#### by John Wermer

#### Introduction

I first came to know Andy Gleason in the early 1950s. I found him friendly, natural, and interesting. Of course, I knew that his work had recently led to the solution of Hilbert’s Fifth Problem. One thing that impressed me strongly about Andy was that he understood, in detail, every colloquium we attended independently of the subject matter.

A link between the Gleason and Wermer families at that time was Philip, Jean and Andy’s Siamese cat. I was a visitor at Harvard in 1959–60, and Andy was going abroad for that year. We rented their apartment. They asked us to take care of Philip for the year, which my two boys and my wife, Christine, and I were happy to do. When spring 1960 came and we knew we should soon have to surrender Philip, it turned out that the Gleasons would not be able to keep him and asked us whether we would take him along to Providence. We accepted with a whoop and a holler. We called him Philip Gleason, and he became a much-valued member of our household. Philip often disappeared for days, but always returned, thinner and wiser, and definitely had more than nine lives.

A mathematical link between Andy and me came out of the former Soviet Union. Gelfand and Silov had recently started a study of commutative Banach algebras and their maximal ideal spaces, and this theory was intimately related to the theory of holomorphic functions. This area aroused the strong interest of a group of young American mathematicians. Andy Gleason was a prominent member of this group and made fundamental contributions to this field of study.

Let __\( \mathcal{A} \)__ be a commutative semisimple Banach algebra with
unit, and let __\( \mathcal{M} \)__ be the space of all maximal ideals of
__\( \mathcal{A} \)__. Gelfand
[e1]
showed that __\( \mathcal{M} \)__ can be endowed
with a topology which makes it a compact Hausdorff space such that
there is an isomorphism
__\( :f\to\hat{f} \)__ which maps __\( \mathcal{A} \)__ to a subalgebra of
the algebra of all continuous functions on __\( \mathcal{M} \)__. Silov
[e2]
then showed
that there exists a minimal closed subset __\( \check{S} \)__ of __\( \mathcal{M} \)__ such that to every
__\( f \)__ in __\( \mathcal{A} \)__ and each point __\( m \)__ in __\( \mathcal{M} \)__ we have the inequality
__\begin{equation}
\label{eqon}
|\hat{f} (m)| \leq \max_{s\in\check{S}}|\hat{f}(s)|.
\end{equation}__
__\( \check{S} \)__ is called the Silov boundary of __\( \mathcal{M} \)__.

The star example of all this is given by the “disk algebra” __\( A \)__,
consisting of all continuous functions on the unit circle __\( \Gamma \)__ which
admit analytic continuation to the open unit disk. We take __\( \|f \| = \max
|f| \)__ taken over __\( \Gamma \)__ for __\( f \)__ in __\( A \)__. Here __\( \mathcal{M} \)__ can be identified with the
closed unit disk __\( \Delta \)__, and __\( \check{S} \)__ becomes the topological boundary of __\( \Delta \)__. For
__\( f \)__ in __\( A \)__, __\( \hat{f} \)__ is the analytic continuation of __\( f \)__ to the interior of __\( \Delta \)__.

Another key example is provided by the bidisk algebra __\( A_2 \)__ which
consists of all functions continuous on the closed bidisk __\( \Delta_2 \)__ in __\( \mathbb{C}^2 \)__
which are holomorphic on the interior of __\( \Delta_2 \)__. The maximal ideal space
of __\( \mathcal{M} \)__ can be identified with __\( \Delta_2 \)__; the Silov boundary is not the
topological boundary of __\( \Delta_2 \)__, but instead the torus __\( T^2 : |z | = 1 \)__, __\( |w | = 1 \)__.

