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JIANG Run-liang, XUE Yi-feng. Some equivalent conditions of commutativity of a textbfC^*-algebra (Chinese)[J]. Journal of East China Normal University (Natural Sciences), 2010, (1): 99-102.
Citation:
JIANG Run-liang, XUE Yi-feng. Some equivalent conditions of commutativity of a textbfC^*-algebra (Chinese)[J]. Journal of East China Normal University (Natural Sciences), 2010, (1): 99-102.
JIANG Run-liang, XUE Yi-feng. Some equivalent conditions of commutativity of a textbfC^*-algebra (Chinese)[J]. Journal of East China Normal University (Natural Sciences), 2010, (1): 99-102.
Citation:
JIANG Run-liang, XUE Yi-feng. Some equivalent conditions of commutativity of a textbfC^*-algebra (Chinese)[J]. Journal of East China Normal University (Natural Sciences), 2010, (1): 99-102.
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’.