Duality Properties in Von Neumann Algebras of Projective Unitary Representations

Let pi be a projective representation of a countable discrete group G on a Hilbert space H. If the set B-pi of Bessel vectors for pi is dense in H, then for any vector x is an element of H, the analysis operator theta(x) makes sense as a densely defined operator from B-pi to l(2)(G)-space. If a projection e is an element of M is equivalent to a projection f(1) is an element of M with f(1) <= f is an element of M, then we write e less than or similar to f. Let P-x (resp. P-y) be the orthogonal projection from l(2)(G) onto [theta(x)(B-pi)] (resp. [theta(y)(B-pi)]). Han and Larson have proved the duality properties of projective unitary representations, i.e. P-x <= P-y is equivalent to Q(x) <= Q(y). In this paper we prove that a similar result is true in the sense of von Neumann equivalence of projections, i.e. P-x less than or similar to P-y in lambda(G)' is equivalent to Q(x) less than or similar to Q(y) in pi(G)''.