C*-代数交换的一些等价条件

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′.

Some equivalent conditions of commutativity of a C~*-algebra

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'.