中文摘要：

对于交换的C^＊-代数，它的每一个遗传子代数（或单侧闭理想）都是它的双侧闭理想．反之，利用C^＊-代数A上的纯态与A中极大左理想的对应关系，得到了：若A中的每一个遗传子代数（或单侧闭理想）都是它的双侧闭理想，则A一定是交换的．因此在非交换的C^＊-代数中必有一个非闭理想的遗传子代数．利用文中的主要结论，还得到了判断C^＊-代数A是交换一个简单条件，即A是交换的当且仅当对A中的任何两个正元a，b存在a’∈A使得ab=ba’．

英文摘要：

Let A be a C^＊-algebra. If A is Abelian, then each hereditary C^＊-subalgebra （or one-sided closed ideal） of A is a closed ideal in A. Conversely, in terms of the correspondence between the pure state and the maximal left idea, we get that if each hereditary C^＊-subalgebra （or one-sided closed ideal） of A is a closed ideal in A, then A must be Abelian. So in a noncommutative C^＊-algebra, there must exist a hereditary C^＊-subalgebra which is not a closed ideal. Using the main result, we also obtain a simple criterion to check if a given C^＊-algebra A is Abelian, that is, A is Abelian if and only for any two positive elements a, b ∈A, there is a＇∈A such that ab=ba＇.