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

Abstract

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 $\Vert f{\Vert }_2:=\Vert {\int }_{{\mathbb{R}}^n}f(x{)}^{\ast }f(x)dx{\Vert }_{\mathcal{A}}^{1/2}$. Also, let $\mathrm{Ad}\mathcal{U}$ 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 $\mathrm{Ad}\mathcal{U}$ 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.