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’.

Key words: left closed ideal, closed ideal, pure state, hereditary mathrm C^*-subalgebra, left closed ideal, closed ideal, pure state