Let ℒ(FN) be the von
Neumann algebra of the free group with N generators x1,…,xN, N ≥ 2 and let A be
the abelian von Neumann subalgebra generated by x1+ x1−1+⋯+ xN+ xN−1 acting
as a left convolutor on l2(FN). The radial algebra A appeared in the harmonic
analysis of the free group as a maximal abelian subalgebra of ℒ(FN), the von
Neumann algebra of the free group. The aim of this paper is to prove that A is
singular (which means that there are no unitaries u in ℒ(FN) excepting those
coming from A such that u∗Au ⊆ A). This is done by showing that the
Pukánszky invariant of A is infinite, where the Pukánszky invariant of A is
the type of the commutant of the algebra 𝒜 in B(l2(FN)) generated by A
and x1+ x1−1+⋯+ xN+ xN−1 regarded also as a right convolutor on
l2(FN).
