Remembering Walter Rudin (1921–2010)

Alexander Nagel and Edgar Lee Stout, Coordinating Editors

alter Rudin, Vilas Professor Emeri- 1945. He entered Duke Uni- tus at the University of Wisconsin- versity, obtaining a B.A. in Madison, died on May 20, 2010, at 1947 and a Ph.D. in math- his home in Madison after a long ematics in 1949. He was battle with Parkinson’s disease. He a C. L. E. Moore Instructor W at the Massachusetts In- was born in Vienna on May 2, 1921. The Rudins were a well-established Jewish stitute of Technology and family which began its rise to prominence in began teaching at the Uni- the first third of the nineteenth century. By the versity of Rochester in 1830s, Walter’s great-grandfather, Aron Pollak, 1952. had built a factory to manufacture matches; he While on leave visiting also became known for his charitable activities, Yale in 1958, Rudin re- including the construction of a residence hall ceived a call from R. H. where seventy-five needy students at the Technical Bing at the University of University in Vienna could live without paying rent. Wisconsin-Madison, ask- As a result, Aron was knighted by Emperor Franz ing if he would be Joseph in 1869 and took the name Aron Ritter interested in teaching sum- Pollak von Rudin. The Rudin family prospered, and mer school. Rudin said Walter’s father, Robert, was a factory owner and that, since he had a Sloan Fellowship, he wasn’t inter- Photograph courtesy of Mary Ellen Rudin. electrical engineer, with a particular interest in Walter Rudin and sister, Vera, in ested in summer teaching. sound recording and radio technology. He married Vienna. Walter’s mother, Natalie (Natasza) Adlersberg, in Then, as he writes in his autobiography, As I Remember It, “my brain slipped 1920. Walter’s sister, Vera, was born in 1925. out of gear but my tongue kept on talking and I After the Anschluss in 1938, the situation for heard it say ‘but how about a real job?’ ” As a result, Austrian Jews became impossible, and the Rudin Walter Rudin joined the Department of Mathemat- family left Vienna. Walter served in the British ics at UW-Madison in 1959, where he remained Army and Navy during the Second World War, and until his retirement as Vilas Professor in 1991. rejoined his parents and sister in New York in late He and his wife, the distinguished mathematician Mary Ellen (Estill) Rudin, were popular teachers at Alexander Nagel is emeritus professor of mathematics at the University of Wisconsin-Madison. His email address is both the undergraduate and graduate level and [email protected]. served as mentors for many graduate students. Edgar Lee Stout is emeritus professor of mathematics at the They lived in Madison in a house designed by Frank University of Washington. His email address is stout@math. Lloyd Wright, and its intriguing architecture and washington.edu. two-story-high living room made it a center for DOI: http://dx.doi.org/10.1090/noti955 social life in the department.

