Spectral synthesis and operator synthesis

K. Parthasarathy ¹   ; R. Prakash ²

¹ Ramanujan Institute for Advanced Study in Mathematics University of Madras Chennai 600 005, India
² Ramanujan Institute for Advanced Study in Mathematics University of Madras Chennai-600 005, India

Studia Mathematica, Tome 177 (2006) no. 2, pp. 173-181

Résumé

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 $\mathop {\rm VN}\nolimits (G)$, $X$-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 $V^{\infty }(G)$-submodule $ \widehat {X}$ of ${\mathcal B}(L^{2}(G))$ (where $V^{\infty }(G)$ is the weak-$*$ Haagerup tensor product $L^{\infty }(G)\otimes _{w^{*}h} L^{\infty }(G)$ ), define the concept of $ \widehat {X}$-operator synthesis and prove that a closed set $E$ in $G$ is of $X$-synthesis if and only if $E^{*}:=\{ (x,y)\in G\times G: xy^{-1}\in E\} $ is of $\widehat {X}$-operator synthesis.

Détail

DOI : 10.4064/sm177-2-6

Keywords: relations between spectral synthesis fourier algebra compact group concept operator synthesis due arveson have studied literature submodule mathop nolimits x synthesis has introduced kaniuth lau studied recently present authors associate infty submodule widehat mathcal where infty weak * haagerup tensor product infty otimes * infty define concept widehat operator synthesis prove closed set x synthesis only * times widehat operator synthesis