A 𝐶*-algebra of entire functions

Abstract

We study the maximal abelian von Neumann algebra corresponding to L ∞ ( R ) L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F 2 F^2 , but it can also be realized as a function algebra contained in F 2 F^2 . This provides an interesting example of a C ∗ C^* algebra whose elements are analytic functions.