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PreprintCC0arXiv

A new perspective on Riemann surfaces and von Neumann algebras via thin Blaschke products

  • Danni Guo
  • Hansong Huang
  • Shan Li
  • Xianfeng Zhao
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Abstract

In 2014, K.Guo and the second author constructed the first example of an infinite Blaschke product B (a thin Blaschke product), for which the associated multiplication operator MB is irreducible on the Bergman space La2(๐”ป) over the unit disk ๐”ป. We take a substantial step forward toward characterizing the irreducibility of the operator MB on La2(๐”ป), or equivalently, the trivialness of the von Neumann algebra {MB,MB*}'. This characterization indicates a close connection with the primeness of B and with the two-component property of the Riemann surface ๐’ฎB associated with B. The approach is built upon a nested family of connected components of the sublevel set {zโˆˆ ๐”ป:|B(z)|<t} for 0<t<1, which captures the function-theoretic and geometric feature of B.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026perspective,
  title = {{A new perspective on Riemann surfaces and von Neumann algebras via thin Blaschke products}},
  author = {Danni Guo and Hansong Huang and Shan Li and Xianfeng Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00933},
  eprint = {2611.00933},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00933},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - A new perspective on Riemann surfaces and von Neumann algebras via thin Blaschke products
AU  - Danni Guo
AU  - Hansong Huang
AU  - Shan Li
AU  - Xianfeng Zhao
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00933
UR  - https://arxiv.org/abs/2611.00933
AB  - In 2014, K.Guo and the second author constructed the first example of an infinite Blaschke product B (a thin Blaschke product), for which the associated multiplication operator M_B is irreducible on the Bergman space L_a^2(๐”ป) over the unit disk ๐”ป. We take a substantial step forward toward characterizing the irreducibility of the operator M_B on L_a^2(๐”ป), or equivalently, the trivialness of the von Neumann algebra {M_B,M_B^*}'. This characterization indicates a close connection with the primeness of B and with the two-component property of the Riemann surface ๐’ฎ_B associated with B. The approach is built upon a nested family of connected components of the sublevel set {zโˆˆ ๐”ป:|B(z)|<t} for 0<t<1, which captures the function-theoretic and geometric feature of B.
ER  -