摘要:
对于交换的 mathrm 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 mathrm 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 mathrm 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 mathrm 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’in A 使得 ab=ba’.
Abstract:
Let Abe a mathrm C^*-algebra. If A is Abelian, then each hereditary mathrm 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 mathrm C^*-subalgebra (or one-sided closed ideal) of A is a closed ideal in A, then A must be Abelian. So in a noncommutative mathrm C^*-algebra, there must exist a hereditarymathrm C^*-subalgebra which is not a closed ideal. Using the main result, we also obtain a simple criterion to check if a given mathrm C^*-algebra A is Abelian, that is, A is Abelian if and only for any two positive elements a, bin A, there is a’in A such that ab=ba’.
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