Larry Baggett

Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝn

An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs that come from multiwavelets are characterized. As an application of this abstract formulation, a constructive procedure is developed, which produces all wavelet sets in ℝnrelative to an integral expansive matrix.

Multiplier representations of abelian groups

Abstract The multiplier representations of a locally compact abelian group are classified, in the case there are only type I representations, and a simple criterion is obtained for determining when only type I representations occur. It is also shown that all the irreducible multiplier representations belonging to a fixed multiplier have the same dimension. To obtain these results we have to extend the little group method of Mackey and Blattner to handle multiplier representations of nonseparable groups.

A separable group having a discrete dual space is compact

Abstract By the dual space of a locally compact group G we mean the set of all equivalence classes of irreducible unitary representations of G equipped with the hull-kernel topology which is derived from the group C∗-algebra of G. A consequence of the Peter-Weyl Theorem is that the dual space of a compact group is discrete. Also, a corollary to the Pontryagin Duality Theorem is that a locally compact abelian group is compact if and only if its dual space (group) is discrete. In this article we obtain a generalization of the corollary to the Pontryagin theorem and therefore a kind of converse to the Peter-Weyl Theorem. We prove that a second countable locally compact group is compact if and only if its dual space is discrete. Examples are given which show that the situation is more complicated in the noncommutative case than it is in the abelian case. For example a non-commutative group need not be discrete even though its dual space is compact.

Smooth cocycles for an irrational rotation

Explicit examples of smooth cocycles not cohomologous to constants are constructed. Necessary and sufficient conditions on the irrational numberθ are given for the existence of such cocycles. It is shown that, depending onθ, the set ofCr cocycles whose skew-product is ergodic is either residual or empty.

A sufficient condition for the complete reducibility of the regular representation

The “Mackey machine” is heavily employed to prove the following theorem. Let G be a separable locally compact group. Suppose that every positive definite function p on G which vanishes at infinity is associated with the regular representation R, i.e., p(g) = (Rgϑ, ϑ) for some L2 function ϑ. Then R decomposes into a direct sum of irreducible representations. This generalizes the theorem of Figa-Talamanca for unimodular groups. Although we use his result several times, our techniques are basically very different, the most difficult part occurring in a connected Lie group context.

The Hausdorff dual problem for connected groups

Abstract It is shown that a connected locally compact group G has a Hausdorff unitary dual space if and only if G is a compact extension of an abelian group. Applications to group C ∗ -algebras are given.

Riemann-Lebesgue subsets of Rn and representations which vanish at infinity

The main theorem of this paper is that if χ is a character of a connected closed normal subgroup of a connected Lie group, then every matrix element of the induced representation Uχ vanishes at infinity modulo the kernel of that representation. As a consequence, it is shown that every faithful irreducible unitary representation of a connected motion group vanishes at infinity. In the course of the development a generalization of the classical Riemann-Lebesgue lemma is proved. Suppose M is an analytic submanifold of Rn which is not contained in any proper hyperplane. Then the Fourier transform of any measure, which is concentrated on M and which is absolutely continuous with respect to the “Lebesgue” measure on M, vanishes at infinity.

ON CIRCLE-VALUED COCYCLES OF AN ERGODIC MEASURE-PRESERVING TRANSFORMATION*

Analytic necessary and sufficient conditions are given for a circle-valued functionf to generate a cocycle which is a multiple of a coboundary. These conditions are then used to derive some other new criteria for cocycles to be coboundaries.

Operators arising from representations of nilpotent Lie groups

Abstract The following two results are obtained for an irreducible multiplier representation T of a connected nilpotent Lie group. First, T f is a Hilbert-Schmidt operator if f is square-integrable with compact support. Second, T f is of trace class if f has derivatives with sufficiently many moments. An application is made of the latter result to show that T f can be of trace class even when f is not continuous.

Equivalence of cocycles under an irrational rotation

This paper describes a method for studying the equivalence relation among cocycles for an irrational rotation. A parameterized family of cocycles is presented, which meets the equivalence class of each piecewise absolutely continuous function whose derivative is L2. The difficulties in describing the equivalence among the elements of this family is shown to reduce to the analogous problem for describing equivalence among step functions, thereby relating this paper to the earlier work of Veech, Petersen, Merrill, and others.