Duality Properties in Von Neumann Algebras of Projective Unitary Representations

Yizheng Li, Chengjun Hou

Let $\pi$ be a projective representation of a countable discrete group $G$ on a Hilbert space $H$. If the set $\Cal B_{\pi}$ of Bessel vectors for $\pi$ is dense in $H$, then for any vector $x\in H$, the analysis operator $\Theta_x$ makes sense as a densely defined operator from $\Cal B_{\pi}$ to $l_2(G)$-space. If a projection $e\in M$ is equivalent to a projection $f_1\in M$ with $f_1\leq f\in M$, then we write $e\lesssim f$. Let $P_x$ (resp. $P_y$) be the orthogonal projection from $\ell^2(G)$ onto [$\Theta_x(\Cal B_{\pi})]$ (resp. [$\Theta_y(\Cal B_{\pi})]$. Han and Larson have proved the duality properties of projective unitary representations, i.e. $P_x\leq P_y$ is equivalent to $Q_x\leq Q_y$. In this paper we prove that a similar result is true in the sense of von Neumann equivalence of projections, i.e., $P_x\lesssim P_y$ in $\lambda (G)′$ is equivalent to $Q_x\lesssim Q_y$ in $\pi (G)''$.