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Spectral synthesis for the operator space projective tensor product of \(C^{\ast}\)-algebras. (English) Zbl 1286.46062

Summary: We study the spectral synthesis for the Banach \(\ast\)-algebra \(A\hat\otimes B\), the operator space projective tensor product of \(C^{\ast}\)-algebras \(A\) and \(B\). It is shown that, if \(A\) or \(B\) has finitely many closed ideals, then \(A\hat\otimes B\) obeys spectral synthesis. The Banach algebra \(A\hat\otimes A\) with the reverse involution is also studied.

MSC:

46L07 Operator spaces and completely bounded maps
46L06 Tensor products of \(C^*\)-algebras
47L25 Operator spaces (= matricially normed spaces)
43A45 Spectral synthesis on groups, semigroups, etc.
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