# Set of tensor products

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?