A computable wandering and tracelike vector for modular orbits in the Bergman space
Abstract
We construct a function Φ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight α=12. Moreover, we show that Φ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra M12(Γ) and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for M12(Γ).
Related
- Global L2 Approximation in the Bergman Spaces with Values in Holomorphic Vector Bundles
- Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity
- Free modules of vector-valued modular forms and applications to Jacobi forms
- On modular vector invariant fields
- On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras
BibTeX
@misc{abreu2026computable,
title = {{A computable wandering and tracelike vector for modular orbits in the Bergman space}},
author = {Luís Daniel Abreu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30014},
eprint = {2609.30014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30014},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A computable wandering and tracelike vector for modular orbits in the Bergman space
AU - Luís Daniel Abreu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30014
UR - https://arxiv.org/abs/2609.30014
AB - We construct a function Φ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight α=12. Moreover, we show that Φ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra M_12(Γ) and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for M_12(Γ).
ER -