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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 $\alpha\ge\beta$.
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M. Chaichian, M. N. Mnatsakanova, K. Nishijima, A.Tureanu, and Yu.A.Vernov, Towards an axiomatic formulation of noncommutative field theories
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