DOCSLIB.ORG

Banach Algebras on Semigroups and on Their Compactifications H. G. Dales, A. T.-M. Lau, and D. Strauss

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Banach Algebras on Semigroups and on Their Compactifications H. G. Dales, A. T.-M. Lau, and D. Strauss Banach Algebras on Semigroups and on their Compactifications H. G. Dales, A. T.-M. Lau, and D. Strauss Author address: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom E-mail address: [email protected] Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada E-mail address: [email protected] Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom E-mail address: [email protected] Contents Chapter 1. Introduction 1 Chapter 2. Banach algebras and their second duals 15 Chapter 3. Semigroups 37 Chapter 4. Semigroup algebras 59 Chapter 5. Stone{Cechˇ compactifications 75 Chapter 6. The semigroup (βS; 2) 79 Chapter 7. Second duals of semigroup algebras 99 Chapter 8. Related spaces and compactifications 121 Chapter 9. Amenability for semigroups 133 Chapter 10. Amenability of semigroup algebras 147 Chapter 11. Amenability and weak amenability for certain Banach algebras 167 Chapter 12. Topological centres 175 Chapter 13. Open problems 199 Bibliography 201 Index 209 Index of Symbols 213 v Abstract Let S be a (discrete) semigroup, and let ` 1(S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` 1(S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are `forbidden values' for this constant. The second dual of ` 1(S) is the Banach algebra M(βS) of measures on the Stone{Cechˇ compactification βS of S, where M(βS) and βS are taken with the first Arens product 2. We shall show that S is finite whenever M(βS) is amenable, and we shall discuss when M(βS) is weakly amenable. We shall show that the second dual of L1(G), for G a locally compact group, is weakly amenable if and only if G is finite. We shall also discuss left-invariant means on S as elements of the space M(βS), and determine their supports. We shall show that, for each weakly cancellative and nearly right cancellative semigroup S, the topological centre of M(βS) is just ` 1(S), and so ` 1(S) is strongly Arens irregular; indeed, we shall considerably strengthen this result by showing that, for such semigroups S, there are two-element subsets of βS n S that are determining for the topological centre; for more general semigroups S, there are finite subsets of βS nS with this property. We have partial results on the radical of the algebras ` 1(βS) and M(βS). We shall also discuss analogous results for related spaces such as W AP (S) and LUC(G). 2000 Mathematics Subject Classification. Primary 43A10, 43A20; secondary 46J10. Key words and phrases. Banach algebras, second duals, dual Banach algebras, Arens products, Arens regular, topological centre, strongly Arens irregular, intro- verted subspace, amenable, weakly amenable, approximately amenable, amenability constant, ultrafilter Stone{Cechˇ compactification, invariant mean, semigroup, Rees semigroup, semigroup algebra, Munn algebra, semi-character, character, cancella- tive semigroup, radical, minimal ideal, idempotent, locally compact group, group algebra, measure algebra, bounded approximate identity, diagonal. vi CHAPTER 1 Introduction Our aim in this memoir is to study the algebraic structure of some Banach algebras which are defined on semigroups and on their com- pactifications. In particular we shall study the semigroup algebra ` 1(S) of a semigroup S and its second dual algebra; this includes the impor- tant special case in which S is a group. Here ` 1(S) is taken with the convolution product ? and the second dual ` 1(S)00 is taken with respect to the first and second Arens products, 2 and 3 ; these sec- ond dual algebras are identified with Banach algebras (M(βS); 2) and (M(βS); 3), which are, respectively, the right and left topological semi- groups of measures defined on βS, the Stone{Cechˇ compactification of S. We shall also study the closed subalgebras ` 1(βS) of M(βS) and some related Banach algebras. Much of our work depends on knowledge of the properties of the semigroup (βS; 2 ), and, in particular, of (βN; 2 ). We wish to stress that (βN; 2 ) is a deep, subtle, and significant mathematical object, with a distinguished history and about which there are challenging open questions; we