# Mark G. Lawrence: New sources of real analyticity in function spaces and function algebras

Time: Wed 2020-02-12 13.15 - 14.15

Lecturer: Mark G. Lawrence

### Abstract

It is a basic fact from approximation theory that given a compact set $$K \subseteq \mathbf{C^n}$$, the closure of the algebra generated by $$z$$ and $$\overline{z}$$ is $$C(K)$$. Another similar result is the Wermer maximality theorem regarding function algebras on the unit circle. In a different direction, the speaker has demonstrated that if an algebra is generated by $$z$$ and $$\overline{z}(g(z))$$, where $$g$$ is entire satisfies some condition on the zero set, then the closed Fréchet algebra generated consists of functions which are real analytic, with infinite radius of convergence. With further restrictions on the zero set of $$g$$, one can construct Bergman space analogues of Fock space. There are analagous results with functions of several variables and functions on come CR manifolds as well. The condition on the zeros of $$g$$ can be fulfilled with explicit functions. The technique is CR wedge extension combined with the theory of the 1-dimensional extension property.

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