity ; the commutativity of this semi-group is related, once again, to Arens regularity of [a6].

It is natural to look for analytic structure in the spectrum . In fact, every fibre over contains a copy of . In many situations, e.g. when is super-reflexive (cf., also Reflexive space), there is an analytic embedding of the unit ball of a non-separable Hilbert space into . Further information has been obtained by J. Farmer [a8], who has studied analytic structure in fibres in -spaces. However, note that there is a peak set (cf. also Algebra of functions) for which is contained in .

There has also been recent (1998) interest in the following areas:

reflexivity of ;

algebras of weakly continuous holomorphic functions; and

Banach-algebra-valued holomorphic mappings.

Basic references on holomorphic functions in infinite dimensions are [a1], [a2], [a3]; a recent (1998) very helpful source, with an extensive bibliography, is [a4].