Resultados motivados por uma caracterização de operadores pseudo-diferenciais conjecturada por Rieffel (Ph. D. thesis, in Portuguese)

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We work with functions defined in Rⁿ with values in a C*- algebra A. We consider the set of the functions of Schwartz (the rapidly decreasing ones) with the usual l₂-norm. We denote ²ⁿA the set of functions of class C∞ with bounded derivatives. We prove, generalizing a result in [10], that pseudodifferential operators with symbol in ²ⁿA are continuous in for the l₂-norm. In[1], Rieffel proves that ⁿA acts on, through a deformed product induced by an anti-symmetric matrix, J (this is the so-called left-regular representation of ²ⁿA). At the end of chapter 4, Rieffel poses the conjecture that all operators adjointable in and that commute with the right-regular representation of 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 L²(Rⁿ) as the pseudodifferential operators with symbol in ²ⁿC (see [17]). We also prove in this work that, if the natural generalization of Cordes characterization holds, then Rieffel's conjecture also holds.