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


Authors: Severino T. Melo and Marcela I. Merklen
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
Published electronically: October 4, 2007
MathSciNet review: 2350407
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Abstract: Given a separable unital C$ ^*$-algebra $ C$ with norm $ \vert\vert\cdot\vert\vert$, let $ E_n$ denote the Banach-space completion of the $ C$-valued Schwartz space on $ \mathbb{R}^{n}$ with norm $ \vert\vert f\vert\vert _2=\vert\vert\langle f,f\rangle\vert\vert^{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(\rn)$, is a operator with symbol $ G(x-J\xi)$, for some $ G\in \mathcal{B}^C(\rn)$. That was conjectured by Rieffel.


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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
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

DOI: https://doi.org/10.1090/S0002-9939-07-09006-5
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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