Freeness and matrix decompositions  

作　　者：GE Liming SHEN Junhao 

机构地区：[1]Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China [2]Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA

出　　处：《Science China Mathematics》2011年第11期2309-2327,共19页中国科学：数学（英文版）

摘　　要：If a semicircular element and the diagonal subalgebra of a matrix algebra are free in a finite von Neumann algebra （with respect to a normal trace）, then, up to the conjugation by a diagonal unitary element, all entries of the semicircular element are uniquely determined in the sense of （joint） distribution. Suppose a selfadjoint element is free with the diagonal subalgebra. Then, in the matrix decomposition of the selfa^tjoint element, any two entries cannot be free with each other unless the selfadjoint element is semicircular. We also define a ＂matricial distance＂ between two elements and show that such distance for two free semicircular elements in a finite von Neumann algebra is nonzero and independent of the properties of the von Neumann algebra.If a semicircular element and the diagonal subalgebra of a matrix algebra are free in a finite von Neumann algebra (with respect to a normal trace), then, up to the conjugation by a diagonal unitary element, all entries of the semicircular element are uniquely determined in the sense of (joint) distribution. Suppose a selfadjoint element is free with the diagonal subalgebra. Then, in the matrix decomposition of the selfadjoint element, any two entries cannot be free with each other unless the selfadjoint element is semicircular. We also define a "matricial distance" between two elements and show that such distance for two free semicircular elements in a finite von Neumann algebra is nonzero and independent of the properties of the von Neumann algebra.

关 键 词：von Neumann algebra FACTOR semicircular element joint distribution 

分 类 号：O151.21[理学—数学] TS77[理学—基础数学]