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AN ORDER CHARACTERIZATION OF COMMUTATIVITY FOR C∗-ALGEBRAS

In this paper, we investigate the problem of when a C∗-algebra is commutative through operator-monotonic increasing functions. The principal result is that the function et, t ∈ [0,∞), is operator-monotonic increasing on a C∗-algebra A if and only if A is commutative. Therefore, C∗-algebra A is commutative if and only if ex+y = exey in A+̇C for all positive elements x, y in A.