Resultados motivados por uma caracteriza\c{c}\~ao de operadores pseudo-diferenciais conjecturada por Rieffel (Ph. D. thesis, in Portuguese)

Abstract

We work with functions defined in R^n with values in a C^*- algebra A. We consider the set \Sa of the functions of Schwartz (the rapidly decreasing ones) with the usual l_2-norm. We denote \CB^{2n}A the set of functions of class C^\infty with bounded derivatives. We prove, generalizing a result in [10], that pseudodifferential operators with symbol in \CB^{2n}A are continuous in \Sa for the l_2-norm. In[1], Rieffel proves that \CB^nA acts on \Sa, through a deformed product induced by an anti-symmetric matrix, J (this is the so-called left-regular representation of \CB^{2n}A). At the end of chapter 4, Rieffel poses the conjecture that all operators adjointable in \Sa and that commute with the right-regular representation of \CB^nA (for the deformed product above) are precisely the operators of the left-regular representation. We prove this for the case A=C (the complex numbers)(see [14]), using Cordes characterization of Heisenberg-smooth operators on L^2(R^n) as the pseudodifferential operators with symbol in \CB^{2n}C (see [17]). We also prove in this work that, if the natural generalization of Cordes characterization holds, then Rieffel's conjecture also holds.