March 2013 Notices of the AMS 295 296 Photograph courtesy of Mary Ellen Rudin. inrfnto ojcue e oatremendous a to led conjecture” function “inner atrsgetdta ahmtcasitouea introduce mathematicians that suggested Walter okrflce i lsia riigadfcsdon focused and training classical his reflected work lkado n ai-ioyLw(91,Walter (1981), Hakim-Sibony-Løw and Aleksandrov 17)ta h eost fdifferent of sets zero the that (1976) 90 hswsavr cieadpplrarea popular and active very a was this 1960s 90,adh ea owr npolm in problems on work to began he and 1960s, eyiflecdb h hnrltvl e td of study new relatively then the by influenced very aibe speetdi he fhsadvanced his of three in presented is variables one-dimensional the of generalization variable and unexplored, and new relatively was variables is is 1969, Polydiscs in in Theory published first, The books. complex several in work Rudin’s of this Much to question. by contributions important solution additional the made after and the research, on of work amount His different. all are functions of ball unit the two he and Walter (1967) example, least polydisk For ball. the both. at for showed the with are work and important There did polydisk several- be. the right should candidates: the disk what unit clear even not was it several in analysis complex of aspects of study analytic the the time that At variables. complex several book, 1962 Groups on his Analysis the in of career transforms mathematical Fourier the L on operate functions that Kahane, the Helson, characterized with which Katznelson, work and 1959 his in in was achievements area major began this Walter’s which seminar of group”, One analysis days. every Abelian those almost compact how locally is “Let a phrase be the replace G to jest, “lgbalcag”, in word, new partially only perhaps and and research, 1950s of late the In groups. locally on Abelian analysis compact harmonic of theory general 1956. the in algebra disk the in ideals closed characterization the complete of the is work Beurling, the Arne on of building area, his this of in One results important algebras. function and also algebras Banach was He variable. complex one holomorphic of and functions series trigonometric of study the 1 ucinTer nteUi alof Ball Unit the in Theory Function agba ui yteie hsapc fhis of aspect this synthesized Rudin -algebra. atrsitrsscagdaani h late the in again changed interests Walter’s was interest Walter’s of area major Another ui os,Madison. house, Rudin h eod ulse n1980, in published second, The . . okdi num- a in worked the of one was oec.Hsearly His each. to contributions jor ma- made he and analysis, mathe- matical of different areas of ber He generation. his of ematicians math- preeminent oie fteASVolume AMS the of Notices atrRudin Walter C n i okon work His . H p Function Fourier classes an tt olg,Nbak;EenrRdn an Rudin, Eleanor Nebraska; College, State Wayne inradGladhdpvdtewy tbecame It way. the paved had Gelfand and Wiener eea rmwr steGladter of theory Gelfand the is framework general A measures. Radon of algebras convolution is Another h oko atrRdno amncaayi is analysis harmonic on Rudin Walter of work The atrRdnadHroi Analysis Harmonic and Rudin Walter Kahane Jean-Pierre Natalie. and four Sofia, by Deniz, survived Adem, also grandchildren: is He Hopkins Baltimore. Johns in the University at oncology of Charles Minnesota; professor and Rudin, Wisconsin; Paul, Madison, St. of in Rudin Robert 3M for working engineer at linguistics and languages Rudin, modern Catherine of children: professor four his by survived is 2006. in Vienna University of the from degree honorary Exposition. an received Mathematical He Society’s for Prize Mathematical Steele P. Leroy American the awarded and Analysis three textbooks: his outstanding for students graduate and undergraduate of Ball Unit the 1986 in published as then were NSF- which of series lectures, a CBMS in summarized was functions inner c tteUiest ePrsSd i mi drs is address email His Paris-Sud. de mathemat- Université [email protected] of the professor at emeritus ics is Kahane Jean-Pierre game. expert this an playing was in Walter and be everywhere, can found be group definitions can Equivalent a group. dual on the on happens expressed what analysis Fourier of concerned, as sense is far the as in Moreover, nineteenth twentieth. modern the the and of abstract sense or and century the problems in same classical the are results: express there to ways speaking, several Generally algebras. Banach series. trigonometric is Fourier matter related of summable A or sequences. made functions algebras integrable of the transforms is, that algebras, the in century. analysis twentieth the harmonic of in middle analysis. tendency of main the the parts from some arising of results structure and algebraic questions cover add analysis. harmonic and of them glance piece a large of a but few at nothing a is celebrated This with complements. his start some of Fourier shall end on I papers the his book. at guide of groups A list mine. on the of is analysis also it and of life most his for of part good a nadto ohswdw ayEln atrRudin Walter Ellen, Mary widow, his to addition In of generations to known also is Rudin Walter ato atrswr el ihteWiener the with deals work Walter’s of Part dis- to was Walter for inspiration general The e osrcin fFntosHlmrhcin Holomorphic Functions of Constructions New ucinlAnalysis Functional (1953), C eladCmlxAnalysis Complex and Real n . rnilso Mathematical of Principles 17) n19 ewas he 1993 In (1973). 60 . Number , (1966), 3 Let me start with the first paper of his mentioned to Lp(G∗) (here G∗ is the in Fourier Analysis on Groups. The title is “Non dual group of G), with p < analytic functions of absolutely convergent Fourier 2 (depending on f ), then F series”, and the year was 1955 [1]. What he dis- is the restriction on (−1, 1) covered was a positive function with an absolutely of an analytic function in a convergent Fourier series, vanishing at 0, such that neighborhood of the closed its square root does not enjoy the same property. interval [−1, 1]. It was the first result of this type, but the question The main event in the do- was in the air, if in different forms. The Wiener-Lévy main after 1958 was the theorem asserts that an analytic function of a func- theorem of Malliavin on tion in the Wiener algebra A(T) (that is, its Fourier spectral synthesis in 1959. series converges absolutely) belongs to A(T). In Spectral synthesis can be ex- other words, the analytic functions operate on pressed in many ways, as A(T); that is, the convolution algebra l1(Z) has a well as nonspectral synthesis. symbolic calculus consisting of analytic functions. The contribution of Walter in Can we replace the analytic functions by a wider that subject was to exhibit a function f in A(G) such that class? The Wiener-Lévy theorem can be translated Photograph courtesy of Mary Ellen Rudin. into Banach algebras via the theory of Gelfand. The the ideals generated by the Rudin in 1956. n problem can be translated as well: which are the powers f are all different functions that operate in a given Banach algebra? (see [4]). Great progress was made on this question between The question on functions operating on A(R) 1955 and 1958. I proved that absolute values do not or A(T) goes back to Paul Lévy (1938) in the operate on the Wiener algebra, then that functions paper where the Wiener-Lévy theorem is stated. Paul Lévy asked a second question, on functions that operate are necessarily infinitely differentiable. operating “below”: which are the changes of Katznelson proved in 1958 the natural conjecture: variables preserving A(R)? The obvious example only analytic functions operate. That occurred in a is affine functions. very informal meeting in Montpellier just before Actually affine functions are the only ones; it the International Congress in Edinburgh; besides is a theorem of Beurling and Helson, published Katznelson and me, Helson, Herz, Rudin, Salem, in 1953. It opened a new and important field, the and others were there, enjoying life and discussing isomorphisms and endomorphisms of the group mathematics. Katznelson’s theorem immediately algebras. Walter entered the subject in 1956 with his had several versions, involving locally compact Acta Mathematica article [5] on the automorphisms abelian groups instead of T. Walter was interested and the endomorphisms of the group algebra of the in convolution algebras of measures or, what is the unit circle. Here is a typical result: a permutation of same, in multiplicative algebras of Fourier-Stieltjes the integers carries Fourier coefficients into Fourier transforms. What we proved in [2] is that only coefficients if and only if the permutation is equal entire functions operate. It is a way to recover to an obvious one, up to a finite number of places; many previous results, going back to the Wiener- an obvious one is a permutation p that satisfies Pitt phenomenon (the inverse of a Fourier-Stieltjes p(n−g)+p(n+g) = 2p(n) for some g. The general transform is not necessarily a Fourier-Stieltjes result extends both the Beurling-Helson and the transform, even when it is bounded), through the Rudin theorems; it deals with homomorphisms discoveries of Schreider in 1950 about the algebra of a group algebra into another group algebra of Fourier-Stieltjes transforms. Our results were and is due to Paul Cohen (1960). Here is a nice published in the form of a series of notes in the particular case, established by Rudin in 1958: the Comptes Rendus, but Walter was incredibly efficient group algebra of a locally compact abelian group in making them known among mathematicians: the G is isomorphic to that of the circle group T if and invited report he made at the Cambridge meeting only if G = T + F, where F is a finite abelian group. of the AMS in August 1958, “Measure Algebras on This question of homomorphism of group Abelian Groups”, contained them all. In the general algebras is linked with an apparently different form about locally compact abelian groups they question, the characterization of idempotent mea- are exposed in Fourier Analysis on Groups. sures. Here again Helson and Rudin paved the Walter’s last contribution to the subject [3] was way, and the final result was obtained by Paul an extension of Katznelson’s theorem, “A strong Cohen, proving a conjecture of Rudin (Cambridge converse of the Wiener-Lévy theorem” in 1962: meeting, 1958): The supports of the Fourier trans- if, for each given f in A(G) with values in the forms of idempotent measures (these Fourier interval (−1, 1), the composed function F(f ) is transforms take values 0 and 1) are the members a Fourier transform of a function which belongs of the “coset ring” of the group G∗, defined as

