In this note we define and explore, à la Godement, spectral subspaces of Banach space representations of the Fourier-Eymard algebra of a (nonabelian) locally compact group.
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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{∞}(G)$-submodule X̂ of ℬ(L²(G)) (where $V^{∞}(G)$ is the weak-* Haagerup tensor product $L^{∞}(G) ⊗_{w*h} L^{∞}(G)$), define the concept of X̂-operator synthesis and prove that a closed set E in G is of X-synthesis if and only if $E*: = {(x,y) ∈ G × G: xy^{-1} ∈ E}$ is of X̂-operator synthesis.
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