Quasi-expectations and amenable von Neumann algebras

Bunce, John W.; Paschke, William L.

doi:10.1090/S0002-9939-1978-0482252-3

Quasi-expectations and amenable von Neumann algebras
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by John W. Bunce and William L. Paschke PDF

Proc. Amer. Math. Soc. 71 (1978), 232-236 Request permission

Abstract:

A quasi-expectation of a ${C^ \ast }$-algebra A on a ${C^\ast }$-subalgebra B is a bounded linear projection $Q:A \to B$ such that $Q(xay) = xQ(a)y \forall x,y \in B,a \in A$. It is shown that if M is a von Neumann algebra of operators on Hilbert space H for which there exists a quasi-expectation of $B(H)$ on M, then there exists a projection of norm one of $B(H)$ on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on $M’ \cap N$. These two facts yield as an immediate corollary the recent result of A. Connes that all amenable von Neumann algebras are injective.

References

Man Duen Choi and Edward G. Effros, Nuclear $C^*$-algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), no. 3, 443–446. MR 430794, DOI 10.1512/iumj.1977.26.26034
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057

On the cohomology of operator algebras

Les algèbres d’opérateurs dans l’espace hilbertien

Edward G. Effros and E. Christopher Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1–34. MR 448092, DOI 10.1016/0001-8708(77)90085-8
Jôsuke Hakeda and Jun Tomiyama, On some extension properties of von Neumann algebras, Tohoku Math. J. (2) 19 (1967), 315–323. MR 222656, DOI 10.2748/tmj/1178243281
Richard V. Kadison and John R. Ringrose, Cohomology of operator algebras. I. Type $I$ von Neumann algebras, Acta Math. 126 (1971), 227–243. MR 283578, DOI 10.1007/BF02392032
Guyan Robertson, States which have a trace-like property relative to a $C^*$-subalgebra of $B(H)$, Glasgow Math. J. 17 (1976), no. 2, 158–160. MR 415337, DOI 10.1017/S0017089500002913
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Masamichi Takesaki, On the conjugate space of operator algebra, Tohoku Math. J. (2) 10 (1958), 194–203. MR 100799, DOI 10.2748/tmj/1178244713
Masamichi Takesaki, On the singularity of a positive linear functional on operator algebra, Proc. Japan Acad. 35 (1959), 365–366. MR 113153
Masamichi Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. MR 438149, DOI 10.1007/BF02392041
Jun Tomiyama, On the projection of norm one in $W^{\ast }$-algebras, Proc. Japan Acad. 33 (1957), 608–612. MR 96140