On the implementability of non\({}^*\)Bogoliubov automorphisms of CAR algebras on Fock spaces (Q1103159)

This paper is devoted to a proof of the following result. Let \(A(K,J)\) be the self-dual \(C^*\)-algebra of the canonical anti-commutation relations over the infinite-dimensional Hilbert space \(K\) with anti-unitary symmetry operator \(J\). Let \(P\) be a projection such that \(JPJ=1-P\), let \(\phi_ p\) be the Fock state corresponding to \(P\), let \(\pi\) be the irreducible GMS-representation corresponding to \(\phi_ p\) and let \(\alpha_ F\) be the Bogoliubov automorphism determined by a bounded invertible operator \(F\) in \(K\). Then \(\alpha_ F\) is implementable in the sense that there exists a bounded invertible operator \(\Gamma\) on \(H_{\pi}\) such that for all \(a\) in \(A(K,J)\), \[ \Gamma\pi(a)\Gamma^{-1}=\pi(\alpha_ F(a)) \] if and only if FP-PF is Hilbert-Schmidt and \(F^*F-1\) is trace class. This result generalises a well known result for the case in which F is unitary.