Locality and differential operators on \(C^*\)-algebras. (English) Zbl 0593.46056
Let \(\delta\) be the generator of a strongly continuous one-parameter group of *-automorphisms of a \(C^*\)-algebra \({\mathcal A}\). It is proved that a densely defined operator K on \({\mathcal A}\) satisfying a certain second order locality condition with respect to \(\delta\) must have the form \(K=L\delta +M\delta^ 2\), where L and M are possibly unbounded central multipliers of the two-sided ideal generated by the range of \(\delta\). If K, in addition, is an invariant dissipation satisfying a certain saturation property, it is proved that K is the generator of a strongly continuous one-parameter semigroup of completely positive contractions.
Since this paper was written, various other concepts of locality have been introduced and interrelated, see e.g. O. Bratteli, T. Digernes and G. A. Elliott, Lect. Notes Math. 1132, 46-83 (1985; Zbl 0568.46051), O. Bratteli, G. A. Elliott and D. W. Robinson [Compositio Math. 58, 279-319 (1986), Publ. Res. Inst. Math. Sci. Kyoto Univ. 21, 1031-1049 (1985) and J. Oper. Theory 16, 213-233 (1986)]. Furthermore, generator properties have been established for some local operators K which are not invariant; for a review of these developments see [D. W. Robinson, Differential operators on \(C^*\)-algebras, In ”Operator Algebras and Mathematical Physics” (eds. P. E. T. Jørgensen and P. Muhly) Contemp. Math. 60, AMS, Providence (1986)].
Since this paper was written, various other concepts of locality have been introduced and interrelated, see e.g. O. Bratteli, T. Digernes and G. A. Elliott, Lect. Notes Math. 1132, 46-83 (1985; Zbl 0568.46051), O. Bratteli, G. A. Elliott and D. W. Robinson [Compositio Math. 58, 279-319 (1986), Publ. Res. Inst. Math. Sci. Kyoto Univ. 21, 1031-1049 (1985) and J. Oper. Theory 16, 213-233 (1986)]. Furthermore, generator properties have been established for some local operators K which are not invariant; for a review of these developments see [D. W. Robinson, Differential operators on \(C^*\)-algebras, In ”Operator Algebras and Mathematical Physics” (eds. P. E. T. Jørgensen and P. Muhly) Contemp. Math. 60, AMS, Providence (1986)].
MSC:
| 46L55 | Noncommutative dynamical systems |
| 47D03 | Groups and semigroups of linear operators |
| 47B47 | Commutators, derivations, elementary operators, etc. |
Keywords:
local operator; generator of a strongly continuous one-parameter group of; *-automorphisms of a \(C^*\)-algebra; second order locality condition; unbounded central multipliers; invariant dissipation; one- parameter semigroup of completely positive contractions; generator of a strongly continuous one-parameter group of *-automorphisms of a \(C^*\)- algebraCitations:
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