A computable wandering and tracelike vector for modular orbits in the Bergman space

Abstract

We construct a function $\Phi $ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $\alpha =12$. Moreover, we show that $\Phi$ 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 $M_{12}(\Gamma )$ and its commutant, which is generated, in R\u{a}dulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(\Gamma )$.