Some Equivalent Conditions Of Commutativity Of A C~*-algebra

There are many characterizations for the commutative of C*-algebras. For examples, Kaplansky showed that a C*-algebra A is commutative iff the only nilpo-tent element in A is 0; Nakamoto gave a spectral characterization for the commu-tativity of a C*-algebra; Ogasawara, Sherman, Wu, and Ji and Tomiyama gave different characterizations of commutative C*-algebras by means of the order struc-ture. Later, Jeang and Ko proved that for non-constant continuous functions f, g defined on a closed interval I1 and I2 respectively, if f(x)g(y)= g(y)f(x) for all self-adjoint elements x and y in A withσ(x) (?) I1 andσ(y) (?) I2, then A is commu-tative. This result is a corollary of the main theorem, that is, f(A) is dense in A. But its proof relies on the Measure Theory.In this paper, first we prove that if each hereditary C*-subalgebra (or one-sided closed ideal) of A is a closed ideal of A, then A must be Abelian.Then we will also prove that the C*-algebra A is not commutative iff there is a C*-subalgebra B in A" (the enveloping Von Neumann algebra of A) such that B is*-isomorphic to M2(C)). In terms of this result, we can recover some characterizations which we have told above for the commutativity of C*-algebras.