Mapping spaces and liftings for operator spaces.

*(English)*Zbl 0814.47053This article contributes to the authors’ ambitious project which is known as ‘quantized functional analysis’. The basic objects of study are the closed subspaces \(V\) of \(L(H)\), the \(C^*\)-algebra of bounded operators on a Hilbert space, equipped with the canonical string of norms on the spaces \(M_ n(V)\) of \((n\times n)\)-matrices with entries from \(V\). (This is called an operator space structure on \(V\).) Mappings \(T\) between operator spaces \(V\) and \(W\) give rise to maps \(T_ n: M_ n(V)\to M_ n(W)\), and saying that \(T\) has a complete version of some property (P) reflects that all the \(T_ n\) have (P).

In the first part of this paper the authors study the complete versions of nuclear and integral maps. Here, operator space theory involves some unexpected subtleties that originate from the failure of the local reflexivity principle in the operator space setting. For instance, as opposed to the Banach space case there is in general only a complete contraction \(S_ I\) from the operator space of integral operators \(I(V, W^*)\) into the dual of the injective operator space tensor product \((V\check\otimes W)^*\) that need not be a completely isometric bijection. In addition it is shown that \(S_ I\) is a completely isometric bijection for all \(V\) if and only if \(W\) is locally reflexive.

The final two sections are devoted to the notion of a complete \(M\)-ideal, meaning a subspace \(J\subset V\) such that \(M_ n(J)\) is an \(M\)-ideal in \(M_ n(V)\) for all \(n\). As an application a lifting theorem for mappings into complete \(M\)-ideal quotients is proved.

The reader should also compare the authors’ paper in J. Funct. Anal. 122, No. 2, 428-450 (1994; Zbl 0802.46014).

In the first part of this paper the authors study the complete versions of nuclear and integral maps. Here, operator space theory involves some unexpected subtleties that originate from the failure of the local reflexivity principle in the operator space setting. For instance, as opposed to the Banach space case there is in general only a complete contraction \(S_ I\) from the operator space of integral operators \(I(V, W^*)\) into the dual of the injective operator space tensor product \((V\check\otimes W)^*\) that need not be a completely isometric bijection. In addition it is shown that \(S_ I\) is a completely isometric bijection for all \(V\) if and only if \(W\) is locally reflexive.

The final two sections are devoted to the notion of a complete \(M\)-ideal, meaning a subspace \(J\subset V\) such that \(M_ n(J)\) is an \(M\)-ideal in \(M_ n(V)\) for all \(n\). As an application a lifting theorem for mappings into complete \(M\)-ideal quotients is proved.

The reader should also compare the authors’ paper in J. Funct. Anal. 122, No. 2, 428-450 (1994; Zbl 0802.46014).

Reviewer: D.Werner (Berlin)