A characterization of rationality in free semicircular operators

Abstract

For free semicircular elements realized 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 P. A. Linnell to densely defined unbounded operators affiliated with the free group factor. While their result was motivated by quantized calculus in noncommutative geometry, we state our results in terms of free probability theory.