tailieunhanh - Báo cáo toán học: "Borel maps on sets of von Neumann algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Borel maps on sets of von Neumann algebras . | J. OPERATOR THEORY 9 1983 319-340 Copyright by INCREST 1983 BOREL MAPS ON SETS OF VON NEUMANN ALGEBRAS EDWARD A. AZOFF 1. INTRODUCTION In 2 E. Effros showed how to make the collection of closed subsets of a Polish space into a standard Borel space. Applying this idea in 3 he introduced a standard Borel structure on the collection sđ of von Neumann algebras acting on a fixed separable Hilbert space H. The subcollection S of factor von Neumann algebras in sS is easily seen to be Borel and it makes sense to ask whether the various subcollections of S connected with type classification theory are Borel as well. In the follow-up paper 4 Effros provided affirmative answers to most of these questions in particular he showed that the collection s of finite factors on H is Borel but did not resolve the issue for the collection s of semi-finite factors. Since a projection e in a factor A is finite if and only if eAe supports a finite trace it is easy to see that s is analytic. In 11 o. Nielsen applied the Tomita-Take-saki theory of modular automorphism groups to show that is also analytic thereby proving that s is Borel. A second proof that S is Borel outlined on pages 136 7 of 12 is based on a representation-theoretic argument of G. Pedersen 13 . The main result of the present paper Theorem states that there is a Borel function defined on s which selects a non-zero finite projection from each factor belonging to S . The key idea in the proof is the application of a selection theorem which asserts that each Borel set in a product of Polish spaces all of whose sections are ơ-compact admits a Borel uniformization. The paper uses only classical results from the theory of von Neumann algebras. In particular a priori knowledge that S is Borel is not required and this fact is established independently. It is the theme of this paper that descriptive set theory especially those parts of it dealing with set-valued maps can be profitably applied to the study of the Efiros .