hope to introduce the power of this semigroup to those primarily interested in Banach algebras. Indeed the questions that we ask about Banach algebras are often resolved by inspecting the properties of this semigroup, and sometimes require new results about it. So we also hope that those primarily interested in topolog- ical semigroups will be stimulated by the somewhat broader questions, arising from Banach algebra theory, that we raise about (βS; 2 ). In brief, we aspire to interest specialists in both Banach algebra theory and in topological semigroups in our work. For this reason we have tried to incorporate general background from each of these theories in our exposition in an attempt to make the work accessible to both communities. This paper is partially a sequel to the earlier memoir [21]. (For a correction to [21], see p. 198 of the present work.) Notation We recall some notation that will be used throughout; for further details of all terms used, see [19] and [21]. We shall use elementary properties of ordinal 1 2 1. INTRODUCTION and cardinal numbers as given in [19, Chapter 1.1], for example. The minimum infinite ordinal is ! and the minimum uncountable ordinal is !1; these ordinals are also cardinals, and are denoted by @0 and @1, respectively, in this case. The cofinality of an ordinal α is cof α; a cardinal α is regular if cof α = α. The Continuum Hypothesis (CH) @0 is the assertion that the continuum c = 2 is equal to @1; this hy- pothesis is independent of the usual axioms ZFC of set theory. Results that are claimed only in the theory ZFC + CH are denoted by `(CH)'. Let S be a set. The cardinality S is denoted by jSj, the family of all subsets of S is P(S), and the family of all finite subsets of S is Pf (S). Let κ be a cardinal. Then [S]κ = fT ⊂ S : jT j = κg and [S]<κ = fT ⊂ S : jT j < κg : The characteristic function of a subset T of S is denoted by χT ; we set δs = χfsg for s 2 S. We set N = f1; 2;::: g, Z = f0; ±1; ±2;::: g, and Z+ = f0; 1; 2;::: g. + The sets f1; : : : ; ng and f0; 1; : : : ; ng are denoted by Nn and Zn , respect- ively. The set of rational numbers is Q, I = [0; 1], and the unit circle and open unit disc in the complex plane C are denoted by T and D, respectively. The complex conjugate of z 2 C is denoted by z. Algebras Let A be an algebra (always over the complex field, C). The product map is mA :(a; b) 7! ab ; A × A ! A: The opposite algebra to A is denoted by Aop; this algebra has the product (a; b) 7! ba; A × A ! A. In the case where A does not have an identity, the algebra formed by adjoining an identity to A is A# (and A# = A # if A has an identity); the identity of A or A is often denoted by eA. The centre of A is Z(A) = fa 2 A : ab = ba (b 2 A)g : An idempotent in A is an element p such that p2 = p; the family of idempotents in A is denoted by I(A). For p; q 2 I(A), set p ≤ q if pq = qp = p, so that (I(A); ≤) is a partially ordered set; a minimal idempotent in A is a minimal element of the set (I(A) n f0g; ≤). 1. INTRODUCTION 3 Let I be an ideal in an algebra A, and let B be a subalgebra of A such that A = B ⊕ I as a linear space. Then A is the semi-direct product of B and I, written A = B n I. Let I be an ideal in an algebra A, and suppose that I has an identity eI . Then we remark that eI 2 Z(A). Indeed, for each a 2 A we have aeI ; eI a 2 I, and so eI (aeI ) = aeI and (eI a)eI = eI a. Thus eI 2 Z(A). Let A be an algebra. We denote by RA, NA, and QA the (Jacobson) radical of A, the set of nilpotent elements of A, and the set of quasi-nilpotent elements of A, respectively; the sets RA, NA, and QA are defined in [19], but RA is denoted by rad A in [19]. We recall that RA is defined to be the intersection of the maximal modular left ideals of A, that RA is an ideal in A, and that A is defined to be semisimple if RA = f0g. We always have the trivial inclusions: RA ⊂ QA;NA ⊂ QA : In general, we have RA 6⊂ NA and NA 6⊂ RA; further, neither QA nor NA is necessarily closed under either addition or multiplication in A. For an ideal I in A, we have RI = I \ RA; in the case where A=I is semisimple, we have RI = RA. A nilpotent element a 2 A has index n if n = minfk 2 N : ak = 0g; a subset S of A is nil if each element of S is nilpotent; the radical RA contains each left or right ideal which is nil. Let A be an algebra.
