Activities This Week

(Part 2 of the talk from last week.)

In this talk, I will discuss a noncommutative (nc) analogue of the Gleason problem and its application in the “NC Cowen-Douglas” class. The Gleason problem was first studied by Andrew Gleason in studying the maximal ideals of a commutative Banach algebra. In particular, he showed that if the maximal ideal consisting of functions in the Banach algebra $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ vanishing at the origin is finitely generated then it has to be generated by the coordinate functions where $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ is the Banach algebra of holomorphic functions on the open unit ball $\mathbb{B} ( 0, 1 )$ at $0$ in $\mathbb{C}^n$ which can be continuously extended up to the boundary. The question – whether the maximal ideals in algebras of holomorphic functions are generated by the coordinate functions – has been named the Gleason problem. It turns out that the existence of a local solution of the Gleason problem in a reproducing kernel Hilbert space provides a sufficient condition for the membership of the tuple of adjoint of multiplication operators by coordinate functions in the Cowen-Douglas class.

After briefly discussing these classical aspects of the Gleason problem, I will introduce its nc counterpart for uniformly analytic nc functions and show that such a problem in the nc category is always locally uniquely solvable unlike the classical case. As an application one obtains a characterization of nc reproducing Hilbert spaces of uniformly analytic nc functions on a nc domain in $\mathbb{C}^d_{ \text{nc} }$ so that the adjoint of the $d$ - tuple of left multiplication operators by the nc coordinate functions are in the nc Cowen-Douglas class. Along the way, I will recall necessary materials from nc function theory.

This is a part of my ongoing work jointly with Professor Vinnikov on the nc Cowen-Douglas class.

Whitney studied the embeddings of (C^\infty) manifolds into R^N. A simple initial idea is: start from a map M-> R^N, and deform it generically. Hopefully one gets an embedding, at least an immersion. This fails totally because of the “stable maps”. They are non-immersions, but are preserved in small deformations. The theory of stable maps was constructed in 50’s-60’s by Thom, Mather and others. The participating groups are infinite-dimensional, and the engine of the theory was vector fields integration. This chained all the results to the real/complex-analytic case. I will discuss the classical case, then report on the new results, extending the theory to the arbitrary field (of any characteristic).