Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-03T14:38:07.544Z Has data issue: false hasContentIssue false
  • English
  • Français

Injective W*-algebras

Published online by Cambridge University Press:  24 October 2008

Simon Wassermann
Simon Wassermann
Affiliation:
University of Glasgow
Get access Rights & Permissions [Opens in a new window]

Extract

Let M be a W*-algebra. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, M is said to be injective [(8), § 5] if there is a projection of norm one from L(H) onto π(M). Tomiyama [(13), theorem 7.2] has shown that this definition is independent of the choice of representation π, and, moreover, that M is injective if and only if the commutant π(M)′ is injective for some such π. M is said to be semidiscrete [(8), § 3] if the identity map on M can be approximated in the topology of simple-weak* convergence by completely positive normal linear maps in L(M) of finite rank which preserve the identity. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, a *-homomorphism η of the algebraic tensor product M ⊙ π (M)′ into L(H) is defined by

If η has a bounded extension to M ⊗ π(M)′ (the completion relative to the spatial C*-cross norm) we shall say that M is amenable.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 39 - 47
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Combes, F.Poids associés a une algèbre hilbertienne à gauche. Compositio Math. 23 (1971), 49–77.Google Scholar
(2)Choi, M. D. and Effros, E. G.Separable nuclear C*-algebras and injectivity. Duke Math. J. 43 (1976), 309–322.CrossRefGoogle Scholar
(3)Chol, M. D. and Effros, E. G. Nuclear C*-algebras and injectivity: the general case (to appear).Google Scholar
(4)Connes, A.Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup. série 4 (1973), 133–252.CrossRefGoogle Scholar
(5)Connes, A.Classification of injective factors. Ann. of Math. 104 (1976), 73–115.CrossRefGoogle Scholar
(6)Connes, A. On the classification of von Neumann algebras and their automorphisme (preprint).Google Scholar
(7)Dixmier, J.Les algèbres d'opérateurs dans l'espace hilbertien 2nd ed. (Paris, Gauthier-Villars, 1969).Google Scholar
(8)Effros, E. G. and Lance, E. C. Tensor products of operator algebras (to appear).Google Scholar
(9)Følner, E.On groups with full Banach mean value. Math. Scand. 3 (1955), 243–254.CrossRefGoogle Scholar
(10)Loomis, L.Note on a theorem of Mackey. Duke Math. J. 19 (1952), 641–645.CrossRefGoogle Scholar
(11)Sakai, S.C*-algebras and W*-algebras (Berlin, Springer-Verlag, 1971).Google Scholar
(12)Takesaki, M.Duality for crossed products and the structure of von Neumann algebras of type 111. Acta Math. 131 (1973), 243–310.CrossRefGoogle Scholar
(13)Tomiyama, J. Tensor products and projections of norm one in von Neumann algebras. Seminar notes, Copenhagen University, 1971.Google Scholar
(14)Wassermann, S.On tensor products of certain group C*-algebras. J. Functional Analysis 23 (1976), 239–254.CrossRefGoogle Scholar