Classical function theory gives us, in the case of
the disk algebra, not only the maximum principle __\eqref{eqon}__ but also the local maximum principle:
For every __\( f \)__ in __\( A \)__, if __\( z_0 \)__ lies in the open unit disk and
__\( U \)__ is a compact neighborhood of __\( z_0 \)__ contained in
__\( \operatorname{int}\Delta \)__ then
__\begin{equation}
\label{eqtw}
|f (z_0)| \leq \max_{z\in\partial U} |f (z)|.
\end{equation}__

It is a fundamental fact, proved by
Rossi
in
[e6],
that the analogue of __\eqref{eqtw}__ holds in general. We have

__(Local maximum modulus principle)__Fix a point

__\( m \)__in

__\( \mathcal{M}\backslash\check{S} \)__and fix a compact neighborhood

__\( U \)__of

__\( m \)__in

__\( \mathcal{M}\backslash\check{S} \)__. Then we have for each

__\( f \)__in

__\( \mathcal{A} \)__

__\begin{equation} \label{eqth} |\check{f} (m)| \leq \max_{u\in\partial U} |\check{f} (u)|. \end{equation}__

This result suggests that for an arbitrary __\( A \)__,
where __\( M\backslash \check{S} \)__ is nonempty, we should look for some
kind of analytic structure in __\( M\backslash \check{S} \)__. In the 1950s
Gleason set out to find such analytic structure.
He focused on a class of Banach algebras he called
“function algebras”.

Let __\( X \)__ be a compact Hausdorff space. The
algebra __\( C (X ) \)__ of all continuous complex-valued
functions on __\( X \)__, with __\( \|f \| = \max|f| \)__ over __\( X \)__, is a
Banach algebra. A closed subalgebra __\( \mathcal{A} \)__ of __\( C (X ) \)__
which separates the points of __\( X \)__ and contains the
unit is called a “function algebra” on __\( X \)__. It inherits
its norm from __\( C (X ) \)__.

Let __\( \mathcal{M} \)__ be the maximal ideal space of __\( \mathcal{A} \)__. Then __\( X \)__
is embedded in the compact space __\( \mathcal{M} \)__ as a closed
subset, and each __\( f \)__ in __\( \mathcal{A} \)__ has __\( \hat{f} \)__ as a continuous
extension to __\( \mathcal{M} \)__.

#### Parts

__\( \mathcal{A} \)__be a function algebra on the space

__\( X \)__, with maximal ideal space

__\( \mathcal{M} \)__. Fix a point

__\( m \)__in

__\( \mathcal{M} . \)__The map:

__\( f\to\hat{f}(m) \)__is a bounded linear functional on

__\( \mathcal{A} \)__. We use this map to embed

__\( \mathcal{M} \)__into

__\( \mathcal{A}^{\star} \)__, the dual space of

__\( \mathcal{A} \)__.

__\( \mathcal{M} \)__then lies in the unit ball of

__\( \mathcal{A}^{\star} \)__.

Hence, if __\( m \)__ and __\( m^{\prime} \)__ are two points in __\( \mathcal{M} \)__,
__\( \|m - m^{\prime}\| \leq 2 \)__. Gleason
[1]
defined a relation on the
points of __\( \mathcal{M} \)__ by writing
__\( m \bullet m^{\prime} \)__ if __\( \|m - m^{\prime} \| < 2 \)__.
He proved:

__\( \bullet \)__” is an equivalence relation on

__\( \mathcal{M} \)__.

Note: At first sight, this proposition is counterintuitive,
since __\( m \bullet m^{\prime} \)__ and __\( m^{\prime} \bullet m^{\prime\prime} \)__ are equivalent
to __\( \|m - m^{\prime}\| < 2 \)__ and __\( \|m^{\prime} - m^{\prime\prime} \| < 2 \)__. The triangle
inequality for the norm yields __\( \|m - m^{\prime\prime} \| < 4 \)__,
whereas we need __\( \|m - m^{\prime\prime} \| < 2 \)__.

For each __\( \mathcal{A} \)__ the space __\( \mathcal{M} \)__ splits into equivalence
classes under __\( \bullet \)__. Gleason called these equivalence
classes the “parts” of __\( \mathcal{M} \)__.

