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On approximation properties for operator spaces. (English) Zbl 0747.46014
It is known that there are remarkable parallels between the category of operator spaces and completely bounded linear maps on one side and the usual category of normed linear spaces on the other. One of the (non- trivial) problems, however, is to compe up with the ‘correct’ definition for objects analogous to those familiar from the normed space situation, but once this is settled, one is frequently rewarded by several deep consequences of the resulting theory.
The present paper is devoted to developing a theory of tensor products for operator spaces and to studying in this context analogues of Grothendieck’s approximation and metric approximation properties. The authors show that such concepts can be defined within the framework of completely bounded mappings. Again the results they obtain are more or less parallel to those known from the “classical” situation, and again there are some striking applications. For example, if \(R\) and \(S\) are von Neumann algebras with preduals \(R_ *\) and \(S_ *\) and spatial tensor product \(R\overline\otimes S\), then \((\omega_ 1,\omega_ 2)\mapsto\omega_ 1\otimes\omega_ 2\) determines an isometric isomorphism (in the sense of c.b. [=completely bounded] maps) of the c.b. projective tensor product \(R_ *\hat\otimes S_ *\) onto \((R\overline\otimes S)_ *\). In particular, if \(G\) and \(H\) are locally compact groups, then the Fourier algebra \(A(G\times H)\) and \(A(G)\hat\otimes A(H)\) are isometrically c.b. isomorphic as operator spaces.

MSC:
46B28 Spaces of operators; tensor products; approximation properties
46A32 Spaces of linear operators; topological tensor products; approximation properties
46L10 General theory of von Neumann algebras
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