Load more

Recommended publications
  • Noncommutative Uniform Algebras
    Noncommutative Uniform Algebras Mati Abel and Krzysztof Jarosz Abstract. We show that a real Banach algebra A such that a2 = a 2 , for a A is a subalgebra of the algebra C (X) of continuous quaternion valued functions on a compact setk kX. ∈ H ° ° ° ° 1. Introduction A well known Hirschfeld-Zelazkoú Theorem [5](seealso[7]) states that for a complex Banach algebra A the condition (1.1) a2 = a 2 , for a A k k ∈ implies that (i) A is commutative, and further° ° implies that (ii) A must be of a very speciÞc form, namely it ° ° must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X)the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A,thatA is isometrically isomorphic with a subalgebra of CH (X) - the algebra of continuous quaternion valued functions on acompactsetX. That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeld-Zelazkoú Theorem, the analogous results can be stated for none unital algebras as one can formally add aunittosuchanalgebra.
    [Show full text]
  • Abelian, Amenable Operator Algebras Are Similar to C∗-Algebras
    ABELIAN, AMENABLE OPERATOR ALGEBRAS ARE SIMILAR TO C∗-ALGEBRAS LAURENT W. MARCOUX1 AND ALEXEY I. POPOV Abstract. Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C∗-algebra. We do this by showing that if A ⊆ B(H) is an abelian algebra with the property that given any bounded representation % : A!B(H%) of A on a Hilbert space H%, every invariant subspace of %(A) is topologically complemented by another invariant subspace of %(A), then A is similar to an abelian C∗-algebra. 1. Introduction. 1.1. Let A be a Banach algebra and X be a Banach space which is also a bimodule over A. We say that X is a Banach bimodule over A if the module operations are continuous; that is, if there exists κ > 0 so that kaxk 6 κkak kxk, and kxbk 6 κkxk kbk for all a; b 2 A and x 2 X. Given a Banach bimodule X over A, we introduce an action of A upon the dual space X∗ of X under which X∗ becomes a dual Banach A-bimodule. This is the so-called dual action: (ax∗)(x) = x∗(xa) and (x∗a)(x) = x∗(ax) for all a 2 A, x 2 X, x∗ 2 X∗. A (continuous) derivation from a Banach algebra A into a Banach A-bimodule X is a continuous linear map δ : A! X satisfying δ(ab) = aδ(b) + δ(a)b for all a; b 2 A.
    [Show full text]
  • On the Beurling Algebras A+ Α(D)—Derivations And
    IJMMS 2004:32, 1679–1701 PII. S0161171204309270 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON THE BEURLING ALGEBRAS + D —DERIVATIONS Aα( ) AND EXTENSIONS HOLGER STEINIGER Received 27 September 2003 + D Based on a description of the squares of cofinite primary ideals of Aα( ), we prove the ≥ + D following results: for α 1, there exists a derivation from Aα( ) into a finite-dimensional module such that this derivation is unbounded on every dense subalgebra; for m ∈ N and ∈ + + D α [m,m 1), every finite-dimensional extension of Aα( ) splits algebraically if and only if α ≥ m+1/2. 2000 Mathematics Subject Classification: 46J15, 46M20, 46H40. 1. Introduction. Let α be a positive real number. By D, we denote the open unit + D D disk. The Beurling algebra Aα( ) is a subalgebra of the classical disk algebra A( ). ∈ D = ∞ n ∈ D For f A( ) with power series expansion f(z) n=0 anz (z ), the function f + D ∞ | | + α ∞ = belongs to Aα( ) if and only if n=0 an (n 1) < . In this case, we define f α : ∞ | | + α + D n=0 an (n 1) . Clearly, Aα( ) is a Banach algebra with respect to this norm. These algebras have been considered in [13] where results on primary ideals were applied to operator theory. More recently, the algebras have appeared in the examina- tion of finite-dimensional extensions of a whole range of commutative Banach algebras + D [4]. The present paper deals with continuity problems of derivations from Aα( ) and with finite-dimensional extensions of this special type of Beurling algebras. Some of the results of the first paper will be the starting point for our investigation.
