Allen Shields*

Banach Algebras, 1939-1989

In a remarkable series of three notes written fifty years ago in the Doklady Akad. Nauk, I. M. Gel'fand [1939] sketched the theory of commutative Banach algebras together with its principal applications. This was ap- parently his doctoral dissertation (in the bibliography of Silov [19401] the dissertation title is given as: The theory of normed rings). According to Mat XL, vol. II, Gel'fand was born 20 August 1913 in the Odessa re- gion; he received the degree of Candidate in 1935 and the degree of Doctor in 1940. The Candidate of Science degree is equivalent to an American Ph.D., while the Doctorate is a more advanced degree. As happens only very rarely in mathematics, this new theory was essentially the creation of one person.

Just preceding these Doklady notes, there was a joint paper by Gel'fand and Kolmogorov [1939] entitled On rings of continuous functions on topological spaces. In it they prove that if S 1 and S 2 are completely regular spaces, and if their rings of real-valued continuous functions are algebraically isomorphic as rings, then S 1 and S 2 are homeomorph ic . The proof begins by showing that there is a one-to-one correspondence be- tween the maximal ideals in the ring of functions and the points in the underlying space. The space is recov- ered by introducing a suitable topology on the set of maximal ideals. Thus one has a hint of the Gel'fand theory of commutative Banach algebras in this context.

A word about terminology. Gel'fand always spoke of "normed rings," although they were really algebras over the field of complex numbers. In Naimark [1956] they are called "Banach rings." The term "Banach al- gebra" seems to have been used first by Ambrose [1945].

In the first of Gel'fand's three notes the basic defini- tions and theorems are stated; no proofs are given. Eight theorems are stated, the last being the following theorem on automatic continuity: Let A,B be complete (commutative) normed rings, B having the additional property that the intersection of all its maximal ideals is {0}. Then any homomorphism of A onto B is contin- uous. (The non-commutative analog of this is much harder; it was finally proved by B. E. Johnson [1967].)

* C o l u m n editor's address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003 USA

The second note proves the theorem that if an ele- ment never vanishes (when viewed as a function on the space of maximal ideals), then it has an inverse. This is used to prove Wiener's theorem that if an abso- lutely convergent trigonometric series never vanishes, then the reciprocal has an absolutely convergent series. (Wiener [1932] proved this originally as a lemma needed in the proof of his Tauberian theorem.) Gel'fand points out that the method can also be used for absolute convergence with weights and for abso- lutely convergent Fourier-Stieltjes integrals on the line (thereby generalizing a result of Wiener and Pitt). He also considers absolutely convergent series, with weights, on general abelian groups (with the discrete topology). This paper had a very great impact on mathematicians the world over and helped lead to the rapid acceptance and development of the new theory.

The third note deals with the ring of almost periodic functions in the sense of Harald Bohr. Gel'fand proves that this ring is isometrically isomorphic to the ring of all continuous functions on the (compact) group of characters of the additive group of real numbers (with the discrete topology), nowadays often called the Bohr group.

Gel'fand [1941] contains the proofs of the theorems announced in the first note, together with the func- tional calculus for analytic functions of one complex variable. In the definition of a normed ring he assumes that there is an element e # 0 that is a multiplicative identity, and that multiplication is continuous in each variable separately. By regarding the elements as mul- tiplication operators on the underlying Banach space and using the operator norm, he shows that there is an equivalent norm in which Ji e II = 1 and II xy I] ~ II x It II y II for all x, y.

This paper runs from page 3 to page 24 in the Matem. Sbornik. Immediately following it are four ad- ditional papers by Gel'fand, the first one jointly with ~ilov (pages 25-39). It is on different ways to topolo- gize the set of maximal ideals. It also discusses the embedding of a completely regular topological space S in a compact space, by means of the maximal ideal space of the ring of bounded continuous real functions on S.

