Geodesic
Star product algebras of test functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 1, pp. 3-17 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that the Gelfand–Shilov spaces $S^{\beta}_{\alpha}$ are topological algebras under the Moyal $\star$-product if and only if $\alpha\ge\beta$. These spaces of test functions can be used to construct a noncommutative field theory. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in $S^{\beta}_{\alpha}$ if and only if $\beta<1/2$.
Keywords: noncommutative quantum field theory, Moyal product, topological $*$-algebra, Gelfand–Shilov space. 
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M. A. Soloviev. Star product algebras of test functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/TMF_2007_153_1_a0/

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