TY - JOUR

T1 - Completely bounded isomorphisms of injective von neumann algebras

AU - Christensen, Erik

AU - Sinclair, Allan M.

PY - 1989/6

Y1 - 1989/6

N2 - Milutin’s Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(T) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach-Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaces [formula ommited] where H is the infinite dimensional separable Hilbert space, R is the injective II1—factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property). The trace preserving normal conditional expectation available for type II1 von Neumann algebras is exploited to obtain the second Theorem (1.2): if M and N are type II1, von Neumann algebras with M↪N, N↪M, and M⊕M↪M, then M and N are completely boundedly normally isomorphic. Here ↪ denotes a normal ⋆-monomorphism from the one von Neumann algebra into another that is not necessarily unital. Our results are not directly related to those on perturbation in von Neumann algebras [1, 2] as the distance between the algebras used here is different from that used in perturbation theory.

AB - Milutin’s Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(T) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach-Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaces [formula ommited] where H is the infinite dimensional separable Hilbert space, R is the injective II1—factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property). The trace preserving normal conditional expectation available for type II1 von Neumann algebras is exploited to obtain the second Theorem (1.2): if M and N are type II1, von Neumann algebras with M↪N, N↪M, and M⊕M↪M, then M and N are completely boundedly normally isomorphic. Here ↪ denotes a normal ⋆-monomorphism from the one von Neumann algebra into another that is not necessarily unital. Our results are not directly related to those on perturbation in von Neumann algebras [1, 2] as the distance between the algebras used here is different from that used in perturbation theory.

UR - http://www.scopus.com/inward/record.url?scp=84976003291&partnerID=8YFLogxK

U2 - 10.1017/S0013091500028716

DO - 10.1017/S0013091500028716

M3 - Journal article

AN - SCOPUS:84976003291

VL - 32

SP - 317

EP - 327

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -