Pointwise Approximate Identities in Banach Function Algebras H
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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. -
Weak and Norm Approximate Identities Are Different
Pacific Journal of Mathematics WEAK AND NORM APPROXIMATE IDENTITIES ARE DIFFERENT CHARLES ALLEN JONES AND CHARLES DWIGHT LAHR Vol. 72, No. 1 January 1977 PACIFIC JOURNAL OF MATHEMATICS Vol. 72, No. 1, 1977 WEAK AND NORM APPROXIMATE IDENTITIES ARE DIFFERENT CHARLES A. JONES AND CHARLES D. LAHR An example is given of a convolution measure algebra which has a bounded weak approximate identity, but no norm approximate identity. 1* Introduction* Let A be a commutative Banach algebra, Ar the dual space of A, and ΔA the maximal ideal space of A. A weak approximate identity for A is a net {e(X):\eΛ} in A such that χ(e(λ)α) > χ(α) for all αei, χ e A A. A norm approximate identity for A is a net {e(λ):λeΛ} in A such that ||β(λ)α-α|| >0 for all aeA. A net {e(X):\eΛ} in A is bounded and of norm M if there exists a positive number M such that ||e(λ)|| ^ M for all XeΛ. It is well known that if A has a bounded weak approximate identity for which /(e(λ)α) —>/(α) for all feA' and αeA, then A has a bounded norm approximate identity [1, Proposition 4, page 58]. However, the situation is different if weak convergence is with re- spect to ΔA and not A'. An example is given in § 2 of a Banach algebra A which has a weak approximate identity, but does not have a norm approximate identity. This algebra provides a coun- terexample to a theorem of J. -
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. -
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. -
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]. -
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 -
Operator Algebras with Contractive Approximate Identities
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 261 (2011) 188–217 www.elsevier.com/locate/jfa Operator algebras with contractive approximate identities David P. Blecher a,∗,1, Charles John Read b a Department of Mathematics, University of Houston, Houston, TX 77204-3008, United States b Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, United Kingdom Received 15 November 2010; accepted 24 February 2011 Available online 21 March 2011 Communicated by S. Vaes Abstract We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for ∗ C -algebras, and we give many other applications. © 2011 Elsevier Inc. All rights reserved. Keywords: Operator algebras; One-sided ideals; Hereditary subalgebra; Approximate identity; Peak set; Pseudo-invertible elements; Completely positive operator 1. Introduction An operator algebra is a closed subalgebra of B(H), for a Hilbert space H . We recall that by a theorem due to Ralf Meyer, every operator algebra A has a unique unitization A1 (see [30] or [10, Section 2.1]). Below 1 always refers to the identity of A1 if A has no identity. -
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. -
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). -
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 . -
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. -
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.