Singularity of the radial subalgebra of ℒ(FN) and the Pukánszky invariant

Abstract

Let ℒ(F N ) be the von Neumann algebra of the free group with N generators x 1 ,⋯,x N , N≥2 and let A be the abelian von Neumann subalgebra generated by x 1 +x 1 -1 +⋯+x N +x N -1 acting as a left convolutor on l 2 (F N ). The radial algebra A appeared in the harmonic analysis of the free group as a maximal abelian subalgebra of ℒ(F N ), 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(F N ) except those coming from A such that u * Au⊆A). This is done by showing that the Pukanszky invariant of A is infinite, where the Pukanszky invariant of A is the type of the commutant of the algebra A in B(l 2 (F N )) generated by A and x 1 +x 1 -1 +⋯+x N +x N -1 regarded also as a right convolutor on l 2 (F N )