Observe what these parts look like when __\( \mathcal{A} \)__ is the bidisk
algebra __\( A_2 \)__. Here __\( \mathcal{M} \)__ is the closed unit bidisk __\( \Delta_2
: |z | \leq 1 \)__, __\( |w | \leq 1 \)__. Some calculation gives the following:
the interior of __\( \Delta_2 \)__, __\( |z | < 1 \)__, __\( |w | < 1 \)__, is a single part.
Each of the disks __\( (e^{i t} , w )\mid 0 \leq t \leq 2\pi \)__, __\( |w | < 1 \)__, __\( (z
, e^{i s} ) \)__, __\( |z | < 1 \)__, __\( 0 \leq s \leq 2\pi \)__ is a part of
__\( \mathcal{M} \)__. Finally, each point __\( (\exp(it), \exp(is)) \)__, __\( s,t \)__ real,
is a one-point part lying on the torus __\( |z | = 1 \)__, __\( |w | = 1 \)__. Thus
__\( \mathcal{M} \)__ splits into the pieces: one analytic piece of complex
dimension 2, two families of analytic pieces of complex dimension 1,
and uncountably many one-point parts on the Silov boundary of the
algebra.

In complete generality, Andy’s hopes that for
each function algebra the parts of __\( \mathcal{M} \)__ would provide
analytic structure of the complement of the
Silov boundary were not fully realized.
Stolzenberg,
in
[e10],
gave an example of a function algebra
__\( \mathcal{A} \)__ such that the complement of the Silov boundary
of __\( \mathcal{A} \)__ in __\( \mathcal{M} \)__ is nonempty but contains no analytic
structure. However, an important class of Banach
algebras, the so-called “Dirichlet algebras”, and
their generalizations behaved as Andy had hoped.
We turn to these algebras in the next section.

#### Dirichlet algebras

Let __\( X \)__ be a compact Hausdorff space and let __\( \mathcal{A} \)__ be
a function algebra on __\( X \)__. In
[1],
Gleason made the
following definition: __\( \mathcal{A} \)__ is a Dirichlet algebra on __\( X \)__
if __\( \operatorname{Re} (\mathcal{A}) \)__, the space of real parts of the functions
in __\( \mathcal{A} \)__, is uniformly dense in the space __\( C_R (X ) \)__ of all
real continuous functions on __\( X \)__.

The name “Dirichlet” was chosen by Gleason
because in the case when __\( \mathcal{A} \)__ is the disk algebra __\( A \)__,
this density condition is satisfied and has as a consequence
the solvability of the Dirichlet problem
for harmonic functions on the unit disk.

He stated, “It appears that this class of algebras is of considerable importance and is amenable to analysis.” This opinion was born out by developments.

A typical Dirichlet algebra is the disk algebra __\( A \)__
on the circle __\( \Gamma \)__. By looking at __\( A \)__ we are led to the
basic properties of arbitrary Dirichlet algebras. __\( A \)__
has the following properties:

- For each point
__\( z \)__in__\( \Delta \)__, there exists a unique probability measure__\( \mu_z \)__on__\( \Gamma \)__such that for all__\( f \)__in__\( A \)____\[ f (z ) = \int^{\pi}_{-\pi} f (\exp(it))\,d \mu_z, \]__ __\( \mu_z=\frac{1}{2\pi}p_zj\,dt \)__, where__\( p_z \)__is the Poisson kernel at__\( z \)__unless__\( |z | = 1 \)__, and then__\( \mu_z \)__is the point mass at__\( z \)__.

