On the local lifting property for operator spaces
A Banach space \(V\) is said to have the lifting property if given any Banach spaces \(W\subset Y\), a linear contraction \(\varphi: V\to Y/w\), and \(\varepsilon>0\), there is a bounded linear map \(\widetilde\varphi: V\to Y\), such that \[ \|\widetilde\varphi\|< 1+\varepsilon\quad\text{and}\quad q\cdot \widetilde\varphi= \varphi, \] where \(q: Y\to Y/w\) denotes the quotient map from \(Y\) and \(Y/w\). It is known from Grothendieck's results that a Banach space \(V\) has the lifting property if and only if \(V\) is isometrically isomorphic to \(\ell_1(I)\), for some index set \(I\); the Banach space \(L_1(X, M,\mu)\) need not have the lifting property, where \((X,M,\mu)\) is a measure space; but the space \(L_1(X,M,\mu)\) has the local lifting property. A Banach space \(V\) is called a \({\mathcal L}_{1,\lambda}\) space for some \(\lambda> 1\) if for every finite-dimensional subspace \(E\) of \(V\), there exists a finite-dimensional subspace \(F\) of \(V\) containing \(E\) such that the Banach-Mazur distance \(d(F,\ell_1(n))< \lambda\), where \(n= \dim(F)\). It is known that \(L_1(X,M,\mu)\) spaces are exactly \({\mathcal L}_{1,1+\varepsilon}\) spaces for every \(\varepsilon> 0\). Therefore, the lifting property of \(\ell_1(n)\) spaces implies the local lifting property of \(L_1\). The authors of this article study the local lifting property for operator spaces. It is shown that an operator space has the \(\lambda\)-local lifting property (\(\lambda\)-LLP) if and only if it is an \({\mathcal L}\Gamma_{1,\lambda}\) space introduced in Junge. Moreover, it is proved that every \({\mathcal L}_{1,\lambda^+}\) space has the \(\lambda\)-LLP, and every operator space with the \(\lambda\)-LLP must be \(\lambda\)-finitely representable in \(\{T_n\}_{n\in\mathbb{N}}\), \(\lambda\)-completely isomorphic to an operator subspace of the operator predual of a von Neumann algebra, and \(\lambda\)-locally reflexive; and it is also shown that an operator space \(V\) has the \(\lambda\)-LLP if and only if its operator space dual \(V^*\) is \(\lambda\)-injective.
scientific article; zbMATH DE number 1389441
scientific article