A 𝐶*-algebra of entire functions

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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.
Funding Information
  • National Natural Science Foundation of China (11801572)
  • National Natural Science Foundation of China (12171484)
  • National Natural Science Foundation of China (2023JJ20056)
  • National Natural Science Foundation of China (2023RC3028)
  • National Natural Science Foundation of China (2023CXQD032)
  • Natural Science Foundation of Hunan Province (11801572)
  • Natural Science Foundation of Hunan Province (12171484)
  • Natural Science Foundation of Hunan Province (2023JJ20056)
  • Natural Science Foundation of Hunan Province (2023RC3028)
  • Natural Science Foundation of Hunan Province (2023CXQD032)
  • Science and Technology Program of Hunan Province (11801572)
  • Science and Technology Program of Hunan Province (12171484)
  • Science and Technology Program of Hunan Province (2023JJ20056)
  • Science and Technology Program of Hunan Province (2023RC3028)
  • Science and Technology Program of Hunan Province (2023CXQD032)
  • Innovation-Driven Project of Central South University (11801572)
  • Innovation-Driven Project of Central South University (12171484)
  • Innovation-Driven Project of Central South University (2023JJ20056)
  • Innovation-Driven Project of Central South University (2023RC3028)
  • Innovation-Driven Project of Central South University (2023CXQD032)

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