非单C~*-代数α-比较性的等价刻画

Equivalent Characterizations for the α-comparison Property of Non-simple C~*-algebras

给出非单C~*-代数α-比较性的等价刻画:当每个τ∈QT(■H)均为忠实时,■具有α-比较性,当且仅当对于任意的〈a〉,〈b〉∈Cu(■)且<a>∝<b>,若α·d_τ(a)<d_τ(b)(_τ∈QT(■H))则<a>≤<b>在Cu(■)中成立;一般地,当QT(■H)≠Φ时,■具有α-比较性,当且仅当对于任意的<a>,<b>∈Cu(■),若存在η>0,使得d_τ(α)≤(α^(-1)-η)d_τ(b)(_τ∈QT(■H)),则<a>≤<b>在Cu(■)中成立.

Abstract This note is to give such characterizations of the a-comparison property for non-simple C＊-algebras： when each τ∈QT（A H）is faithful, A has the a-comparison property, if and only if, for any 〈a〉,〈b〉∈Cu（A） with 〈a〉∝〈b〉,a·dτ（a）〈dτ（b）（∨τ∈QT（A H））implies （a） ≤（b） in Cu（A）;in general, when QT（A H）≠θ,A has the a-comparison property, if and only if, for any 〈a〉,〈b〉∈Cu（A）,if there is some η〉0such that dτ（a）≤（a^-1-η）dτ（b）（Vτ）∈QT（A H））,then 〈a〉≤〈b〉 in Cu（A）.