Toeplitz algebra and Symbol map via Berezin transform on H²(\mathbbDⁿ)

2024/05/01 by Mo Javed, Amit Maji, Javed, Mo +1
Mathematics · Physics and Astronomy · #32A65 #47A13 #47B35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Waves and Solitons #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2405.10967

openalex publication_date 2024/05/01 · openalex created_date 2024/05/22 · openalex updated_date 2026/07/28

Abstract

Let \mathscrT(L^∞(\mathbbT)) be the Toeplitz algebra, that is, the C^*-algebra generated by the set \T_ϕ : ϕ∈ L^∞(\mathbbT)\. Douglas's theorem on symbol map states that there exists a C^*-algebra homomorphism from \mathscrT(L^∞(\mathbbT)) onto L^∞(\mathbbT) such that T_ϕ↦ ϕ and the kernel of the homomorphism coincides with commutator ideal in \mathscrT(L^∞(\mathbbT)). In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space H²(\mathbbDⁿ) over the open unit polydisc \mathbbDⁿ for n≥ 1. We further obtain a class of bigger C^*-algebras than the Toeplitz algebra \mathscrT(L^∞(\mathbbTⁿ)) for which the analog of symbol map still holds true.

Related

• The Berezin Transform on the Toeplitz Algebra
• Localizations and Essential Commutant of Toeplitz Algebra on Polydisk
• Invertibility of Bergman Toeplitz operators
• The C*-algebra generated by irreducible Toeplitz and composition\n operators
• Analytic continuation of Toeplitz operators and commuting families of C^*-algebras