    [Show full text]
  • AUTOMATIC CONTINUITY of CERTAIN ISOMORPHISMS BETWEEN REGULAR BANACH FUNCTION ALGEBRAS by JUAN J
    AUTOMATIC CONTINUITY OF CERTAIN ISOMORPHISMS BETWEEN REGULAR BANACH FUNCTION ALGEBRAS by JUAN J. FONTt (Received 9 May, 1996) Abstract. Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if/.g = 0 implies Tf.Tg = 0, for all f,g e A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B. Finally, we show that linear isometries between regular uniform algebras are separating. As corollaries, a classical theorem of Nagasawa ([19]) and the Banach-Stone theorem (both for regular uniform algebras) are easily inferred. 1. Introduction. The basic problem in the theory of automatic continuity of linear operators consists of giving algebraic conditions on two Banach algebras si and 38 which ensure that every algebra homomorphism !T:s4-*2ft is necessarily continuous. The most important positive result in this context is due to B. E. Johnson ([15]). Every algebra homomorphism of a Banach algebra onto a semisimple Banach algebra is continuous. In this paper we shall study the automatic continuity of a special type of linear mapping which extends the concept of algebra homomorphism: Let A and B be two Banach algebras. Then a linear map T denned from A into B is said to be separating or disjointness preserving (also called d-homomorphisms, Lamperti operators,...) if f.g — 0 implies Tf.Tg=0, for all/,g e A.
    [Show full text]
  • Regularity Conditions for Banach Function Algebras
    Regularity conditions for Banach function algebras Dr J. F. Feinstein University of Nottingham June 2009 1 1 Useful sources A very useful text for the material in this mini-course is the book Banach Algebras and Automatic Continuity by H. Garth Dales, London Mathematical Society Monographs, New Series, Volume 24, The Clarendon Press, Oxford, 2000. In particular, many of the examples and conditions discussed here may be found in Chapter 4 of that book. We shall refer to this book throughout as the book of Dales. Most of my e-prints are available from www.maths.nott.ac.uk/personal/jff/Papers Several of my research and teaching presentations are available from www.maths.nott.ac.uk/personal/jff/Beamer 2 2 Introduction to normed algebras and Banach algebras 2.1 Some problems to think about Those who have seen much of this introductory material before may wish to think about some of the following problems. We shall return to these problems at suitable points in this course. Problem 2.1.1 (Easy using standard theory!) It is standard that the set of all rational functions (quotients of polynomials) with complex coefficients is a field: this is a special case of the “field of fractions" of an integral domain. Question: Is there an algebra norm on this field (regarded as an algebra over C)? 3 Problem 2.1.2 (Very hard!) Does there exist a pair of sequences (λn), (an) of non-zero complex numbers such that (i) no two of the an are equal, P1 (ii) n=1 jλnj < 1, (iii) janj < 2 for all n 2 N, and yet, (iv) for all z 2 C, 1 X λn exp (anz) = 0? n=1 Gap to fill in 4 Problem 2.1.3 Denote by C[0; 1] the \trivial" uniform algebra of all continuous, complex-valued functions on [0; 1].
