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Spectra of independent power measures

Published online by Cambridge University Press:  24 October 2008

W. J. Bailey ,
G. Brown  and
W. Moran
W. J. Bailey
Affiliation:
University of Liverpool and University of York
G. Brown
Affiliation:
University of Liverpool and University of York
W. Moran
Affiliation:
University of Liverpool and University of York
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Extract

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 27 - 35
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

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