Projective repesentations of the Hilbert Lie group U (H)2 via quasifree statres on the CAR algebra

The group $\text{U}$ (H)₂ of unitary operators (on a Hilbert space H) which differ from the identity by a Hilbert-Schmidt operator may be imbedded in the group of Bogoliubov automorphisms of the CAR algebra over H in such a way as to be weakly inner in any gauge-invariant quasifree representation. Consequently each such quasifree representation determines a projective representation of $\text{U}$ (H)₂. If 0 ? A ? I is the operator on H determining the quasifree representation π_A and ?_A denotes the cyclic projective representation of $\text{U}$ (H)₂ generated from the G.N.S. cyclic vector $\text{Ω}{}_{\mathrm{A}}\text{for}\text{π}{}_{\mathrm{A}}$, then the 2-cocycle in $\text{U}$ (H)₂ determined by ?_A can be given explicitly. We prove that this 2-cocycle is a coboundary if any only if A or 1 ? A is Hilbert-Schmidt. The representations ?_A, on restriction to the group $\text{U}$ (H)₁ consisting of unitaries which differ from the identity by a trace class operator, always determine 2-cocycles which are coboundaries. These representations of $\text{U}$ (H)₁ have already been investigated by 21., 22., 87–110). Thus the Stratila-Voiculescu representations of $\text{U}$ (H)₁ always extend to projective representations of $\text{U}$ (H)₂ and to ordinary representations when A or 1 ? A is Hilbert-Schmidt. This fact enables exploitation of the type analysis of Stratila and Voiculescu to determine the type of the von Neumann algebra ρ_A($\text{U}$(H)₂)″. In the special case where 0 and 1 are not eigenvalues of $\text{A, Ω}{}_{\mathrm{A}}$ is cyclic and separating for ρ_A($\text{U}$(H)₂)″ and hence determines a K.M.S. state on this algebra. It is shown that for special choices of A, type III_λ (0 < λ ? 1) factors ρ_A($\text{U}$(H)₂)″ may be constructed.