The next paper is on ideals and primary ideals in

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 3 �9 1989 Springer-Verlag New York 15

normed rings (pages 41-47). A generalized nilpotent (or quasinilpotent) element in a normed ring is an ele- ment x such that II x" It TM ~ 0. (This is equivalent to saying that x, viewed as a function on the space of maximal ideals, is the constant function 0.) The first theorem states that if x is a generalized nilpotent and if [[ (e - ~x) -1 [I <~ c(r/cos~)', ~ = rei% q~ ~ ~r/2, then x "+1 = 0. The proof uses a Phragm6n-Lindel6f theorem in a form due to Nevanlinna. Then Gel'fand gives criteria for each closed ideal to be the intersection of maximal ideals. A primary ideal is defined to be a closed ideal contained in only one maximal ideal. For a given non- negative integer n, the last theorem gives a criterion, for a special class of normed rings, that each maximal ideal contain at most n primary ideals.

Next comes a paper (pages 49-50) entitled On the theory of characters of abelian topological groups.

THEOREM 1. If y is a generalized nilpotent, if x = e - y, and if ll x" l] <<- M for n = 0 ,+-1 ,+2 . . . . , then y = O.

Note that x is invertible since, viewed as a function on the maximal ideal space, it is the constant function 1 and thus never vanishes. The proof uses the first theorem of the preceding paper (see above). This theorem attracted a lot of attention, and generaliza- tions were published by Hille [1944] and Stone [1948]. Hille showed that the conclusion remains true under the weaker hypothesis that n-lIr x, H ~ 0 (n ~ + ~). Stone generalized this further: yk = 0 if and only if II xn r[ = o(I n Ik). Also, if o is replaced by O then the condition is necessary and sufficient for yk+l = O. Recent improvements in this result will be found in Pytlik [1987]. Silov [1950] pointed out that all of these results could be obtained from the last theorem of the preceding paper of Gel'fand on bounding the number of primary ideals contained in a maximal ideal (see above). Gel'fand states that the method of proof of Theorem 1 will also show that if y is any element in a normed ring (y need not be a generalized nilpotent), and i f l l e + y ] ] = ] [ e - y l ] = 1, t h e n y = 0. Usingthis one can show that e is an extreme point of the unit ball.

THEOREM 2. Let R be a commutative normed ring and let G be a bounded subset that is a multiplicative group. Then for each two distinct elements x 1 and x 2 of G there is a con- t inuous character of G that takes different values at x 1 and x 2.

The last paper in this remarkable series (pages 51-66) is entitled On absolutely convergent trigonometric series and integrals. This paper expands the material in the second Doklady note of 1939 and provides full proofs.

The first phase of development of the theory was summarized in a long expository article, Gel'fand,

Raikov, and ~ilov (GRS) [1946] had written in 1940; publication was delayed by the war. Subsequently, GRS [1960], the article was expanded and rewritten as a book.

Among the major contributors to the theory was G. E. Silov. In particular he is remembered for the Silov boundary and the Silov idempotent theorem. For a commutative normed ring he proved the existence of a smallest closed subset (now called the ~ilov boundary) of the maximal ideal space with the property that every element in the ring, when viewed as a function on the maximal ideal space, attains its maximum mod- ulus on this set. This was published in GRS [1946], where it was credited to him. ~ilov showed that the boundary arises in another context. The Gel ' fand theory shows that multiplicative linear functionals on a commutative Banach algebra are automatically contin- uous and have norm 1. Further, there is a one-to-one correspondence between the maximal ideals and the kernels of multiplicative linear functionals. The Hahn- Banach theorem guarantees that continuous linear functionals can be extended from a closed subspace of a Banach space to the whole space, preserving the norm. However, if one algebra is contained as a closed subalgebra in another, with the same identity, there is no-guarantee that a multiplicative linear functional on the smaller algebra can be extended to be multiplica- tive on the larger algebra. Silov showed that multipli- cative functionals corresponding to maximal ideals in the boundary of the smaller algebra can always be so extended.

