Correspondence theory on $p$-Fock spaces with applications to Toeplitz algebras

TL;DR

This paper develops a correspondence theory for Toeplitz algebras over p-Fock spaces, generalizing known results for p=2, and characterizes their structure using translation invariant symbols and operator spaces.

Contribution

It introduces a new correspondence framework for p-Fock spaces and extends the characterization of Toeplitz algebras beyond the classical case.

Findings

Full Toeplitz algebra is the norm closure of bounded uniformly continuous symbol operators.

Generalizes Xia's result from p=2 to arbitrary p in Fock spaces.

Toeplitz algebras with translation invariant C*-subalgebras are generated by operators with the same symbols.

Abstract

We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia (J. Funct. Anal. 269:781-814, 2015) in the case $p = 2$, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant $C^\ast$ subalgebra of the bounded uniformly continuous functions as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols.