Celebratio Mathematica

Andrew Mattei Gleason

Gleason’s work on Banach Algebras

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 $$\label{eqon} |\hat{f} (m)| \leq \max_{s\in\check{S}}|\hat{f}(s)|.$$ $$\check{S}$$ 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 $$\label{eqtw} |f (z_0)| \leq \max_{z\in\partial U} |f (z)|.$$

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

The­or­em 1: (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 $$\mathcal{A}$$ $$\label{eqth} |\check{f} (m)| \leq \max_{u\in\partial U} |\check{f} (u)|.$$

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

Let $$\mathcal{A}$$ 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:

Pro­pos­i­tion: The re­la­tion “$$\bullet$$” 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:

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

The­or­em 2: Let $$\mathcal{A}$$ 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]:

The­or­em 3: Let $$\mathcal{A}$$ 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

Let $$\mathcal{A}$$ 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]:

The­or­em 4: Let $$\mathcal{A}$$, $$\mathcal{M}$$, $$m_0$$ 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).

Works

[1]A. M. Gleason: “Func­tion al­geb­ras,” pp. 213–​226 in Sem­inars on ana­lyt­ic func­tions (Prin­ceton, NJ, 1958), vol. 2. 1958. Zbl 0095.​10103 incollection

[2]A. M. Gleason: “Fi­nitely gen­er­ated ideals in Banach al­geb­ras,” J. Math. Mech. 13 : 1 (1964), pp. 125–​132. MR 0159241 Zbl 0117.​34105 article