中文摘要：

对于C^＊-代数A的正元a和b而言，知道[a]≤[b]，[b]≤[a]是不能得出[a]=[b]的结论，但在实际应用中，常常需要找出关于[a]=[b]成立的充分条件来解决一些问题．为此，引入了性质PH，得出了在[a]≤[b]和[b]≤[a]的条件下[a]=[b]成立的一些充分条件，并且做出了严格的证明．

英文摘要：

It is well known that the conclusion [a]=[b] cannot be obtained only with the conditions [a]≤[b] and [b]≤[a] for positive elements a and b in a C^＊-algebra A. But it is necessary in solving practical problems in studying some sufficient conditions for [a]=[b]. For this reason, the property （PH） is introduced and some sufficient conditions with detailed proofs for [a]=[b]are obtained.