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, $E$ will denote a complex Banach space with open unit ball $B_E$.

Definition and basic properties.

Let $\mathcal{P} ( \square ^ { n } E )$ denote the space of complex-valued $n$-homogeneous polynomials $P : E \rightarrow \bf C$, i.e. functions $P$ to which is associated a continuous $n$-linear function $A : E \times \ldots \times E \rightarrow \mathbf{C}$ such that $P ( z ) = A ( z , \dots , z )$ for all $z \in E$. Each such polynomial is associated with a unique symmetric $n$-linear form via the polarization formula. For an open subset $U \subset E$, one says that $f : U \rightarrow \bf C$ is holomorphic, or analytic, if $f$ has a complex Fréchet derivative at each point of $U$ (cf. also Algebra of functions). Equivalently, $f$ is holomorphic if at each point $z _ { 0 } \in U$ there is a sequence of $n$-homogeneous polynomials $( P _ { n } ) = ( P _ { n } ( z _ { 0 } ) )$ such that $f ( z ) = \sum _ { n = 0 } ^ { \infty } P _ { n } ( z - z _ { 0 } )$ for all $z$ in a neighbourhood of $z_0$. If $ \operatorname {dim} E = \infty$, then the algebra $\mathcal{H} ( U )$ of holomorphic functions from $U$ to $\mathbf{C}$ always contains as a proper subset the subalgebra $\mathcal{H} _ { b } ( U )$ of holomorphic functions which are bounded on bounded subsets $B \subset U$ such that $\operatorname { dist } ( B , U ^ { c } ) > 0$. The latter space is a Fréchet algebra with metric determined by countably many such subsets, whereas there are a number of natural topologies on $\mathcal{H} ( U )$.

The natural analogues of the classical Banach algebras of analytic functions are the following:

$\mathcal{H} ^ { \infty } ( B _ { E } ) \equiv \{ f \in \mathcal{H} ( B _ { E } ) : f \, \text { bounded on } \, B _ { E } \}$;

$\mathcal{A} _ { b } ( B _ { E } ) \equiv$

\begin{equation*} \{ f \in \mathcal{H} ^ { \infty } ( B _ { E } ) : f \, \text{continuous and bounded on}\,\overline{B_E}\}; \end{equation*}

$\mathcal{H} _ { uc } ^ { \infty } ( B _ { E } ) \equiv$

\begin{equation*} \{ f \in \mathcal{H} ^ { \infty } ( B _ { E } ) : f \ \text { uniformly continuous on } B _ { E } \}. \end{equation*}

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 $\mathcal{A}$ of analytic functions, let $\mathcal{M} ( \mathcal{A} )$ denote the set of homomorphisms $\phi : \mathcal{A} \rightarrow \mathbf{C}$. Since the Michael problem has an affirmative solution [a5], every homomorphism is automatically continuous. For each such $\phi$, define $\Pi ( \phi ) \equiv \phi | _ { E ^{ *}} \subset E ^ { * * }$ (noting that, always, $E ^ { * } \subset \mathcal{A}$). Basic topics of interest here are the relation between the "fibres" $\Pi ^ { - 1 } ( w )$, $w \in E ^ { * * }$, and the relation between the geometry of $E$ and of $\mathcal{M} ( \mathcal{A} )$.

The spectrum $\mathcal{M}$ displays very different behaviour in the infinite-dimensional setting, in comparison with the finite-dimensional situation. As an illustration, every element $z \in E ^ { * * }$ corresponds to a homomorphism on $\mathcal{H} _ { b } ( E )$. Indeed, for each $n$ there is a linear extension mapping from $\mathcal{P} ( \square ^ { n } E ) \rightarrow \mathcal{P} ( \square ^ { n } E ^ { * * } )$. Applying this mapping to the Taylor series of a holomorphic function yields a multiplicative linear extension operator, mapping $f \in \mathcal{H} _ { b } ( E )$ to $\tilde { f } \in {\cal H} _ { b } ( E ^ { * * } )$; similar results hold for $\mathcal{A} = H ^ { \infty } ( B _ { E } )$ and $\mathcal{A} = \mathcal{H} _ { uc } ^ { \infty } ( B _ { E } )$. For example, each $z \in E ^ { * * }$ yields an element of $\mathcal{M} ( \mathcal{H} _ { b } ( E ) )$ via $\tilde { \delta _ { z } } : f \in \mathcal{H} _ { b } ( E ) \rightarrow \tilde { f } ( z ) \in \mathbf{C}$. A complete description of $\mathcal{M} ( \mathcal{H} _ { b } ( E ) )$ is unknown (1998) for general $E$, although it is not difficult to see that $\mathcal{M} ( \mathcal{H} _ { b } ( c _ { 0 } ) ) = \{ \widetilde { \delta _ { z } } : z \in \operatorname{l} _ { \infty } \}$. The question of whether the fourth dual of $E$ also provides points of the spectrum is connected with Arens regularity of $E$ [a7]. In any case, $\mathcal{M} ( \mathcal{H} _ { b } ( E ) )$ can be made into a semi-group with identity $\delta _ { 0 }$; the commutativity of this semi-group is related, once again, to Arens regularity of $E$ [a6].

It is natural to look for analytic structure in the spectrum $\mathcal{M} ( \mathcal{H} ^ { \infty } ( B _ { E } ) )$. In fact, every fibre $\mathcal{M} _ { z } \equiv \Pi ^ { - 1 } ( z )$ over $z \in \overline { B } _ { E ^{* *}}$ contains a copy of $( \beta \mathbf{N} \backslash \mathbf{N} ) \times \Delta$. In many situations, e.g. when $E$ is super-reflexive (cf., also Reflexive space), there is an analytic embedding of the unit ball of a non-separable Hilbert space into $\mathcal{M} _ { 0 }$. Further information has been obtained by J. Farmer [a8], who has studied analytic structure in fibres in $\text{I} _ { p }$-spaces. However, note that there is a peak set (cf. also Algebra of functions) for $\mathcal{H} ^ { \infty } ( B _ { \text{l}_p } )$ which is contained in $\mathcal{M} _ { 0 }$.

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

reflexivity of $\mathcal{P} ( \square ^ { n } E )$;

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].

References

[a1]	S. Dineen, "Complex analysis in localy convex spaces" , North-Holland (1981)
[a2]	S. Dineen, "Complex analysis on infinite dimensional spaces" , Springer (1999)
[a3]	J. Mujica, "Complex analysis in Banach spaces" , North-Holland (1986)
[a4]	T. Gamelin, "Analytic functions on Banach spaces" , Complex Potential Theory (Montreal 1993) , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 439 , Kluwer Acad. Publ. (1994) pp. 187–233
[a5]	B. Stensones, "A proof of the Michael conjecture" preprint (1999)
[a6]	R. Aron, B. Cole, T. Gamelin, "Spectra of algebras of analytic functions on a Banach space" J. Reine Angew. Math. , 415 (1991) pp. 51–93
[a7]	R. Aron, P. Galindo, D. Garcia, M. Maestre, "Regularity and algebras of analytic functions in infinite dimensions" Trans. Amer. Math. Soc. , 384 : 2 (1996) pp. 543–559
[a8]	J. Farmer, "Fibers over the sphere of a uniformly convex Banach space" Michigan Math. J. , 45 : 2 (1998) pp. 211–226