Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra
Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra
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有限冯·诺依曼代数的正圆锥中的非正曲度规
DOI：
10.1112/s0024610706022848
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
G. Larotonda
中科院分区：
文献类型：
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作者：
E. Andruchow;G. Larotonda
本文研究了具有有限正规忠实迹态τ的von Neumann代数A的正可逆元空间的度量几何。迹导出了一个不完全黎曼度量<$x，y <$a = τ（ya−1xa−1），尽管所涉及的技巧非常不同，但这里的情况在许多相关方面类似于n × n矩阵被视为对称空间时的情况。例如，我们证明测地线是最短路径的度量诱导，测地距离是一个凸函数;我们给出了一个内在的（代数）特征的测地线凸子流形M的M;并在一个适当的假设下，我们证明了一个因式分解定理的代数元素，类似于岩泽分解矩阵。这个分解是通过一个非线性正交投影<$M：<$M → M得到的，这个映射对于测地距离是压缩的。
In this paper we study the metric geometry of the space Σ of positive invertible elements of a von Neumann algebra A with a finite, normal and faithful tracial state τ. The trace induces an incomplete Riemannian metric 〈x,y〉a = τ (ya−1xa−1), and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the n × n matrices when they are regarded as a symmetric space. For instance, we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds M of Σ; and under a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection ΠM : Σ → M, a map which turns out to be contractive for the geodesic distance.