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{\Large Department of Mathematics, BGU}

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{\Huge Operator Algebras and Operator Theory}\\[0.2\baselineskip]

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\textbf{On} \emph{Monday, December  5, 2022}
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\textbf{At} \emph{16:00 -- 17:00}
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\textbf{In} \emph{-101 (basement)}

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{\large\scshape Prahllad Deb 
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  (BGU)
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will talk about
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{\Large\bfseries NC Gleason problem and its application in the NC Cowen-Douglas class - ctd.\par}
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\textsc{Abstract:}
(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.








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