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Let $A,B$ be operators that come with an argument from $L^2(\mathbb{R})$. Then we consider the set

$$ \mathcal{A} = \{ e^{iA(f)}\otimes e^{iB(g)}, \ f,g\in L^2(\mathbb{R})\}$$

and in particular the algebra it generates by its double commutant $\mathcal{A}’’$. I’m wondering if there is any nice way of expressing either the set or algebra, $\mathcal{A}$ or $\mathcal{A}’’$, in terms of a Cartesian product, perhaps. Does anybody know any results handling a kind of decomposition of set of tensor products of bounded operators?

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