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Subalgebras of the dual of the Fourier algebra of a compact group

Published online by Cambridge University Press:  24 October 2008

Charles F. Dunkl  and
Donald E. Ramirez
Charles F. Dunkl
Affiliation:
University of Virginia, Charlottesville, Virginia, U.S.A.
Donald E. Ramirez
Affiliation:
University of Virginia, Charlottesville, Virginia, U.S.A.
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Extract

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 2 , March 1972 , pp. 329 - 333
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Dunkl, C. and Ramirez, D.Topics in harmonic analysis (Appleton–Century–Crofts, New York, 1971).Google Scholar
(2)Dunkl, C. and Ramirez, D.Weakly almost periodic functionals on the Fourier algebra, to appear in Trans. Amer. Math. Soc.Google Scholar
(3)Dunford, N. and Schwartz, J.Linear operators, Part I (Interscience, New York, 1958).Google Scholar