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

TL;DR

This paper characterizes certain smooth operators in a deformed algebra setting, confirming a conjecture by Rieffel about their structure as left multiplication operators in a noncommutative deformation context.

Contribution

It provides a complete characterization of Rieffel's conjecture, identifying smooth vectors commuting with specific right multiplication operators as left multiplication operators in a deformed algebra.

Findings

Characterization of smooth vectors as left multiplication operators

Confirmation of Rieffel's conjecture in the context of deformation quantization

Identification of algebraic structure of operators commuting with twisted right multiplications

Abstract

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 $\|f\|_2 := \left\| \int_{\mathbb{R}^n} f(x)^*f(x) \, dx \right\|_\mathscr{A}^{1 / 2}$. 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 $h \in \mathcal{S}^\mathscr{A}(\mathbb{R}^n)$, 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.