The characterization of differential operators by locality: Dissipations and ellipticity (Q1075567)

Let \(\delta\) be the generator of a strongly continuous one-parameter group of *-automorphisms of a \(C^*\)-algebra \({\mathcal A}\), and let H be a differential operator of the form \(H=\sum^{n}_{m=0}\lambda_ m\delta^ m\) where \(\lambda_ m\in {\mathbb{C}}\). It is known from previous work that if \({\mathcal A}\) is abelian and H is a dissipation, i.e. \(H(x^*x)\leq x^*H(x)+H(x^*)x\), then H is elliptic in the sense that \(\lambda_ m=0\) for \(m>2\), \(\lambda_ 2\leq 0\) and \(\lambda_ 0\geq 0\). This conclusion is not longer true if e.g. \({\mathcal A}\) is a matrix algebra, but in this paper it is established that the conclusion is true in a variety of special cases where \({\mathcal A}\) is non-abelian. One special case is that there is a factor representation \(\pi\) of \({\mathcal A}\) and a strongly continuous one-parameter unitary group \(t\in {\mathbb{R}}\to U_ t=e^{itK}\in \pi ({\mathcal A})''\) such that U implements the automorphism group defined by \(\delta\) and K is unbounded.