He proved in [1]:

__\( \mathcal{A} \)__be a Dirichlet algebra on the space

__\( X \)__, and let

__\( \mathcal{M} \)__be its maximal ideal space.

__(i \( ^{\prime} \))__ Fix

__\( m \)__in

__\( \mathcal{M} \)__. There exists a unique probability measure

__\( \mu_m \)__on

__\( X \)__such that

__\[ \hat{f} (m ) =\int_X f \,d \mu_m, \quad \text{for all }f \text{ in } A. \]__

__(ii \( ^{\prime} \))__ Fix points

__\( m \)__and

__\( m^{\prime} \)__in

__\( \mathcal{M} \)__. Then

__\( m \)__and

__\( m^{\prime} \)__lie in the same part of

__\( \mathcal{M} \)__if and only if the measures

__\( \mu_m \)__and

__\( \mu_{m^{\prime}} \)__are mutually absolutely continuous. In this case, the corresponding Radon–Nikodym derivative is bounded above and below on

__\( X \)__.

Note: For __\( m \)__ in __\( \mathcal{M} \)__, __\( \mu_m \)__ is called “the representing
measure for __\( m \)__”.

It turned out that when __\( \mathcal{A} \)__ is a Dirichlet algebra
with maximal ideal space __\( \mathcal{M} \)__, then each part of __\( \mathcal{M} \)__ is
either a single point or an analytic disk. Explicitly,
it is proved in Wermer
[e7]:

__\( \mathcal{A} \)__be a Dirichlet algebra with maximal ideal space

__\( \mathcal{M} \)__. Let

__\( \Pi \)__be a part of

__\( \mathcal{M} \)__. Then either

__\( \Pi \)__consists of a single point or there exists a continuous one-one map

__\( \tau \)__of the open unit disk onto

__\( \Pi \)__such that for each

__\( f \)__in

__\( \mathcal{A} \)__the composition

__\( \hat{f}\circ \tau \)__is holomorphic on the unit disk.

#### Examples

*Example 1*: Let __\( K \)__ be a compact set in the complex
plane __\( C \)__ with connected complement, and let
__\( X \)__ be the boundary of __\( K \)__. The uniform closure __\( P (X ) \)__
of polynomials on __\( X \)__ is a Dirichlet algebra on __\( X \)__.

*Example 2*: Fix __\( \alpha > 0 \)__. __\( A_{\alpha} \)__ denotes the space of
all continuous functions __\( f \)__ on the torus __\( T^2 \)__ consisting
of all points __\( (e^{i \theta} , e^{i\phi} ) \)__ in __\( C^2 \)__ such that __\( f \)__ has the
Fourier expansion on __\( T^2 \)__:
__\[
\sum_{n+m\alpha\geq 0} c_{nm}e^{in\theta}e^{im\phi}.
\]__

These algebras are studied by
Helson
and
Lowdenslager
in
[e4]
and by
Arens
and
Singer
in
[e3].
Each __\( A_{\alpha} \)__ is a Dirichlet algebra on __\( T^2 \)__.

*Example 3*: Let __\( \gamma \)__ be an arc on the Riemann
sphere __\( S \)__. Let __\( B (\gamma ) \)__ denote the algebra of all continuous
functions on __\( \gamma \)__ which have a continuous
extension to the full sphere __\( S \)__ which is holomorphic
on __\( S \)__ outside of __\( \gamma \)__. For a certain class of arcs,
studied by
Browder
and Wermer in
[e11],
__\( B (\gamma ) \)__ is a
Dirichlet algebra on __\( \gamma \)__.

It turned out that substantial portions of the theory of Hardy spaces
__\( H^p \)__ on the unit disk have natural generalizations when the disk
algebra is replaced by an arbitrary Dirichlet algebra.This was pointed
out by
Bochner
in
[e5]
in a slightly different context. It was carried
out in
[e4]
for Example 2, and in an abstract context by various
authors. (See
Gamelin
[e13].)

Further, Hoffman in [e9] introduced a generalization of Dirichlet algebras, called “logmodular algebras”, to which the theory of Dirichlet algebras has a natural extension. In particular, parts of the maximal ideal space of such an algebra are either points or disks.

