A characterization of rationality for operators in free semicircular elements

TL;DR

This paper characterizes rational operators in free semicircular elements by linking their rationality to the finite rank of certain commutators, bridging free probability and noncommutative geometry.

Contribution

It establishes an equivalence between operator rationality and commutator rank finiteness in free semicircular elements, extending previous results to a new setting.

Findings

Rationality of operators corresponds to finite commutator rank.

Provides a free probability analogue of known results in free group $C^*$-algebras.

Connects rationality with noncommutative geometric concepts.

Abstract

Realizing free semicircular elements on the full Fock space, we prove an equivalence between rationality of operators obtained from them and finiteness of the rank of their commutators with right annihilation operators. This is an analogue of the result for the reduced $C^*$-algebra of the free group by G. Duchamp and C. Reutenauer which was extended by PA. Linnel to densely defined unbounded operators affiliated with the free group factor. Although their result was motivated from quantized calculus in noncommutative geometry, we state our results in terms of free probability theory.