Direct integrals on selfdual cones and standard forms of von Neumann algebras

Abstract

Connes [6] and Haagerup [11] established the relation between von Neumann algebras (abbreviate: VNA) and selfdual, homogenous and orientated cones in Hilbert space. The ideal center ofa selfdual cone in a Hilbert space is the hermitian part of an abelian VNA [3]. This gives rise to direct integral decompositions of selfdual cones with respect to von Neumann subalgebras of the VNA generated by the ideal center (see w Theorem 3.17). The existence of a "central" decomposition has been shown in [18], too. The purpose of the present paper is the study of the properties of such decompositions (w 3). Especially, we determine the facial structure of a direct integral of selfdual cones (w 3, Proposition 3.5, Lemma 3.6, 3.7, Corollary 3.8). It follows rough ly spokenthat all properties of selfdual cones, which are expressed by means of facial projections, hold for a direct integral of selfdual cones iff they hold for the fibres (w 3, Corollary 3.9, 3.10, Remark 3.11). Further, this is used to study the connection between direct integrals of VNA and the associated selfdual, homogenous and orientated cones, i.e. the associated standard forms (w Proposition 3.14, Theorem 3.15). Another consequence is the decomposibility of certain linear maps between direct integrals of VNA (w Theorem 4.1). Direct integrals of standard forms of VNA have also been considered in [13, 15] and [21]. But the approach to direct integrals of standard forms of VNA given there is based on the decomposition theory for left Hilbert algebras, which isn't used in this paper. The paper is subdivided into four sections. In the preliminaries (w 1) we recall basis facts on selfdual cones in Hilbert space and the terminology needed here. w 2 is devoted to the theory of direct sums. In w 3 and w 4 we develop the decomposition theory for selfdual cones and present as mentioned above applications to von Neumann algebra theory.