单C~*-代数α-比较性的一种等价刻画

An Equivalent Characterization for the α-comparison Property of Simple C~*-algebras

给出C~*-代数α-比较性的等价刻画:对于单的含单位元的稳定有限的C~*-代数A而言,A具有α-比较性,当且仅当对于任意的<a>,<b>∈W(A),若α·d_r(a)<d_τ(b)(_τ∈QT(A)),则<a>≤<b>在Cuntz半群W(A)中成立.利用此刻画,证明了具有α-比较性的C~*-代数一定具有弱比较性;若A具有α-比较性,其中α=m+1,则A具有正元的强迹m-比较性;对于满足Kirchberg-R?rdam条件的C~*-代数,E-稳定、严格比较、α-比较性(α=m+1)、强迹m-比较性、弱比较性以及局部弱比较性彼此等价;若α:=inf{α′∈(1,∞)|A具有α′-比较}<∞,则A具有α-比较性.

We give an equivalent characterization for the α-comparison property of C＊-algebras： any simple unital stably finite C＊-algebra ~ has the a-comparison prop- erty, if and only if, for any （a）, （b） e W（~a＇）, a. d^-（a） 〈 d^（b）（VT ~ QT（~）） implies that （a） 〈 （b） holds in W（A）. Using this characterization, we prove the following results： C＊-algebras with a-comparison property have weak comparison; C＊-algebras with α-comparison property for α = m ＋ 1 have strong tracial m-comparison of posi- tive elements; α-stability. strict comparison, α-comparison property for α= m＋l, strong tracial m-comparison, weak comparison and local weak comparison all agree for the C＊-algebras satisfying the conditions given by Kirchberg-RCrdam; if α-：= inf{c∈ （1, ∞）A has the α＇-comparison property} 〈 ∞, then α＇ has the a-comparison prop- erty.