=========================================================================== Technion SPECIAL MATHEMATICS COLLOQUIUM Please note the unusual day and unusual time. (In fact we will have two special colloquia on this day. See a separate announcement for details of Prof. Pesenson's talk earlier, at 14:30.) Speaker: Alexander Brudnyi, University of Calgary Title: Stein-like theory for Banach-valued holomorphic functions on the maximal ideal space of H^\infty and new developments in the Sz-Nagy operator corona problem. Date: Thursday June 2 at 16:00. Place: Technion. Amado 232 is now definitely reserved for this colloquium. Apology: Because of some rather special circumstances we are not entirely sure at this stage that Prof. Brudnyi will be available to give this talk. Please check the Techmath website to obtain updated information about this. Abstract: In this talk I describe a new approach to the study of the Banach algebra H^\infty of bounded holomorphic functions on the unit disk D with pointwise multiplication and supremum norm. It is based on a new method for solving Banach-valued d-bar equations on D which allows one to develop a Stein-like theory for Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of H^\infty. Specifically, as in the classical theory of Stein spaces, I establish the vanishing of the cohomology of sheaves of germs of such functions, solve the second Cousin problem and prove Runge-type approximation theorems for them. Then I apply the developed technique to the study of the algebra of holomorphic functions on D with relatively compact images in a commutative unital complex Banach algebra A. In particular, solving a Banach-valued corona problem, I prove that the maximal ideal space of such algebra is the direct product of maximal ideal spaces of H^\infty and A. This generalizes the famous Carlseson Corona theorem and solves a major problem posed in the mid of 60th. The same result would also follow if we knew that H^\infty has the Grothendieck approximation property (which is still an open problem). In the second part of my talk I apply the developed technique to establish the following version of the Oka principle: if a Banach holomorphic vector bundle on the maximal ideal space of H^\infty is topologically trivial, then it is holomorphically trivial as well. This leads to new positive results in the area of the Sz.-Nagy operator valued corona problem (a noncommutative analog of the Carleson theorem) posed in 1978. This talk does not require any preliminary specialized knowledge because all basic definitions and results will be formulated there. --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Michael Cwikel <mcwikel@math.technion.ac.il>