On the split property for inclusions of 𝑊*-algebras

Fidaleo, Francesco

doi:10.1090/S0002-9939-01-06154-8

On the split property for inclusions of $W^{*}$-algebras
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Proc. Amer. Math. Soc. 130 (2002), 121-127 Request permission

Abstract:

A characterization of the quasi–split property for an inclusion of $W^*$–algebras in terms of the metrically nuclear maps is established. This result extends the known characterization relative to inclusions of $W^*$–factors. An application to type $I$ von Neumann algebras is also presented.

References

D. Buchholz, C. D’Antoni, and K. Fredenhagen, The universal structure of local algebras, Comm. Math. Phys. 111 (1987), no. 1, 123–135. MR 896763
Detlev Buchholz, Claudio D’Antoni, and Roberto Longo, Nuclear maps and modular structures. I. General properties, J. Funct. Anal. 88 (1990), no. 2, 233–250. MR 1038440, DOI 10.1016/0022-1236(90)90104-S
Detlev Buchholz, Claudio D’Antoni, and Roberto Longo, Nuclear maps and modular structures. II. Applications to quantum field theory, Comm. Math. Phys. 129 (1990), no. 1, 115–138. MR 1046280
A. Connes and V. Jones, Property $T$ for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57–62. MR 766450, DOI 10.1112/blms/17.1.57
Claudio D’Antoni and Roberto Longo, Interpolation by type $\textrm {I}$ factors and the flip automorphism, J. Functional Analysis 51 (1983), no. 3, 361–371. MR 703083, DOI 10.1016/0022-1236(83)90018-6
Edward G. Effros and Zhong-Jin Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), no. 2, 163–187. MR 1060634, DOI 10.1142/S0129167X90000113
Edward G. Effros and Zhong-Jin Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. (3) 69 (1994), no. 1, 171–197. MR 1272425, DOI 10.1112/plms/s3-69.1.171
Francesco Fidaleo, Operator space structures and the split property, J. Operator Theory 31 (1994), no. 2, 207–218. MR 1331773
F. Fidaleo, Some operator ideals in non-commutative functional analysis, Z. Anal. Anwendungen 17 (1998), no. 3, 759–776. MR 1649409, DOI 10.4171/ZAA/849
Uffe Haagerup, $L^{p}$-spaces associated with an arbitrary von Neumann algebra, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977) Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 175–184 (English, with French summary). MR 560633
Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
Zhong-Jin Ruan, Subspaces of $C^*$-algebras, J. Funct. Anal. 76 (1988), no. 1, 217–230. MR 923053, DOI 10.1016/0022-1236(88)90057-2
Şerban Strătilă and László Zsidó, Lectures on von Neumann algebras, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1979. Revision of the 1975 original; Translated from the Romanian by Silviu Teleman. MR 526399
Stephen J. Summers, On the independence of local algebras in quantum field theory, Rev. Math. Phys. 2 (1990), no. 2, 201–247. MR 1090281, DOI 10.1142/S0129055X90000090
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728