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Generalized F. and M. Riesz Theorems

Throughout this section, we adopt the notation of Section 5, specifically, the notation and assumptions of Theorem 5.1.

Suppose that is a sup path attaining representation of by isomorphisms of . In [4], we proved the following result concerning bounded operators from into that commute with the representation in the following sense:

for all .

Theorem 6.1   Suppose that is a representation of that is sup path attaining, and that commutes with . Let be weakly analytic. Then is also weakly analytic.

To describe an interesting application of this theorem from [4], let us recall the following.

Definition 6.2   Let be a sup path attaining representation of in . A weakly measurable in is called quasi-invariant if and are mutually absolutely continuous for all . Hence if is quasi-invariant and , then if and only if for all .

Using Theorem 6.1 we obtained in [4] the following extension of results of de Leeuw-Glicksberg [9] and Forelli [12], concerning quasi-invariant measures.

Theorem 6.3   Suppose that is a sup path attaining representation of by isometries of . Suppose that is weakly analytic, and is quasi-invariant. Write for the Lebesgue decomposition of with respect to . Then both and are weakly analytic. In particular, the spectra of and are contained in .

Our goal in this section is to extend Theorems 6.1 above to representations of a locally compact abelian group with ordered dual group . More specifically, we want to prove the following theorems.

Theorem 6.4   Suppose that is a sup path attaining representation of by isomorphisms of such that is sup path attaining for each . Suppose that commutes with in the sense that

for all . Let be weakly analytic. Then is also weakly analytic.

As shown in [4, Theorem (4.10)] for the case , an immediate corollary of Theorem 6.4 is the following result.

Theorem 6.5   Suppose that is a sup path attaining representation of by isometries of , such that is sup path attaining for each . Suppose that is weakly analytic with respect to , and is quasi-invariant with respect to . Write for the Lebesgue decomposition of with respect to . Then both and are weakly analytic with respect to . In particular, the -spectra of and are contained in .

Proof of Theorem 6.4.     Write

as in (5.1), where the series converges unconditionally in . Then

 (42)

It is enough to show that the -spectrum of each term is contained in . Consider the measure . We have . Hence by Theorem 4.4, is -analytic. Applying Theorem 6.1, we see that

 (43)

Since commutes with , it is easy to see from Proposition 3.10 and Corollary 3.11 that

Hence by (43) and Theorem 4.4,

which shows the desired result for the first term of the series in (42). The other terms of the series (42) are handled similarly.

Acknowledgments The second author is grateful for financial support from the National Science Foundation (U.S.A.) and the Research Board of the University of Missouri.

Next: Bibliography Up: Decomposition of analytic measures Previous: Decomposition of Analytic Measures
Stephen Montgomery-Smith 2002-10-30