Vol. 45, No. 1, 1973

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Cyclic vectors for representations associated with positive definite measures: nonseparable groups

Frederick Paul Greenleaf and Martin Allen Moskowitz

Vol. 45 (1973), No. 1, 165–186

Let μ be any positive definite measure on a locally compact group, and let (πμ,μ) be the associated unitary representation of G. Previous work of the authors’ showed that a cyclic vector exists for πμ if G is second countable; there is now a simple proof of this result, due to Hulanicki. Rather elementary conditions on the way μ is related to the geometry of G are examined which are necessary, or sufficient, for the existence of a cyclic vector. These conditions require μ to be “constant” on cosets (or double cosets) of certain subgroups of G. A conjectured necessary and sufficient conditions is presented. These results are adequate to decide whether or not πu is cyclic for various nontrivial measures. As a special case it is shown that the left regular representation of G is cyclic G is first countable.

Mathematical Subject Classification 2000
Primary: 22D10
Received: 23 December 1971
Published: 1 March 1973
Frederick Paul Greenleaf
Martin Allen Moskowitz