March 2013 Notices of the AMS 297 generated by all the main conjecture is still unsolved; it says that cosets of sub- Sidon sets are nothing but a finite union of quasi- groups of G∗ by independent sets, quasi-independent meaning that means of com- there is no linear relation with coefficients 1, −1, plementation and or 0 between its elements. Sidon sets are still a finite intersection. subject of interest, and the subject, including the There are a name of Sidon sets, was introduced by Rudin. number of other A question was raised by Rudin on the Λ(p) sets: results of Rudin There is a natural inclusion between the collections in Fourier analysis, of Λ(p) sets, since every Λ(p) is Λ(q) when q < p. about factoriza- Is this inclusion strict? The answer is negative for 1 n tion in L (R ), la- indices < 2: all Λ(p) are the same for 1 < p < 2. cunary sequences, When p is an even integer > 2, Rudin exhibited Rudin with Jaap Korvaar. 0 0 thin sets, weak a Λ(p) set that is not Λ(p ) for any p > p. Only almost periodic in 1989 was Bourgain able to extend this to all functions, posi- p > 2; therefore, the inclusion is strict for p > 2. tive definite se- The situation for p = 2 is not yet settled. quences, and abso- The Rudin sets on R or T are independent sets lutely monotonic over the rationals which carry measures whose functions (another Fourier transform tends to 0 at infinity. Because of example of oper- the Kronecker theorem, this cannot happen with ating functions). I countable sets. They are thin, but not too thin. The shall restrict my- construction of Rudin is clever (1960). It can be self to lacunary replaced by a random construction. I believe that I sequences and thin am responsible for the name of Rudin sets. sets; the name of I may be responsible also for the name of the Rudin is attached Rudin-Shapiro sequence. The description and the

