===========================================================================

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.

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>