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Journal of Lie Theory 19 (2009), No. 2, 275--290
Copyright Heldermann Verlag 2009

Smooth and Weak Synthesis of the Anti-Diagonal in Fourier Algebras of Lie Groups

B. Doug Park
Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Ebrahim Samei
Dept. of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada


Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal ${\check\Delta}_G = \{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[{n\over2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property of J. Ludwig and L. Turowska [Growth and smooth spectral synthesis in the Fourier algebras of Lie groups, Studia Math. 176 (2006) 139--158]. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed by B. E. Forrest, E. Samei and N. Spronk [Convolutions on compact groups and Fourier algebras of coset spaces, Studia Math. to appear; arXiv:0705.4277]. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.

Keywords: Locally compact groups, Lie groups, Fourier algebras, smooth synthesis, weak synthesis.

MSC: 43A30, 43A45; 22E15, 43A80

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