Photos courtesy of Mary Ellen Rudin. to some of them. history are well described in the article [7] of Walter Rudin with Lipman Bers. The name of Rudin Rudin, “Some theorems on Fourier coefficients” of appears frequently in relation to automatic se- 1959. It is a beautiful and very useful automatic quences and their role in Fourier series; I shall sequence, and I used it as soon as Walter told me explain the use of Rudin-Shapiro sequences. about it. It answered a question asked by Raphaël Though lacunary sequences and thin sets can Salem in our informal Montpellier meeting. Salem be considered in general groups, let me restrict had overlooked the fact that Harold Shapiro had myself to the integers and the circle. already answered the question already in 1951. In 1957 Walter defined the Paley sequences Shapiro deserves recognition, and a few colleagues as sequences (n(k)) such that the coefficients of would prefer to change the name to Shapiro-Rudin. order n(k) of a function of the Hardy class H1 Likely it is too late. There are abuses of that sort in (the subspace of L1(T) generated by the imaginary all parts of mathematics, and they usually benefit exponentials with positive frequencies) belong strong and well-known mathematicians. to l2. The theorem of Paley is that Hadamard The name of Rudin will stay in the history of sequences (meaning n(k + 1)/n(k) > q > 1) are mathematics by the importance of his contributions Paley. Obviously this extends to finite unions of to many parts of analysis and by his exceptional Hadamard sequences. The theorem of Rudin is talent for exposition, in his articles as well as in that it is a characterization of Paley sequences. his books. Fourier analysis corresponds to part The Sidon sets have very many definitions, and of his life, but only to a part. His whole life as a they were studied by Rudin in the important paper mathematician and as a human being deserves to of 1960 “Trigonometric series with gaps” [6]. He be known. introduced the Λ(p) sets, E, defined by the fact that the Lp norm of a trigonometric polynomial whose References frequencies lie in E are dominated by the Ls norms [1] Walter Rudin, Nonanalytic functions of absolutely for an s < p, up to a constant factor depending on convergent Fourier series, Proc. Nat. Acad. Sci. USA 41 s. He studied the relation between Sidon sets and (1955), 238–240. (p) sets, proved an inequality saying that a Sidon [2] Jean-Pierre Kahane and Walter Rudin, Caractéri- Λ sation des fonctions qui opèrent sur les coefficients set is a (p) set for all values of p, and conjectured Λ de Fourier-Stieltjes, C. R. Acad. Sci. Paris 274 (1958), that this inequality was optimal. This is the case, as 773–775. proved by Pisier in 1978 using Gaussian processes. [3] Walter Rudin, A strong converse of the Wiener-Levy Though much is known now about Sidon sets, theorem, Canad. J. Math. 14 (1962), 694–701.

