The Comparison Of Positive Elements In C～*-algebras

Let A is a C~* -algebra, a, b is positive. In this article, we define the 'equal' relation between a and b, discuss the symmetry and delivery of this relation. On this foundation, we define the comparison of positive elements in C~*-algebras, prove a important equivalent condition of this relation, and make summary of some conclusions .Then we show that if A is finite, then [a] = [b](?)[a]≤[b],[b]≤[a]; If a, b is well—supported, [a]≤[b], then(?)c∈A_+, s.t. [b] = [a]+[c]. At last, we make use of this 'equal' relation to construct the K. -group, and get some of it's basic properties.