Orientation in operator algebras

E. M. Alfsen, F. W. Shultz
1998 Proceedings of the National Academy of Sciences of the United States of America  
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. A problem that dates back to the 1950s is to characterize the ordered linear spaces that are the self-adjoint parts of C*algebras or of von Neumann algebras. This
more » ... oblem was implicit in Kadison's paper (1), and it was explicitly raised for von Neumann algebras by Sakai (2) and for C*-algebras by Sherman (3). It follows from Kadison's results (1) that the self-adjoint part of a C*-algebra is isometrically isomorphic, as an ordered normed linear space, to the space A(K) of all w*-continuous affine functions on the state space K. Similarly, the self-adjoint part of a von Neumann algebra is isometrically isomorphic to the space of all bounded affine functions on the normal state space. In view of this, characterizing the selfadjoint part of a C*-algebra (respectively von Neumann algebra) is equivalent to characterizing the state space of a C*-algebra (respectively the normal state space of a von Neumann algebra). Connes gave a solution of the ordered linear space version of this problem for a -finite von Neumann algebra in ref. 4 by first characterizing the associated cone P of Tomita-Takesaki theory. The state spaces of C*-algebras were characterized by Alfsen, Hanche-Olsen, and Shultz (5) and the normal state spaces of von Neumann algebras by Iochum and Shultz (6). By a theorem of Kadison (7) the ordering and the norm of a C*-algebra (or equivalently, its state space) determine the symmmetrized (Jordan) product 1 ⁄2(ab ϩ ba). However, they do not determine the product itself, because the opposite algebra has the same ordering and norm. Thus some additional structure is needed to determine the associative product. It was Connes (4) who first realized that a concept of orientation is relevant for this purpose (and his concept was later used in the axiomatic context of JBW-algebras by Bellissard and Iochum in refs. 8 and 9). Alfsen, Hanche-Olsen, and Shultz (5) also introduced a concept with the same name and for the same purpose. However, the definitions had little in common. Connes' notion was algebraic, global in nature, and applied to von Neumann algebras. That of Alfsen, Hanche-Olsen, and Shultz was geometric, local in nature, and applied to state spaces of C*-algebras. One purpose of our current work is to generalize both notions so that they apply to both C* and von Neumann algebras. (There are some significant obstacles to overcome to accomplish this as we will discuss later.) Our second purpose is to relate the two concepts and to explain how they both relate to dynamics. We will build a bridge between these two concepts of orientation by introducing a third concept: that of a dynamical correspondence. This paper is a survey of the results with brief comments on the
doi:10.1073/pnas.95.12.6596 pmid:9618457 pmcid:PMC22571 fatcat:zfcx7rpxkzfipham4e6lnhrbjy