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{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

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 \( \mathcal{M} \).

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:

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.

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]:

(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]:

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