Some representations of the automorphism group of an infinite continuous homogeneous measure algebra

Abstract

Let (X, r, m) be an infinite continuous homogeneous measure space. Let A be the measure algebra of (X, τ, m) and G be the automorphism group of A. The canonical representation of G on the subspace of all elements of ®nL\X, r, m) of some fixed maximal symmetry type is irreducible. Two such representations are equivalent iff they correspond to the same ne N and to the same partition of n. l Introdution and notation* Let H be a Hubert space. If neN, let (gΓ H denote the tensor product of H with itself n times. Let Sn be the symmetric group on the first n natural numbers. Let Θ be the unique representation of Sn on ®w H such that θ(g)(v1 (x) v2 (x) • ® O = vgω (g) vgW (x) (x) ^(Λ) if geS n and ^.e If for 1 ^ ί ^ w. If is a set of operators on a Hubert space and contains the adjoint of each of its members, then the commutant S' of is a von Neumann algebra; the double commutant S of is the smallest von Neumann algebra which contains [1]. Θ(Sn) is generated by its mutually orthogonal minimal projections [9]. These projections are in a 1-1 correspondence with the maximal symmetry types. The vectors in the range of a minimal projection are said to be of the corresponding maximal symmetry type. Let U{H) be the unitary group on H. Let I be the canonical representation of U(H) on H. If ne N, let <gΓ / be the tensor product of I with itself n times. The restriction of ® % I to the subspace of all vectors of any fixed maximal symmetry type is irreducible. If H is finite dimensional, this result is classical [9]; if H is infinite dimensional, it is due to Segal [8]. The author has obtained similar results [3] for the symmetric groups on an infinite set S, where is an orthonormal basis for H. In all three cases, these representations can be explicitly characterized. Below we prove that the analogous representations of the automorphism group of an infinite homogeneous measure algebra are irreducible. No characterization of these representations is known.