298 Notices of the AMS Volume 60, Number 3 [4] , Closed ideals in group algebras, Bull. Amer. his book Fourier Math. Soc. 66 (1960), 81–83. Analysis on [5] , The automorphisms and the endomorphisms Groups. I was of the group algebra of the unit circle, Acta Math. 95 present at this (1956), 39–55. beginning and [6] , Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. recall it rather [7] , Some theorems on Fourier coefficients, Proc. clearly. During Amer. Math. Soc. 10 (1959), 855–859. the academic year 1963–64 a Jean-Pierre Rosay working semi- nar dedicated to trying to I came to Madison for a one-year visit in 1986 with

learn some- Photo by Yvonne Nagel. the hope of working with Walter. I admired Walter’s thing about the Mary Ellen and Walter Rudin, 1991. work in several complex variables. I especially liked then not-widely- his book Function Theory on the Unit Ball in Cn, known subject of several complex variables was a stimulating book of supreme elegance where run in Madison by a group of students and originality is to be found in the least details. some faculty members, including Walter. It was Our collaboration soon began. Working with a question of the blind leading the blind. The Walter was pure enjoyment, with daily exchanges. volume of Fuks [1] had just appeared in an English He always came with challenges that gave rise to translation published by the AMS, so we in the an immediate desire to work. That cannot surprise seminar set out to read through it systematically, readers of his books. Nothing was rushed. Pieces but before long we recognized that this was not were kept, without any rushing to premature global really what was desired. About that time someone writing, and at the end the last pleasure was the found the beautiful Tata lectures [2] of Malgrange magic of his elegant writing (not simply cut and which give a concise introduction to the modern paste!). theory of higher-dimensional complex analysis, As soon as I arrived in Madison, I noticed including the important notions of coherent a very special quality of life in the mathematics analytic sheaves and the associated fundamental department. It is obvious that Mary Ellen and Walter Theorems A and B of Cartan and Serre. The Rudin contributed largely to the atmosphere, and seminar turned to these notes and became a great they have been wonderful hosts for many. When success. unexpectedly (the move was never planned) I was Walter’s principal research efforts soon turned invited to stay at Madison, I quickly accepted. Walter to multivariate complex analysis. Not unnaturally, was the main reason, but having great colleagues his efforts in this direction began with some around Walter, such as P. Ahern, A. Nagel, and function-theoretic questions on the unit polydisc S. Wainger also played a role. In addition to a in Cn, which is the n-fold Cartesian product collaboration with Walter, a true friendship with of the unit disc in the plane with itself. For a Mary Ellen and Walter developed. classically trained analyst who is approaching multidimensional complex analysis for the first Edgar Lee Stout time, it is entirely natural to begin by studying the possible extensions of classical results on the unit disc in the plane to analogous results on the Walter Rudin and Several Complex Variables. polydisc. One soon realizes that some classical The Beginning results have direct and often easy analogues on the Although Walter Rudin began his mathematical polydisc and that the analogues of some classical career with work in Euclidean harmonic analysis results are simply false. The most interesting kinds in a thesis on uniqueness problems concerning of results are those that present new phenomena. Laplace series, from very early on he also pursued The first paper of Rudin’s about function theory investigations in complex analysis. Most of his on the polydisc [5] was written jointly with me and work in complex analysis until the early 1960s was comprises two rather disparate kinds of results. concerned with one-dimensional theory. The first is a characterization of the rational inner Rudin’s main work in several complex variables functions on the polydisc, i.e., the rational functions began in the early 1960s after the publication of of n complex variables that are holomorphic on the polydisc and that are unimodular on the distinguished boundary Tn (which is the Jean-Pierre Rosay is emeritus professor of mathematics at the University of Wisconsin-Madison, retired in Anchorage. Cartesian product of n copies of the unit circle His email address is [email protected]. in the plane). These are natural n-dimensional