    [Show full text]
  • Uniform Algebras As Banach Spaces
    The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Uniform Algebras as Banach Spaces TW Gamelin SV Kislyakov Vienna Preprint ESI June Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat UNIFORM ALGEBRAS AS BANACH SPACES TW Gamelin SV Kislyakov June Abstract Any Banach space can b e realized as a direct summand of a uniform algebra and one do es not exp ect an arbitrary uniform algebra to have an abundance of prop erties not common to all Banach spaces One general result concerning arbitrary uniform algebras is that no prop er uniform algebra is linearly homeomorphic to a C K space Nevertheless many sp ecic uniform algebras arising in complex analysis share or are susp ected to share certain Banach space prop erties of C K We discuss the family of tight algebras which includes algebras of analytic functions on strictly pseudo con vex domains and algebras asso ciated with rational approximation theory in the plane Tight algebras are in some sense close to C K spaces and along with C K spaces they have the Pelczynski and the DunfordPettis prop erties We also fo cus on certain prop erties of C K spaces that are inherited by the disk algebra This includes a dis p cussion of interp olation b etween H spaces and Bourgains extension of Grothendiecks theorem to the disk algebra We conclude with a brief description of linear deformations of uniform algebras and a brief survey of the known classication results Supp orted in part by the Russian
    [Show full text]
  • 1. Introduction
    FACTORIZATION IN COMMUTATIVE BANACH ALGEBRAS H. G. DALES, J. F. FEINSTEIN, AND H. L. PHAM Abstract. Let A be a (non-unital) commutative Banach algebra. We consider when A has a variety of factorization properties: we list the (ob- vious) implications between these properties, and then consider whether any of these implications can be reversed in various classes of commu- tative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products. 1. Introduction Let A be a (non-unital) commutative Banach algebra. We wish to examine when A factors in a variety of senses. Our main results are counter-examples to a number of questions that have been raised. Indeed, we shall list seven such factorization properties, called (I)(VII), and note that each of these immediately implies the next one. We shall also, in x4, discuss two other `lo- cal' factorization properties, called (A) and (B) (where (A) ) (B)); these properties are relevant for Esterle's classication of commutative, radical Banach algebras that is given in [14]. We shall then discuss whether or not any of these implications can be reversed when we restrict attention to par- ticular classes of commutative Banach algebras. We shall show that several cannot be reversed, but we leave open other possible reverse implications. A summary in x6 describes our knowledge at the present time. We shall concentrate on two particular classes of commutative Banach algebras A: rst, on the case where A is semi-simple (so that A is a Banach function algebra), and in particular when A is a maximal ideal in a uniform algebra on a compact space, and, second, on the other extreme case where A 2010 Mathematics Subject Classication.
    [Show full text]
  • Order Structure, Multipliers, and Gelfand Representation of Vector-Valued Function Algebras
    ORDER STRUCTURE, MULTIPLIERS, AND GELFAND REPRESENTATION OF VECTOR-VALUED FUNCTION ALGEBRAS JORMA ARHIPPAINEN, JUKKA KAUPPI∗ AND JUSSI MATTAS Abstract. We continue the study begun by the third author of C∗-Segal algebra valued function algebras, with an emphasis on the order structure. Our main result is a characterization theorem for C∗-Segal algebra valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra valued function algebras. We also consider Gelfand representation of these algebras in the commutative case. Introduction The concept of a Segal algebra originated in the work of Reiter on subalgebras of the L1-algebra of a locally compact group; cf. [19]. It was generalized to arbitrary Banach algebras by Burnham in [8]. A C∗-Segal algebra is a Banach algebra which is a dense (not necessarily self-adjoint) ideal in a C∗-algebra. Despite many important examples in analysis, such as the Schatten classes for instance, little has been known about the general structure and properties of C∗-Segal algebras. In particular, most results have relied on the additional assumption of an approximate identity. The multiplier algebra and the bidual of self-adjoint C∗-Segal algebras were described in [2, 14], and the form of the closed ideals of C∗-Segal algebras with an approximate identity was given in [6]. Recently, a new approach to Segal algebras was introduced in [5, 12], which allows a more detailed study of their structure and properties, particularly in the absence of an approximate identity ([5] the commutative case and [12] the noncommutative case).
    [Show full text]
  • Function Representation of a Noncommutative Uniform Algebra
    Function representation of a noncommutative uniform algebra Krzysztof Jarosz Abstract. We construct a Gelfand type representation of a real noncommu- tative Banach algebra A satisfying f 2 = f 2, for all f A: k k 2 1. Introduction A uniform algebra A is a Banach algebra such that (1.1) f 2 = f 2 , for all f A: k k 2 The commutative, complex uniform algebras constitute the most classical class of Banach algebras, one which has been intensely studied for decades. It is well known that any such algebra is isometrically isomorphic with a subalgebra of CC (X) - the algebra of all continuous complex valued functions de…ned on a compact set X and equipped with the sup-norm topology. Hirschfeld and Zelazko· [6] proved that the assumption that the algebra is commutative is super‡uous - commutativity al- ready follows from (1.1). Algebras of analytic functions serve as standard examples of complex uniform algebras. Commutative, real uniform algebras have also been studied for years [9]. Any such algebra A is isometrically isomorphic with a real subalgebra of CC (X) for some compact set X; furthermore X can be often divided into three parts X1;X2; and X3 such that A X1 is a complex uniform algebra, A X2 consists of complex conjugates j j of the functions from A X1 , and A X3 is equal to CR (X3). Any commutative, real uniform algebra can bej naturally identi…edj with a real subalgebra of df C (X; ) = f C (X): f ( (x)) = f (x) for x X C 2 C 2 n 2 o where : X X is a surjective homeomorphism with = idX .
