Title: Some Banach space properties of ideals in uniform algebras
Speaker: Serguei Kisliakov (Steklov Mathematical Institute St.Petersburg, Russia)
Time: 15:00-17:30, May 6
Location: Room 201，Ming De Building
Abstract: It is well known that the orthogonal projection from the space L^2 on the unit circle onto the Hardy class H^2 is discontinuous in the sup-norm. This classical statement has been generalized in various ways in different times. In my lectures I'll concentrate on two such generalizations. In 1961, Glicksberg conjectured that an arbitrary nonselfajoint closed subalgebra of C(K) is not a complemented subspace of C(K). This conjecture was justified eventually, but this happened some 25 years after it had been announced. On the other hand, fairly recently, I. Zlotnikov and myself proved that, under very mild and natural conditions, given two ideals in a proper uniform algebra, the sum of one of them and the complex conjugate of the other is not closed in the uniform norm. This statement and the Glicksberg conjecture seem to be independent from each other, but the methods of proof are somewhat similar. The conceptual difficulty is that, in the two cases, it is necessary to find some weak traces of an analytic structure linked with a general uniform algebra, and that this analyticity should emerge basically from nothing.
