View source for Banach space of analytic functions with infinite-dimensional domains

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 75 formulas, 75 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
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|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|Fréchet derivative]] at each point of $U$ (cf. also [[Algebra of functions|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 $\operatorname { dist } ( B , U ^ { c } ) &gt; 0$. The latter space is a [[Fréchet algebra|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|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 [[#References|[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|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|Arens regularity]] of $E$ [[#References|[a7]]]. In any case, $\mathcal{M} ( \mathcal{H} _ { b } ( E ) )$ can be made into a [[Semi-group|semi-group]] with identity $\delta _ { 0 }$; the commutativity of this semi-group is related, once again, to Arens regularity of $E$ [[#References|[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|Reflexive space]]), there is an analytic embedding of the unit ball of a non-separable [[Hilbert space|Hilbert space]] into $\mathcal{M} _ { 0 }$. Further information has been obtained by J. Farmer [[#References|[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|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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]; a recent (1998) very helpful source, with an extensive bibliography, is [[#References|[a4]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  S. Dineen,   "Complex analysis in localy convex spaces" , North-Holland  (1981)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Dineen,   "Complex analysis on infinite dimensional spaces" , Springer  (1999)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Mujica,   "Complex analysis in Banach spaces" , North-Holland  (1986)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  B. Stensones,   "A proof of the Michael conjecture"  ''preprint''  (1999)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J. Farmer,   "Fibers over the sphere of a uniformly convex Banach space"  ''Michigan Math. J.'' , '''45''' :  2  (1998)  pp. 211–226</td></tr></table>