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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multipliers on modules over the Fourier algebra


Authors: Charles F. Dunkl and Donald E. Ramirez
Journal: Trans. Amer. Math. Soc. 172 (1972), 357-364
MSC: Primary 43A22
MathSciNet review: 0324318
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Abstract: Let $ G$ be an infinite compact group and $ \hat G$ its dual. For $ 1 \leq p < \infty ,{\mathfrak{L}^p}(\hat G)$ is a module over $ {\mathfrak{L}^1}(\hat G) \cong A(G)$, the Fourier algebra of $ G$. For $ 1 \leq p,q < \infty $, let $ {\mathfrak{M}_{p,q}} = {\operatorname{Hom} _{A(G)}}({\mathfrak{L}^p}(\hat G),{\mathfrak{L}^q}(\hat G))$. If $ G$ is abelian, then $ {\mathfrak{M}_{p,p}}$ is the space of $ {L^P}(\hat G)$-multipliers. For $ 1 \leq p < 2$ and $ p'$ the conjugate index of $ p$,

$\displaystyle A(G) \cong {\mathfrak{M}_{1,1}} \subset {\mathfrak{M}_{p,p}} = {\mathfrak{M}_{p',p'}} \subsetneqq {\mathfrak{M}_{2,2}} \cong {L^\infty }(G).$

Further, the space $ {\mathfrak{M}_{p,p}}$ is the dual of a space called $ \mathcal{A}_p$, a subspace of $ {\mathcal{C}_0}(\hat G)$. Using a method of J. F. Price we observe that

$\displaystyle \cup \{ {\mathfrak{M}_{q,q}}:1 \leq q < p\} \subsetneqq {\mathfrak{M}_{p,p}} \subsetneqq \cap \{ {\mathfrak{M}_{q,q}}:p < q < 2\} $

(where $ 1 < p < 2$). Finally, $ {\mathfrak{M}_{q,p}} = \{ 0\} $ for $ 1 \leq p < q < \infty $.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0324318-3
PII: S 0002-9947(1972)0324318-3
Keywords: Fourier algebra, modules over the Fourier algebra, multipliers
Article copyright: © Copyright 1972 American Mathematical Society