William Moran

An elementary proof of Gleason's theorem

Gleasons theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an elementary proof of Gleasons theorem which is accessible to undergraduates having completed a first course in real analysis.

The Pompeiu problem for groups

This question has been first studied in some detail in [8] where partial results are obtained: principally, semisimple Lie groups are not Pompeiu in general while the Heisenberg group and the motion group of

A decomposition theorem for numbers in which the summands have prescribed normality properties

IR^{2}

Ergodic measures for the irrational rotation on the circle

where shown to be Pompeiu. Certain extensions have recently been obtained in [9]: semidirect products of vector groups with vector groups are Pompeiu, and the motion group of

Non-monomial Multiplier Representations of Abelian Groups

IR^{n},

Some groups with T 1 primitive ideal spaces

\uparrow\iota\geq 4

Raikov Systems and Radicals in Convolution Measure Algebras

, is not. It turns out that there are far reaching generalizations of these results (joint work with A. Carey and W. Moran [6]).

The inhomogeneous minimum of binary quadratic forms

If A is a set of integers, each exceeding unity, then every real number can be expressed as a sum of four numbers, each of which is non-normal with respect to every base belonging to A and is normal to every base which is not multiplicatively dependent on any element of A. This result is proved and generalized to allow noninteger bases.

On the Group Theoretic Approach to the Canonical Anticommutation Relations

Riesz products are employed to give a construction of quasi-invariant ergodic measures under the irrational rotation of T. By suitable choice of the parameters such measures may be required to have Fourier-Stieltjes coefficients vanishing at infinity. We show further that these are the unique quasi-invariant measures on T with their associated Radon-Nikodym derivative.