On the Ergodic Mixing Theorem

H. A. Dye
1965 Transactions of the American Mathematical Society  
Introduction. An invertible measure-preserving transformation T of a probability space (T, S, m) is called weakly mixing if y n-1 (1.1) lim -S I m(E n T~kF) -m(E)m(F) \ = 0, n 1 = 0 or what is equivalent, if for k in the complement of a set of density 0, m(E O T~*F)-> m(£)m(F), for each pair of measurable sets E, F. (The so-called mixing theorem of ergodic theory (see Halmos [5, p. 39] ) establishes the equivalence of (1.1) and each of the following two conditions: first, that the unitary
more » ... t the unitary operator l/r on L2 (T, S, m) induced by T has continuous spectrum on the subspace of L2 orthogonal to the constant functions; and second, that the Cartesian square T x T on the product space (T x T, S x S, m x m) is ergodic. This theorem appeared early in the modern development of ergodic theory, in works of von Neumann and Hopf, and unlike most of these early results, it appears to have resisted subsequent generalization. For example, no variant of the theorem seems to be known in case the measure-preserving transformation Tis noninvertible. In this situation the induced operator UT is, in general, a nonunitary isometry, to which ordinary spectral theory does not apply. In this note we prove an abstract mixing theorem in a setting considerably more general than the above. In place of the semigroup of non-negative integers we consider an arbitrary topological semigroup G which admits both a leftinvariant and a right-invariant Banach mean. This includes all abelian or compact topological semigroups, and an extensive class of discrete semigroups (see Day [1]). And, in place of the operator UT, we consider an arbitrary weakly continuous isometric representation Ux of G on a complex Hubert space. The operation lim (1 /n) 1Z"Zo is supplanted by "almost convergence," a notion introduced by G. Lorentz [6] , and in turn, the condition that UT have continuous spectrum is supplanted by the condition that the representation Ux have no finite-dimensional subrepresentations. The abstract mixing theorem (proved in §3) specializes in the measure-theoretic case to a direct generalization of the classical mixing theorem (see §4). Our proof is quite elementary, consisting of a refinement for amenable semigroups of the traditional Peter-Weyl theory.
doi:10.2307/1993948 fatcat:icfozxysivbudjf5lzxdmivsaq