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

Let $\mathcal{A}$ be a unital C${}^{\ast }$-algebra and $E_n$ be the Hilbert $\mathcal{A}$-module defined as the completion of the $\mathcal{A}$-valued Schwartz function space ${\mathcal{S}}^{\mathcal{A}}({\mathbb{R}}^n)$ with respect to the norm $\mathrm{\parallel }f{\mathrm{\parallel }}_2:={\parallel {\int }_{{\mathbb{R}}^n}f(x{)}^{\ast }f(x)dx\parallel }_{\mathcal{A}}^{1\mathrm{/}2}$. Also, let $\text{Ad }\mathcal{U}$ 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 $\text{Ad }\mathcal{U}$ which commute with a certain algebra of right multiplication operators $R_h$, with $h\in {\mathcal{S}}^{\mathcal{A}}({\mathbb{R}}^n)$, 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.