×

The curvature and index of completely positive maps. (English) Zbl 1057.46053

The concept of curvature for a commuting \(d\)-tuple (a tuple of \(d\) commuting operators on a Hilbert space) was introduced by W. Arveson, [J. Reine Angew. Math. 552, 173–236 (2000; Zbl 0951.47008)]. D. Kribs, [Integral Equations Oper. Theory 41, No. 4, 426–454 (2001; Zbl 0994.47007)] and G. Popescu, [Adv. Math. 158, No.2, 264-309 (2001; Zbl 1002.46029)] presented a version for not necessarily commuting \(d\)-tuples that is different from Arveson’s, but Popescu clarified the relation.
Considering the members of the tuple as coefficients of the Kraus decomposition of a CP-map, every tuple determines a CP-map. CP-maps on a von Neumann algebra \(M\) lead in two ways to \(W^*\)-correspondences (that is, Hilbert bimodules over a \(W^*\)-algebra plus technical conditions): The GNS-correspondence over \(M\) as constructed by W. Paschke [Trans. Am. Math. Soc. 182, 443–468 (1973; Zbl 0239.46062)] and the Arveson-Stinespring correspondence over the commutant \(M'\) of \(M\) as constructed by the authors [cf. P.S. Muhly and B. Solel, Int. J. Math. 13, No. 8, 863–906 (2002; Zbl 1057.46050), reviewed above]. The relation among the two has been clarified in M. Skeide [Contemp. Math. 335, 253–262 (2003; Zbl 1057.46056), reviewed below]. In fact, the Arveson-Stinespring correspondence is the commutant of the GNS-correspondence.
Using Jones’s index theory, the authors define the index of a general \(W^*\)-correspondence over a semifinite factor. Generalizing Kribbs’s form of the definition for tuples, the authors define the curvature of such a correspondence. To achieve these goals, a couple of known and new results have to be combined appropriately. In the end, they define the curvature of a CP-map on a semifinite factor as the curvature of its associated Arveson-Stinespring correspondence, that is, of the commutant of the GNS-correspondence. Then they investigate in how far the CP-map (and the associated correspondences) is determined by its index and curvature in terms of outer conjugacy.

MSC:

46L55 Noncommutative dynamical systems
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
46L08 \(C^*\)-modules
46L07 Operator spaces and completely bounded maps
PDFBibTeX XMLCite
Full Text: DOI arXiv