Celebratio Mathematica

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 $$\begin{array}{}\text{(1)}& |\hat{f}(m)|\le \underset{s\in \check{S}}{max}|\hat{f}(s)|.\end{array}$$ $\check{S}$ 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 $\mathrm{\Gamma }$ which ad­mit ana­lyt­ic con­tinu­ation to the open unit disk. We take $\parallel f\parallel =max|f|$ taken over $\mathrm{\Gamma }$ for $f$ in $A$. Here $\mathcal{M}$ can be iden­ti­fied with the closed unit disk $\mathrm{\Delta }$, and $\check{S}$ be­comes the to­po­lo­gic­al bound­ary of $\mathrm{\Delta }$. For $f$ in $A$, $\hat{f}$ is the ana­lyt­ic con­tinu­ation of $f$ to the in­teri­or of $\mathrm{\Delta }$.

An­oth­er key ex­ample is provided by the bid­isk al­gebra $A_2$ which con­sists of all func­tions con­tinu­ous on the closed bid­isk ${\mathrm{\Delta }}_2$ in ${\mathbb{C}}^2$ which are holo­morph­ic on the in­teri­or of ${\mathrm{\Delta }}_2$. The max­im­al ideal space of $\mathcal{M}$ can be iden­ti­fied with ${\mathrm{\Delta }}_2$; the Silov bound­ary is not the to­po­lo­gic­al bound­ary of ${\mathrm{\Delta }}_2$, 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 $\text{(1)}$ 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 $\mathrm{int}\mathrm{\Delta }$ then $$\begin{array}{}\text{(2)}& |f(z_0)|\le \underset{z\in \partial U}{max}|f(z)|.\end{array}$$

It is a fun­da­ment­al fact, proved by Rossi in [e6], that the ana­logue of $\text{(2)}$ holds in gen­er­al. We have

This res­ult sug­gests that for an ar­bit­rary $A$, where $M\mathrm{\setminus }\check{S}$ is nonempty, we should look for some kind of ana­lyt­ic struc­ture in $M\mathrm{\setminus }\check{S}$. 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 $\parallel f\parallel =max|f|$ 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 $\mathcal{M}$.

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

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

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 ${\mathrm{\Delta }}_2:|z|\le 1$, $|w|\le 1$. Some cal­cu­la­tion gives the fol­low­ing: the in­teri­or of ${\mathrm{\Delta }}_2$, $|z|<1$, $|w|<1$, is a single part. Each of the disks $(e^{it},w)\mid 0\le t\le 2\pi$, $|w|<1$, $(z,e^{is})$, $|z|<1$, $0\le s\le 2\pi$ 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.

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 $\mathrm{Re}(\mathcal{A})$, 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 $\mathrm{\Gamma }$. By look­ing at $A$ we are led to the ba­sic prop­er­ties of ar­bit­rary Di­rich­let al­geb­ras. $A$ has the fol­low­ing prop­er­ties:

1. For each point $z$ in $\mathrm{\Delta }$, there ex­ists a unique prob­ab­il­ity meas­ure ${\mu }_z$ on $\mathrm{\Gamma }$ such that for all $f$ in $A$ $$f(z)={\int }_{-\pi }^{\pi }f(\exp (it))d{\mu }_z,$$
2. ${\mu }_z=\frac{1}{2\pi }{p}_{z}jdt$, 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]:

(i${}^{\prime }$) Fix $m$ in $\mathcal{M}$. There ex­ists a unique prob­ab­il­ity meas­ure ${\mu }_m$ on $X$ such that $$\hat{f}(m)={\int }_{X}fd{\mu }_m,\text{for all }f\text{ in }A.$$

(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]:

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 $T^2$: $$\sum _{n+m\alpha \ge 0}{c}_{nm}{e}^{in\theta }{e}^{im\phi }.$$

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 $H^{\mathrm{\infty }}$ de­note the al­gebra of all bounded ana­lyt­ic func­tions on the unit disk, with $\parallel f\parallel =sup|f|$, taken over the unit disk. Then $H^{\mathrm{\infty }}$ is a Banach al­gebra. Let $X$ de­note the Silov bound­ary of this al­gebra. The re­stric­tion of $H^{\mathrm{\infty }}$ 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 $H^{\mathrm{\infty }}$ are points or ana­lyt­ic disks.

Let $M$ be the max­im­al ideal space of $H^{\mathrm{\infty }}$, 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 $M\mathrm{\setminus }D$.

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 proved the fol­low­ing in [2]:

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 $f(z)=\sum _{j=1}^n(z_j-a_j)g_j(z)$ 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).