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March 25, 2015 By

Cordes characterization for pseudodifferential operators with symbols valued in a noncommutative C$^*$-algebra

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 24–32

March 30, 2009

Severino T. Melo, Instituto de Matematica e Universidade de Sao Paulo, Rua do Matao 1010, 05508-090 Sao Paulo, Brazil; email: toscano@ime.usp.br

Marcela I. Merklen, Instituto de Matematica e Universidade de Sao Paulo, Rua do Matao 1010, 05508-090 Sao Paulo, Brazil; email: marcela@ime.usp.br

Abstract/Résumé:

Given a separable unital C\(^*\)-algebra \({\mathcal A}\) with norm \(\|\cdot\|\), let \(E\) denote the Banach-space completion of the \({\mathcal A}\)-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 \(B=b(x,D)\) with \({\mathcal A}\)-valued symbol \(b(x,\xi)\) to each smooth function with bounded derivatives \(b\in\mathcal{B}^{\mathcal{A}(\mathbb{R}^{2n})}\) defines an injective mapping \(O\) from \(\mathcal{B}^{\mathcal{A}(\mathbb{R}^{2n})}\) 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 C\(^*\)-module \(E\). It is known that \(O\) is surjective if \({\mathcal A}\) is commutative. In this paper, we show that if \(O\) is surjective for \({\mathcal A}\), then it is also surjective for \(M_k({\mathcal A})\).

Étant donné une C\(^*\)-algèbre \(A\), séparable et avec unité, soit \(E\) l’espace de Banach obtenu par complétation de l’espace de Schwartz sur \(\mathbb{R}^n\) avec valeurs dans \(A\) par la norme induite par le produit interne à valeurs dans A: \(\langle f,g\rangle=\int f(x)^*g(x)\,dx\). L’association de l’opérateur pseudo-différentiel \(B=b(x,D)\), ayant symbol \(b(x,\xi)\) à valeurs dans A, à chaque fonction smooth \(b\), à derivées bornées, define une application injective \(O\) de l’ensemble de tous ces symbols dans l’ensemble de tous les opérateurs ayant orbite lisse par l’action du group de Heisenberg sur l’algèbre de tous les opérateurs adjointables sur le C\(^*\)-module de Hilbert \(E\). Il est bien connu que, si \(A\) est commutatif, alors \(O\) est surjective. Dans cet article nous montrons que, si \(O\) est surjective pour une algèbre quelconque \(A\), alors elle est surjective aussi pour l’algèbre des matrices \(k\) par \(k\) à coefficients dans \(A\).

Keywords: Hilbert C∗-modules, pseudodifferential operators
AMS Subject Classification: Pseudodifferential operators 47G30

[This journal is open access except for the current year and the preceding 5 years]

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