Calvin C. Moore

Asymptotic properties of unitary representations

Abstract If ϱ is a unitary representation of a locally compact group G, one is very often interested in the asymptotic behavior of the operators ϱ(g) as g tends to ∞ in G. Put another way one is interested in what operators not in ϱ(G) can be (weak) limits of the ϱ(g) as g tends to ∞. The first part of the paper deals with the question of what kinds of unitary operators can be limits of ϱ(g), and we show that for connected G the possibilities are very limited. The second part of the paper deals with the question of when one can conclude that essentially the only operator of any kind not in ϱ(G) obtained as a limit of the ϱ(g) is 0. This amounts to showing that the matrix coefficients of ϱ “vanish at ∞,” and we establish affirmative results for irreducible representations of connected algebraic groups over local fields (archimedian and non-archimedian). Such results turn out to have applications to the theory of automorphic forms and to ergodic theory.

Orbit structure and countable sections for actions of continuous groups

Abstract It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space ( S , μ ), then there is a Borel subset E ⊂ S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ∼ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackeys theory of measure groupoids. A natural notion of “approximate finiteness” ( AF ) is introduced for nonsingular actions of G , and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF . Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.

Exponential Decay of Correlation Coefficients for Geodesic Flows

We obtain exponential decay bounds for correlation coefficients of geodesic flows on surfaces of constant negative curvature (and for all Riemannian symmetric spaces of rank one), answering a question posed by Marina Ratner. The square integrable functions on the unit sphere bundle of M are allowed to satisfy weak differentiability conditions. The methods are those of unitary representation theory and invoke the notion of Sobelev vectors of representations. In the course of the discussion we obtain a new characterization of tempered irreducible representation of semi-simple groups.

Groups with T1 primitive ideal spaces

Abstract We give a simple necessary and sufficient condition for the group C ∗ -algebra of a connected locally compact group to have a T 1 primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C ∗ -algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C ∗ -algebra of a discrete finitely generated solvable group is T 1 if and only if the group is a finite extension of a nilpotent group.

Amenable subgroups of semi-simple groups and proximal flows

We present a classification of maximal amenable subgroups of a semi-simple groupG. The result is that modulo a technical connectivity condition, there are precisely 2′ conjugacy classes of such subgroups ofG and we shall describe them explicitly. Herel is the split rank of the groupG. These groups are the isotropy groups of the action ofG on the Satake-Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action ofG on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable.

Groups admitting ergodic actions with generalized discrete spectrum

The class of groups admitting an effective ergodic action with generalized discrete spectrum is a natural generalization of the class of maximally almost periodic groups. H. Freudenthal has given a complete characterization of the connected maximally almost periodic groups, and here we give a complete characterization of the almost connected groups admitting an effective ergodic action with generalized discrete spectrum. Namely, we show that an almost connected group is in this class if and only if it is typeR. It is known that this is equivalent to the group being of polynomial growth, and for Lie groups is just the condition that all eigenvalues of the adjoint representation lie on the unit circle.