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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 Foundation for Basic Research grant Contents Uniform Algebras Analytic Functions on Banach Spaces Characterization of Prop er Subalgebras Tight Subspaces and Subalgebras of C K The Pelczynski and DunfordPettis Prop erties Absolutely Summing and Related Op erators on the Disk Algebra Interp olation of HardyTyp e Subspaces Bourgain Pro jections Perturbation of Uniform Algebras The Dimension Conjecture Uniform Algebras A uniform algebra is a closed subalgebra A of the complex algebra C K that contains the constants and separates p oints Here K is a compact Hausdor space and A is endowed with the supremum norm inherited from C K The algebra A is said to b e proper if A C K Uniform algebras arise naturally in connection with problems in approximation theory The main examples of prop er uniform algebras come from complex analysis The prototypical prop er uniform algebra is the disk algebra which we denote simply by C consisting of the A analytic functions on the op en unit disk in the complex plane that extend continuously to n the b oundary More generally if K is a compact subset of C we denote by AK or by C K the algebra of functions continuous on K and analytic on the interior of K Also we A may consider the uniform closure of the restriction to K of some algebras of elementary holomorphic functions such as analytic p olynomials or rational functions with singularities o K The uniform closure of the rational functions with singularities o K is denoted by RK If K is a compact subset of the complex plane Runges approximation theorem asserts that RK includes the functions that are analytic in a neighb orho o d of K It may o ccur that RK is a prop er subalgebra of C K even when K has empty interior Other uniform n algebras asso ciated with a domain D in C are the algebra AD of analytic functions on D that extend continuously to the closure of D and the algebra H D consisting of all b ounded analytic functions on D Endowed with the supremum norm on D the algebra H D b ecomes a uniform algebra on the smallest compactication of D to which the functions extend continuously In the case of the op en unit disk fjz j g we may identify H with a closed subalgebra H d of L d via nontangential b oundary values where d is the arclength measure on the unit circle One of the goals in studying uniform algebras is to use the to ols of functional analysis and the Gelfand theory in order to prove approximation theorems or to understand why ap proximation fails Mergelyans theorem that RK AK whenever K is a compact subset of the complex plane whose complement has a nite numb er of comp onents was eventually given a pro of by Glicksb erg and Wermer that dep ends on uniformalgebra techniques and a less dicult theorem of Walsh on approximation by harmonic functions We quote two other approximation theorems whose pro ofs dep end on the algebra structure and on techniques of functional analysis Theorem Wermer We Let A be a not necessarily closed algebra of analytic functions on the unit circle in the complex plane Suppose that A separates the points of and that each function in A extends to be analytic in a neighborhood of Then either A is dense in C or else there is a nite bordered Riemann surface with border such that the functions in A extend to be analytic on the surface Theorem Davie Da Let be the area measure on a compact subset K of the K complex plane and let H be the weakstar closure of RK in L Then each K K f H is approximable pointwise ae on K by a sequence of functions f RK K n such that jjf jj jjf jj n In some sense uniform algebra theory can b e regarded as an abstract study of the maximum principle for algebras For an incisive account of this asp ect of the theory see AWe Uniform algebra theory has also served as a source of interesting problems One such problem originally raised by S Kakutani asked whether the op en unit disk is dense in the sp ectrum of the algebra H This problem b ecame known as the corona problem It was answered armatively by L Carleson see Gar While the corona theorem p er se has not played a really signicant role in analysis the techniques that were devised to solve the problem have played an imp ortant role in function theory The question arises as to the extent that prop erties of various uniform algebras dep end only on their linear structure From the p oint of view of Banach spaces how sp ecial are uniform algebras We will see in Section that generic uniform algebras are as bad as generic Banach spaces in the sense that any Banach space is isometric to a complemented subspace of a uniform algebra On the other hand we will show in Section that a prop er uniform algebra is distinct as a Banach space from C K In fact no prop er uniform algebra is linearly isomorphic to a complemented subspace of a C K space or even to a quotient of a C K space It is of interest to know which prop erties of C K are inherited by uniform algebras Towards answering this question a great deal of eort has gone into determining which prop erties of C are passed down to the disk algebra C Some of the known results are A summarized in the table b elow At present there is no unied theory but rather only fragmented results on uniform algebras as Banach spaces Our aim in this article is to present a selection of results in order to give an idea of what has b een studied and what problems are currently op en Sometimes pro ofs or indications of pro ofs are also given to convey the avor of the techniques that are employed Section includes a brief intro duction to algebras of analytic functions on domains in Banach spaces In Section we show how some basic facts ab out psumming and pintegral op erators lead to several characterizations of prop er uniform algebras Sections and are devoted to certain prop erties that involve weak compactness The prop erties are wellknown for C K spaces and they are inherited by algebras generated by analytic functions on planar sets and on strictly pseudo convex domains where the key ingredient is the solvability of a problem Prop erty of X C K or X C C K C Reference A A X is a linear quotient of C S Yes No x X is complemented in a Banach lattice Yes No x X has the DunfordPettis prop erty Yes Yes x X has the Pelczynski prop erty Yes Yes x X is weakly sequentially complete Yes Yes x X has a basis Yes if K is metric Yes Wo j X veries Grothendiecks theorem Yes Yes x X has a complemented copy of C Yes if K is metric Yes Wo j and uncountable In Sections through we talk of prop erties that are more sp ecic to the disk algebra C We fo cus on Bourgains extension of Grothendiecks theorem to the disk algebra obtained A in B B This sub ject was treated in detail in the survey K The exp osition here will follow the ideas of K with slight simplications at some p oints and with emphasis on the interp olatory nature of the pro ofs The extension of Grothendiecks theorem is discussed in Section and the results on interp olation used in the pro ofs are dealt with in Section Section includes a brief account of Bourgain pro jections the main technical to ol used by Bourgain to transfer results from continuous to analytic functions A go o d deal of the material presented here has already b een discussed in various mono graphs and exp ository pap ers We mention particularly the early lecture notes of Pelczynski Pe and the research monographs of Wo jtaszczyk Wo j and of Diestel Jarchow and Tonge DJT For background on uniform algebra theory see Gam Sto and AWe For Davies theorem see also Gam The basic Hardy space theory that we refer to is covered in Du Gar and Hof One reference for several complex variables and pseudo convexity is Ra The exp ository pap ers K and K also cover in part the material of the present article We collect here some standard notation and conventions We denote by the unit circle fjz j g in the complex plane and by dm d the normalized arclength measure 1 on The
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