March 2013 Notices of the AMS 299 analogues of the There is some overlap in themes, e.g., inner func- finite Blaschke tions, peak-interpolation sets, Hp-theory, zero-set products. The sec- problems. At the time this book was written, it ond kind of result was not at all clear that any nonconstant inner the paper contains functions—bounded holomorphic functions with concerns the exten- radial limits almost everywhere of modulus one— sion to the poly- exist on the ball. The book contains several results disc of the Rudin- of the general sort that a nonconstant inner func- Carleson Theorem, tion on the ball must behave in very complicated which character- ways, but it neither proves nor disproves their With John-Eric Fornaess. izes as the closed existence. It was eventually shown by Alexandroff subsets of length and by Hakim-Sibony-Løw that nonconstant inner zero of the unit functions do exist. It is curious that although inner circle the peak- functions on the disc play a fundamental role interpolation sets in many function-theoretic considerations, so far for the disc alge- those on the ball have not found many applications. bra, that is, the When it appeared, this book gave a panoramic view sets on which of almost the entire known theory of functions every continuous on the ball, and it has served as the standard function can be reference since. matched by a func- Walter’s interests were much broader than tion continuous on suggested so far. For example, in [3] he gives a Photos by Yvonne Nagel. characterization of the algebraic varieties in Cn, Rudin and Pat Ahern, who wrote eight the closed unit and in [4] he obtains the structure of a proper joint papers. disc and holomor- phic on its interior. holomorphic map from the ball in Cn to a domain This paper contains some examples of peak- in Cn. He also gave two series of CBMS lectures, interpolation sets in the torus Tn and was the first the first on the edge-of-the-wedge theorem, the paper to address the question of this kind of inter- second on constructions of functions on the ball. polation in n dimensions. In spite of considerable Both were very well received. subsequent work, there is still no characterization Quite aside from the breadth and depth of his of the peak-interpolation sets for the polydisc or research efforts, one must notice the clarity and for any other domain in Cn with n > 1. elegance of Walter’s expository work. His style is After this paper was written, Rudin continued concise yet clear and can well serve as a model of to think about function theory on the polydisc mathematical exposition. and published in 1969 his book Function Theory in I count myself very fortunate to have been Polydiscs, which contained many results obtained a student of Walter Rudin, first in a wonderful in the preceding few years by him and others. The year-long course in complex variables—his Real book contains the foundations of Hardy space and Complex Analysis had not yet appeared—and theory on the polydisc; a discussion of the zero subsequently as a doctoral student. While I studied sets of bounded holomorphic functions and, more with him for my doctorate, Walter was always very generally, of functions of the Hardy class; the helpful, always ready with a technical suggestion theory of peak-interpolation sets; and a discussion about some point under consideration, and, more of inner functions. All of these subjects are direct importantly always very encouraging, which is analogues of well-understood topics in classical essential for beginning students. He and his wife, function theory, but to this day not one of them Mary Ellen, herself a distinguished mathematician, exhibits the refined completion enjoyed by the were very generous with their hospitality, both to classical theory. their students and to their former students. What About the time the polydisc book was published, a pleasure it has been to enjoy their friendship Rudin’s attention turned to function theory on the over the last half-century. unit ball in Cn. This is a considerably different subject from analysis on polydiscs, in good mea- References sure because of the greater degree of symmetry [1] B. A. Fuks, Theory of Analytic Functions of Several on the ball. Analysis on the ball is a rich subject in Complex Variables, translated by A. A. Brown, J. M. Dan- itself and also serves as a model for analysis on skin, and E. Hewitt, American Mathematical Society, Providence, RI, 1963. the general strictly pseudoconvex convex domain. [2] B. Malgrange, Lectures on the Theory of Functions Walter published in 1980 in Springer’s Grundlehren of Several Complex Variables, volume 13 of Tata n series his Function Theory on the Unit Ball of C , Institute of Fundamental Research Lectures on Mathe- which is much longer than the polydisc book. matics and Physics. Distributed for the Tata Institute

