Non-commutative Banach Function Spaces

Abstract

In this paper we survey some aspects of the theory of non-commutative Banach function spaces, that is, spaces of measurable operators associated with a semi- finite von Neumann algebra. These spaces are also known as non-commutative symmetric spaces. The theory of such spaces emerged as a common generalization of the theory of classical (“commutative”) rearrangement invariant Banach function spaces (in the sense of W.A.J. Luxemburg and A.C. Zaanen) and of the theory of symmetrically normed ideals of bounded linear operators in Hilbert space (in the sense of I.C. Gohberg and M.G. Krein). These two cases may be considered as the two extremes of the theory: in the first case the underlying von Neumann algebra is the commutative algebra L _∞ on some measure space (with integration as trace); in the second case the underlying von Neumann algebra is B (ℌ), the algebra of all bounded linear operators on a Hilbert space ℌ (with standard trace). Important special cases of these non-commutative spaces are the non-commutative L _p-spaces, which correspond in the commutative case with the usual L _p-spaces on a measure space, and in the setting of symmetrically normed operator ideals they correspond to the Schatten p-classes $\mathfrak{S}_p$ .