Some Operator Ideals in Non-Commutative Functional Analysis

Abstract

We study classes of linear maps between operator spaces $E$ and $F$ which factorize through maps arising in a natural manner by the Pisier vector-valued non-commutative $L^p$-spaces $S_p[E]$ based on the Schatten classes on the separable Hilbert space ${\ell }^2$. These classes of maps, firstly introduced in [28] and called p-nuclear maps, can be viewed as Banach operator ideals in the category of operator spaces, that is in non-commutative (quantized) functional analysis. We also discuss some applications to the split property for inclusions of $W\ast$-algebras such as those describing the physical observables in Quantum Field Theory.