300 Notices of the AMS Volume 60, Number 3 of Fundamental Research, Bombay, 1962. Notes by the corresponding problem for the disk algebra in R. Narasimhan. his paper [3]. [3] Walter Rudin, A geometric criterion for algebraic A smooth manifold in Cn is called totally real varieties, J. Math. Mech. 17 (1967/1968), 671–683. if no tangent space at a point of M contains a [4] , Proper holomorphic maps and finite reflection complex line (e.g., Rn is totally real in Cn). Totally groups, Indiana Univ. Math. J. 31, no. 5 (1982), 701–720. [5] Walter Rudin and E. L. Stout, Boundary properties of real submanifolds play an important role in the n functions of several complex variables, J. Math. Mech. complex geometry of C . Walter studied totally 14 (1965), 991–1005. real embeddings of the 3-sphere in C3 jointly with Pat Ahern in [4], and totally real embeddings of John Wermer the Klein bottle in C2 in [5]. Walter is well known in the mathematical world Walter Rudin has made important contributions for his many basic textbooks, some of which are: to a very wide range of problems in analysis. Of • Functional Analysis. Second Edition, Mc- special interest to me is some of Walter’s work Graw Hill (1991) on problems linking the theory of commutative • Real and Complex Analysis. Third Edition, Banach algebras and complex function theory. McGraw Hill (1987) The following question was raised by Isadore • Function Theory of the Unit Ball of Cn, Singer in the 1950s. The disk algebra A = A(D) Springer Verlag (1980) is the algebra of continuous complex-valued func- • Principles of Mathematical Analysis. Third tions on the closure D of the open unit disk Edition, McGraw Hill (1976)

D = {z ∈ C |z| < 1} which are holomorphic on • Function Theory in Polydiscs, W. A. Ben- the open disk. Singer asked, to what extent is A(D) jamin (1969) characterized by the following two properties: I own several of them and have found them (1) A is an algebra of functions continuous on the excellent and of great value. closed unit disk, Walter Rudin has contributed to so many (2) the maximum principle holds with respect to subjects that the above has touched on only a the boundary of the disk. small part of his work. Walter proved the following theorem: Suppose that One nonmathematical book of Walter’s is his B is an algebra of continuous functions on the disk autobiography, The Way I Remember It, published such that by the American Mathematical Society. I found it of particular interest, since Walter and I are fellow (i) the function f (z) = z lies in B; refugees from Nazi-occupied Vienna in the late (ii) for every function f in B and every point thirties, and so his fascinating story was of special z ∈ D, 0 concern to me. I remember one line from him. It

|f (z0)| ≤ max{|f (w)| |w| = 1 was oral and may not be in the book. Walter said, so w belongs to the boundary of the disk}. “During the war, I served in the British Army. After that, nothing ever bothered me....” Then B is the disk algebra. He gave extensions of this result, using the local References maximum principle and replacing the disk by other [1] Walter Rudin, Analyticity, and the maximum modulus domains, in [1]. principle, Duke Math. J. 20 (1953), 449–457. Walter’s work was influential in stimulating the [2] , Subalgebras of spaces of continuous functions, development of the theory of function algebras. Proc. Amer. Math. Soc. 7 (1956), 825–830. In the study of polynomial approximation in Cn, [3] , The closed ideals in an algebra of analytic functions, Canad. J. Math. 9 (1957), 426–434. one asks, let K be a compact set in Cn and let P(K) [4] Walter Rudin and Patrick Ahern, Totally real em- denote the uniform closure on K of polynomials in beddings of S3 in C3, Proc. Amer. Math. Soc. 94 (1985), the complex coordinates. When is P(K) = C(K)? 460–462. In 1926 J. L. Walsh proved (in Math. Ann. 96) that [5] , Totally real Klein bottles in C2, Proc. Amer. Math. for each continuous arc J in the complex plane, Soc. 82 (1981), 653–654. P(J) = C(J). In [2] Walter gave a counterexample to the corresponding statement in C2. For each commutative Banach algebra, one would like to identify all the closed ideals. This is usually difficult. Using results of Beurling on the invariant closed subspaces on H2, Walter solved

John Wermer is emeritus professor of mathematics at Brown University. His email address is [email protected].

March 2013 Notices of the AMS 301