Let __\( H^{\infty} \)__ denote the algebra of all bounded analytic functions on the
unit disk, with __\( \|f \| = \sup |f | \)__, taken over the unit disk. Then
__\( H^{\infty} \)__ is a Banach algebra. Let __\( X \)__ denote the Silov boundary of
this algebra. The restriction of __\( H^{\infty} \)__ to __\( X \)__ is a function
algebra on __\( X \)__. This restriction is not a Dirichlet algebra on __\( X \)__,
but it is a log-modular algebra on __\( X \)__. By what was said above, the
parts of the maximal ideal space of __\( H^{\infty} \)__ are points or analytic disks.

Let __\( M \)__ be the maximal ideal space of __\( H^{\infty} \)__, taken with the Gelfand
topology. __\( M \)__ is compact and contains the open unit disk __\( D \)__ as a subset.
Lennart Carleson
proved in 1962 the so-called Corona Theorem, which
implies that __\( D \)__ is dense in __\( M \)__. The question had arisen earlier as to
the (possible) analytic structure in the complement __\( M \backslash D \)__.

Partial results on this question were obtained in 1957 by a group of people talking at a conference, and this result was published under the pseudonym “I. J. Schark”1 in the paper [e8]. Hoffman and Gleason were prominent participants in this enterprise.

#### Gleason’s problem

__\( \mathcal{A} \)__be a function algebra and

__\( \mathcal{M} \)__be its maximal ideal space. Fix a point

__\( m_0 \)__in

__\( \mathcal{M} \)__. As a subset of

__\( \mathcal{A} \)__,

__\( m_0 \)__is the set of

__\( f \)__such that

__\( \hat{f} (m_0 ) = 0 \)__. We ask: when does

__\( m_0 \)__have a neighborhood in

__\( \mathcal{M} \)__which carries structure of a complex-analytic variety? By this we mean the following: there exists a polydisk

__\( \Delta^{n} \)__in

__\( \mathbb{C}^n \)__and an analytic variety

__\( V \)__in

__\( \Delta^n \)__, and there exists a homeomorphism

__\( \tau \)__of a neighborhood

__\( \mathcal{N} \)__of

__\( m_0 \)__on

__\( V \)__such that for all

__\( f \)__in

__\( \mathcal{A} \)__the composition of

__\( \hat{f} \)__with the inverse of

__\( \tau \)__has an analytic extension from

__\( V \)__to

__\( \Delta^n \)__.

Gleason proved the following in [2]:

__\( \mathcal{A} \)__,

__\( \mathcal{M} \)__,

__\( m_0 \)__be as above. Assume that

__\( m_0 \)__, as an ideal in

__\( \mathcal{A} \)__, is finitely generated (in the sense of algebra). Then there exists a neighborhood

__\( \mathcal{N} \)__of

__\( m_0 \)__which has the structure of a complex-analytic variety.

This result leads naturally to the following question, raised by Gleason:

Let __\( D \)__ be a bounded domain in __\( \mathbb{C}^n \)__ and denote
by __\( A(D) \)__ the algebra of continuous functions on
the closure of __\( D \)__ which are analytic on __\( D \)__. Fix a
point __\( a = (a_1 ,\dots, a_n) \)__ in __\( D \)__. Given __\( f \)__ in __\( A(D ) \)__ with
__\( f (a ) = 0 \)__, do there exist functions __\( g_1 ,\dots, g_n \)__ in
__\( A(D) \)__ such that __\( f (z ) =\sum^n_{j =1} (z_j - a_j )g_j (z) \)__ for
every __\( z \)__ in __\( D \)__?

It is now known that the answer is yes if __\( D \)__
is a strictly pseudo-convex domain in __\( \mathbb{C}_n \)__. A history
of the problem is given by
Range
in
([e14],
Chapter VII, paragraph 4).