Indice d’une esperance conditionnelle. (Index of a conditional expectation). (French) Zbl 0657.46041
H. Kosaki [J. Funct. Anal. 66, 123-140 (1986; Zbl 0607.46034)] extended Jones’ index theory for a type \(II_ 1\)-factor by defining the index of a conditional expectation E:M\(\to N\) for arbitrary factors M and subfactors N. In the paper under review the authors introduce and study the index of a conditional expectation E for arbitrary von Neumann algebras M and subalgebras N. This index Ind(E) is an element of the center of M and coincides with Kosaki’s index when M and N are factors. Moreover, when Ind(E) is scalar-valued its possible values are those calculated by V. F. R. Jones [Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)].
Reviewer: M.B.Bekka
MSC:
| 46L10 | General theory of von Neumann algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 46L51 | Noncommutative measure and integration |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
| 46L35 | Classifications of \(C^*\)-algebras |
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
| 58J22 | Exotic index theories on manifolds |
Keywords:
correspondences; Jones’ index theory; index of a conditional expectation; von Neumann algebrasReferences:
| [1] | P.L. Aubert : Théorie de Galois pour une W*-algèbre , Comment Math. Helvetici, 51 (1975) 411-433. · Zbl 0339.46044 · doi:10.1007/BF02568166 |
| [2] | J. Bion-Nadal : Von Neumann subalgebras of a Type II1 factor: correspondances and property T , Preprint. |
| [3] | N. Bourbaki : Espaces vectoriels topologiques, Chapitre IV , Hermann, Paris. · Zbl 0482.46001 |
| [4] | F. Combes : Systèmes hilbertiens à gauche et représentation de Gelfand-Segal. Operator algebras and groups representations , Monographs and Studies in Mathematics 17, Pitman (1984) 77-108. · Zbl 0524.46031 |
| [5] | F. Combes and C. Delaroche : Groupe modulaire d’une espérance conditionnelle dans une algèbre de von Neumann , Bull. Soc. Math. France 103 (1975) 385-426. · Zbl 0321.46050 · doi:10.24033/bsmf.1808 |
| [6] | A. Connes : On the spatial theory of von Neumann algebras , J. Funct. Analysis, 35 (1980) 153-164. · Zbl 0443.46042 · doi:10.1016/0022-1236(80)90002-6 |
| [7] | A. Connes : Correspondances , Notes manuscrites (1980). |
| [8] | J. Dixmier : Les algèbres d’opérateurs dans l’espace hilbertien , Gauthier-Villars, Paris, (1969). · Zbl 0175.43801 |
| [9] | U. Haagerup : Normal weights on W*-algebras , J. Funct. Analysis 19 (1975) 302-317. · Zbl 0304.46043 · doi:10.1016/0022-1236(75)90060-9 |
| [10] | U. Haagerup : Operator valued weights in von Neumann algebras I , J. Funct. Analysis 32 (1979) 175-206; II, J. Funct. Analysis 33 (1979) 339-361. · Zbl 0426.46047 · doi:10.1016/0022-1236(79)90072-7 |
| [11] | J.F. Havet : Calcul fonctionnel continu dans les modules hilbertiens autoduaux , Preprint, Orléans. |
| [12] | J.F. Havet : N-algèbres hilbertiennes , Preprint, Orléans. |
| [13] | V.F.R. Jones : Index for subfactors , Inventionnes Math. 72 (1983) 1-25. · Zbl 0508.46040 · doi:10.1007/BF01389127 |
| [14] | H. Kosaki : Extension of Jones’ theory on index to arbitrary factors , J. Funct. Analysis 66 (1986) 123-140. · Zbl 0607.46034 · doi:10.1016/0022-1236(86)90085-6 |
| [15] | W.L. Paschke : Inner product modules over B*-algebras , Trans. Am. Math. Soc. 182 (1973) 443-468. · Zbl 0239.46062 · doi:10.2307/1996542 |
| [16] | W.L. Paschke : Inner product modules arising from compact automorphism groups on von Neumann algebras , Trans Am. Math. Soc. 224 (1976) 87-102. · Zbl 0339.46048 · doi:10.2307/1997417 |
| [17] | M. Pimsner and S. Popa : Entropy and index for subfactors , Ann. Scient. Ec. Norm. Sup. 19 (1986) 57-106. · Zbl 0646.46057 · doi:10.24033/asens.1504 |
| [18] | M.A. Rieffel : Induced representations of C*-algebras , Adv. Math. 13 (1974) 176-257. · Zbl 0284.46040 · doi:10.1016/0001-8708(74)90068-1 |
| [19] | M.A. Rieffel : Morita equivalence for C*-algebras and W*-algebras , J. Pure Appl. Algebra 5 (1974) 51-96. · Zbl 0295.46099 · doi:10.1016/0022-4049(74)90003-6 |
| [20] | J.L. Sauvageot : Sur le produit tensoriel relatif d’espaces de Hilbert , J. Operator Theory 9 (1983) 237-252. · Zbl 0517.46050 |
| [21] | H. Zettl : A Characterisation of Ternary rings of operators , Adv. Math. 48 (1983) 117-143. · Zbl 0517.46049 · doi:10.1016/0001-8708(83)90083-X |
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