On Rieffel’s conjecture characterizing a deformed algebra as Heisenberg smooth operators

Let \mathscr{A} be a unital C ^{*} -algebra and E_{n} be the Hilbert \mathscr{A} -module defined as the completion of the \mathscr{A} -valued Schwartz function space \mathcal{S}^{\mathscr{A}}(\mathbb{R}^{n}) with respect to the norm \|f\|_{2} := \|{ \int_{\mathbb{R}^n} f(x)^*f(x) \, dx}\|_\mathscr{A}^{1 / 2} . Also, let \operatorname{Ad}\mathcal{U} be the canonical action of the (2n + 1) -dimensional Heisenberg group by conjugation on the algebra of adjointable operators on E_{n} , and let J be a skew-symmetric linear transformation on \mathbb{R}^{n} . We characterize the smooth vectors under \operatorname{Ad}\mathcal{U} which commute with a certain algebra of right multiplication operators R_{h} , with h \in \mathcal{S}^{\mathscr{A}}(\mathbb{R}^{n}) , where the product is “twisted” with respect to J according to a deformation quantization procedure introduced by M.A. Rieffel. More precisely, we establish that they coincide with an algebra of left multiplication operators and show that this solves, in particular, a conjecture posed by Rieffel.