tailieunhanh - Báo cáo toán học: "Positive cones and $L^p$-spaces associated with a von Neumann algebra "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tích cực tế bào hình nón và $ L ^ p $-không gian liên kết với một đại số von Neumann. | J. OPERATOR THEORY 6 1981 25-37 Copyright by INCREST 1981 NORM-CLOSE GROUP ACTIONS ON c -ALGEBRAS RICHARD H. HERMAN and JONATHAN ROSENBERG INTRODUCTION This paper is in many ways an outgrowth of 5 We deal with many of the same questions but work in the C -algebraic setting with groups more general than the real line. One aspect of the results of 5 was to determine conditions under which two representations of R as automorphisms of an operator algebra were cohomologous more specifically exterior equivalent in Connes s sense. This in fact is the case 23 Lemma if and only if the two representations appear as opposite corners of a representation on the 2x2 matrices over the algebra. We are however interested in determining analytical conditions under which this happens. As in 5 we see that one such condition is that the representations be norm close in a neighborhood of the identity. In 5 Theorem under the closeness condition just mentioned it was shown that an obstruction in a certain twisted second cohomology group with coefficients in the unitary group of the center of the von Neumann algebra was zero. This gave the exterior equivalence of the two automorphism groups and in turn a specific relation between the two generators. Since JT2 R T 0 this result may not be surprising. But since for many commonly occurring groups G . R with n 2 H2 G T 0 vanishing of the cohomological obstruction is not so automatic for general groups and it is less clear whether positive results can be obtained. Thus instead of appealing to general cross-section theorems or cohomology vanishing theorems we directly choose certain unitary operators so that the relevant 2-cocycle is cohomologous to 0. We are able to do this for a finite separable simple c -algebra 9Í and G a connected simply connected Lie group one of whose representations on 91 leaves a trace invariant. The results are better in the unital case and there we show at least for separable c -algebras that for G R the .