    [Show full text]
  • Arxiv:Math/0203199V1 [Math.FA] 19 Mar 2002
    Amenability for dual Banach algebras Volker Runde Abstract ∗ We define a Banach algebra A to be dual if A = (A∗) for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Ba- nach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability — Connes-amenability and strong Connes-amenability — which take the w∗-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of A∗∗; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for C∗-algebras. 2000 Mathematics Subject Classification: 43A10, 46B28, 46H25, 46H99 (primary), 46L10, 46M20. 1 Introduction arXiv:math/0203199v1 [math.FA] 19 Mar 2002 Amenable Banach algebras were introduced by B. E. Johnson in [Joh 2], and have since then turned out to be extremely interesting objects of research.
    [Show full text]
  • Mathematical Work of Franciszek Hugon Szafraniec and Its Impacts
    Tusi Advances in Operator Theory (2020) 5:1297–1313 Mathematical Research https://doi.org/10.1007/s43036-020-00089-z(0123456789().,-volV)(0123456789().,-volV) Group ORIGINAL PAPER Mathematical work of Franciszek Hugon Szafraniec and its impacts 1 2 3 Rau´ l E. Curto • Jean-Pierre Gazeau • Andrzej Horzela • 4 5,6 7 Mohammad Sal Moslehian • Mihai Putinar • Konrad Schmu¨ dgen • 8 9 Henk de Snoo • Jan Stochel Received: 15 May 2020 / Accepted: 19 May 2020 / Published online: 8 June 2020 Ó The Author(s) 2020 Abstract In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence. Keywords Szafraniec Á Mathematical work Á Biography Mathematics Subject Classification 01A60 Á 01A61 Á 46-03 Á 47-03 1 Biography Professor Franciszek Hugon Szafraniec’s mathematical career began in 1957 when he left his homeland Upper Silesia for Krako´w to enter the Jagiellonian University. At that time he was 17 years old and, surprisingly, mathematics was his last-minute choice. However random this decision may have been, it was a fortunate one: he succeeded in achieving all the academic degrees up to the scientific title of professor in 1980. It turned out his choice to join the university shaped the Krako´w mathematical community. Communicated by Qingxiang Xu. & Jan Stochel [email protected] Extended author information available on the last page of the article 1298 R. E. Curto et al. Professor Franciszek H. Szafraniec Krako´w beyond Warsaw and Lwo´w belonged to the famous Polish School of Mathematics in the prewar period.
    [Show full text]
  • Banach Function Algebras with Dense Invertible Group
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 4, April 2008, Pages 1295–1304 S 0002-9939(07)09044-2 Article electronically published on December 21, 2007 BANACH FUNCTION ALGEBRAS WITH DENSE INVERTIBLE GROUP H. G. DALES AND J. F. FEINSTEIN (Communicated by N. Tomczak-Jaegermann) Abstract. In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov bound- ary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that tsr(A) ≥ tsr(C(ΦA)) whenever A is approximately regular. 1. Introduction Let A be a commutative, unital Banach algebra. The character space of A is denoted by ΦA,asin[8].Fora ∈ A, we denote the Gel’fand transform of a by a.WesaythatΦA contains analytic structure if there is a continuous injection τ from the open unit disk D to ΦA such that, for all a ∈ A, a ◦ τ is analytic on D. In [16], Stolzenberg gave a counter-example to the conjecture that, whenever a uniform algebra has proper Shilov boundary, its character space must contain analytic structure (see also [17, Theorem 29.19], [19] and [1]). Cole gave an even more extreme example in his thesis [3], where the Shilov boundary is proper and yet every Gleason part is trivial. It is elementary to show that the invertible group of A cannot be dense in A whenever ΦA contains analytic structure.
    [Show full text]