The ~ilov idempotent theorem states that the max- imal ideal space of a commutative Banach algebra is disconnected if and only if the identity is the sum of two nontrivial idempotents, whose product is 0. This happens if and only if the algebra is the direct sum of two closed ideals, each of which has an identity. In this case the maximal ideal space of the original al- gebra may be identified with the disjoint union of the maximal ideal spaces of the two subalgebras. This re- sult is in Silov [1953]. For the special case of a sym- metric algebra (to each x in the algebra there corre- sponds an element x* in the algebra such that, viewed as functions on the maximal ideal space, each is the complex conjugate of the other) the result had been obtained earlier by Gel'fand [1941].

In addition we mention two papers of Gel'fand and Naimark [1943] and [1948]. They proved that a normed ring with an involution can be isometrically *-isomorphically embedded into the ring of bounded linear transformations on Hilbert space. If the ring is commutative, then it is isometrically *-isomorphic to the ring of all continuous functions on the maximal ideal space. They needed the assumptions: J[ x*x il = Jp x* Jl II x tt, I1 x 1t = tl x* Pf, and e + x*x is invertible, for all x. They conjectured that the last two were conse- quences of the first. This was eventual ly sett led

16 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO, 3, 1989

a r o u n d 1960 by J. Gl imm, R. Kadison and T. Ono. (The reader had best consult MR 22 #5906, 5905, and 47 #2379 for bibliographic data and priorities.) The au- thor is indebted to R. B. Burckel for the reference to R. Doran and V. Belfi [1986]. In this beautiful book the reader will find a his tory together with the most effi- c i e n t m o d e r n p r o o f s of t he G e l ' f a n d - N a i m a r k theo rems u n d e r still weaker hypo theses , and page- size photos of the two men.

Gel'fand must have been a strong candidate for a Fields Medal on the basis of this work and his work on group representations.

Gel ' fand must have been a strong candidate for a Fields Medal on the basis of this work and his work on group representat ions. The first Fields medals were given (to L. Ahlfors and J. Douglas) at the Interna- tional Congress of Mathematicians in 1936. The next congress had been scheduled for 1940, but because of the war it was pos tponed , and was finally held in 1950 in Cambr idge , Massachuse t t s . Un fo r tuna t e ly , the Cold War was then in full swing, and this may have worked against Gel ' fand 's chances; at any rate, he did not receive a medal. Shortly before the opening of the Congress a te legram was received f rom the Soviet a c a d e m y stat ing tha t no ma thema t i c i ans f rom the USSR would at tend the Congress.

Gel ' fand was p romoted to professor at Moscow Uni- versity in 1943; in addit ion he had held a position at the Mathematics Institute of the Academy of Sciences since 1939. I. M. Yaglom told me that a n u m b e r of J e w i s h m a t h e m a t i c i a n s , i n c l u d i n g G e l ' f a n d a n d Yaglom, were dismissed from the university; I believe it was in 1949. The list was pos ted and ins tead of mere ly listing initials for the first name and patro- nymic these were wri t ten out in full: Israel Moiseevi~ Gel ' fand, Israel Moiseevi~ Yaglom. On the other hand, Gel ' fand was awarded a Stalin Prize in 1951, which was a high honor and included a substantial amoun t of money.

There is one more result about general commutat ive Banach algebras that we wish to ment ion, namely, the local maximum modulus principle of Rossi [1960]. By the Gel ' fand theory we may view the elements of the algebra as cont inuous functions on the maximal ideal space. If U is an open subset of this space, disjoint f rom the Silov boundary , then the maximum, on the closure of U, of the absolute value of each funct ion in the algebra is at tained on the boundary of U.

Subsequent work t ended to be on uniform algebras (closed subalgebras of the cont inuous functions on a compact Hausdorf f space, separating the points of the space and conta in ing the constants) . This is a rich theory wi th m a n y applicat ions, bu t that is ano the r story; see, for example, Gamelin [1969].

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