Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and thenfor all .

A theorem of Hausdorff Young type is proved for integral operators in the setting of gage spaces. This theorem is used to show that the norm of the Lp-Fourier transform on unimodular groups is stable under compact extension. © 1978.

Let T be a unitary operator on a Hubert space H. Then in particular, (i) T is a contraction, i.e. ∥ T ∥ ≦ 1; and (ii) The spectrum of T is a subset of the unit circle, i.e. Sp (T)⊂C, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator T a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras. © 1968 by Pacific Journal of Mathematics.

We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real line as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian. © 1974 American Mathematical Society.

A sharp inequality of Hausdorfu-Young type is proved for integral operators. Applications are made in operator theory and in harmonic analysis on locally compact groups. © 1977 Pacific Journal of Mathematics. All rights reserved.