Pseudodifferential operators with C$^*$-algebra-valued symbols: Abstract characterizations

Abstract:

Given a separable unital C$^*$-algebra $C$ with norm $||\cdot ||$, let $E_n$ denote the Banach-space completion of the $C$-valued Schwartz space on $\mathbb {R}^{n}$ with norm $||f||_2=||\langle f,f\rangle ||^{1/2}$, $\langle f,g\rangle =\int f(x)^*g(x)dx$. The assignment of the pseudodifferential operator $A=a(x,D)$ with $C$-valued symbol $a(x,\xi )$ to each smooth function with bounded derivatives $a\in \mathcal {B}^C (\mathbb {R}^{2n})$ defines an injective mapping $O$, from $\mathcal {B}^C(\mathbb {R}^{2n})$ to the set $\mathcal {H}$ of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module $E_n$. In this paper, we construct a left-inverse $S$ for $O$ and prove that $S$ is injective if $C$ is commutative. This generalizes Cordes’ description of $\mathcal {H}$ in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric $n\times n$ matrix $J$ and if $C$ is commutative, then any $A\in \mathcal {H}$ which commutes with every pseudodifferential operator with symbol $F(x+J\xi )$, $F\in \mathcal {B}^C(\mathbb {R}^{2n})$, is a pseudodifferential operator with symbol $G(x-J\xi )$, for some $G\in \mathcal {B}^C(\mathbb {R}^{2n})$. That was conjectured by Rieffel.