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Finite rank approximation and semidiscreteness for linear operators. (English) Zbl 0989.46033
A $$C^*$$-algebra $$B$$ is called to be nuclear if $$A\otimes_{\min}B= A\otimes_{\max}B$$ holds true for all $$C^*$$-algebras $$A$$. This is equivalent to the existence of a net $$B@>\alpha_i>> M_{n_i} @>\beta_i>> B$$ of completely positive contractions $$\alpha_i$$, $$\beta_i$$ with $$\lim\beta_i\alpha_i= I_B$$ pointwise. G. Pisier has introduced an operator version of this concept: A completely bounded map $$u: Y\to B$$, $$B$$ any $$C^*$$-algebra and $$Y$$ any operator space, is called to be $$c$$-nuclear, if for every $$C^*$$-algebra $$A$$ the mapping $$I_A\otimes u: A\otimes_{\min} Y\to A\otimes_{max}B$$ is bounded with a norm less than $$c$$. There is again an equivalent characterization by finite rank approximation $$Y@> \alpha_i>> M_{n_i} @>\beta_i>> B$$ of $$u$$. For the case of von Neumann algebras the notion of nuclearity has to be replaced by that of semidiscreteness: a von Neumann algebra $$M$$ is called to be semidiscrete, if $$A\otimes_{\min} M= A\otimes_{\text{nor}}M$$ for each $$C^*$$-algebra $$A$$. ($$A\otimes_{\text{nor}}M$$ denotes the normal tensor product introduced by E. Effros.) Again in this case there is an equivalent characterization of nets of finite rank operators $$M@> \alpha_i>> M_{n_i}@> \beta_i>> M$$.
In the paper under review the author introduces an operator version of semidiscreteness as follows: a map $$u: Z\to M$$, $$Z$$ an operator space, is called to be $$c$$-semidiscrete, if for any $$C^*$$-algebra $$A$$ the tensor map $$I_A\otimes u: A\otimes_{\min} Z\to A\otimes_{\text{nor}}M$$ is bounded with a norm less than $$c$$. As a main result, an equivalent characterization of the $$c$$-discreteness by finite rank approximation of $$u$$ in the above sense is given, and the relationship to the $$c$$-nuclearity is studied. Since a $$C^*$$-algebra $$B$$ is nuclear iff $$B^{**}$$ is semidiscrete, one has to compare $$c$$-nuclear maps $$u: Y\to B$$ with properties of $$u^{**}$$. The author shows that for completely bounded mappings $$u: Y\to B$$ the $$c$$-semidiscreteness of $$u^{**}$$ implies the $$c$$-nuclearity of $$u$$. The converse holds true if $$Y$$ is locally reflexive.

##### MSC:
 46L06 Tensor products of $$C^*$$-algebras 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.) 46B28 Spaces of operators; tensor products; approximation properties 47L20 Operator ideals 46L10 General theory of von Neumann algebras 47L30 Abstract operator algebras on Hilbert spaces
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