Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Boundedness of pseudodifferential operators of a C*-algebra-valued symbol

Marcela I. Merklena1

a1 Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05315-970 São Paulo, Brazil (marcela@ime.usp.br)

Abstract

Let us consider the set SA(Rn) of rapidly decreasing functions G: RnA, where A is a separable C*-algebra. We prove a version of the Calderón–Vaillancourt theorem for pseudodifferential operators acting on SA(Rn) whose symbol is A-valued. Given a skew-symmetric matrix, J, we prove that a pseudodifferential operator that commutes with G(x + JD), G xs2208 SA(Rn), is of the form F(xJD), for F a C-function with bounded derivatives of all orders.

(Received October 12 2003)

(Accepted April 06 2005)