Pseudodifferential operators with C$^*$-algebra-valued symbols: Abstract characterizations
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- by Severino T. Melo and Marcela I. Merklen PDF
- Proc. Amer. Math. Soc. 136 (2008), 219-227 Request permission
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.References
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Additional Information
- Severino T. Melo
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05311-970 São Paulo, Brazil
- MR Author ID: 294301
- Email: toscano@ime.usp.br
- Marcela I. Merklen
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05311-970 São Paulo, Brazil
- Email: marcela@ime.usp.br
- Received by editor(s): October 18, 2006
- Published electronically: October 4, 2007
- Additional Notes: The first author was partially supported by the Brazilian agency CNPq (Processo 306214/ 2003-2)
The second author had a postdoctorol position sponsored by CAPES-PRODOC - Communicated by: Andreas Seeger
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 219-227
- MSC (2000): Primary 47G30; Secondary 46L65, 35S05
- DOI: https://doi.org/10.1090/S0002-9939-07-09006-5
- MathSciNet review: 2350407