Automorphic Group Representations: A New Proof of Blattner's Theorem

To each of these ways of obtaining 51 correspond automorphisms of 5(. The construction that will interest us is the fourth one. The automorphisms in question arise naturally in the following way. The vectors in E generate 51 as a von Neumann algebra; the full orthogonal group O(E) of E acts on E, hence on 5t as automorphisms. The normal subgroup G2 of O(E) which acts as inner automorphisms of 51 is of particular interest. The group G2 was determined by Blattner [2]. The normal subgroup Gi of O(E) which acts as inner automorphisms of the CAR algebra is also of interest, and was determined by Shale and Stinespring [21]. The purpose of this article is to give new, simpler proofs of these two results. We quote a result of the first-named author according to which each element of a proper normal subgroup of 0(E) is a compact perturbation either of the identity operator / or of —/. We prove directly that Gx and G2 are proper normal subgroups of O(E) and then exploit the spectral theory of compact operators. Although our proofs are new, they make liberal use of original ideas of Blattner, Shale and Stinespring. It is illuminating to prove the two basic results simultaneously. Our proofs are naive and constructive: corresponding to the spectral decomposition of an orthogonal operator in G2, the implementing unitary operator in 51 is explicitly constructed as an infinite product. Let ^ denote the CAR algebra. Since the centres of 51 and # comprise scalar multiples of the identity, each implementing unitary operator is determined up to such a scalar. Consequently we have projective unitary representations of Gx and G2 as follows: G2