Characterizations of tracial functionals on C∗-algebras

We establish several characterizations of tracial functionals [Formula: see text] on the finite [Formula: see text]-algebra [Formula: see text] (that is, [Formula: see text] for some number [Formula: see text]) via any one of the inequalities [Formula: see text] and [Formula: see text], which are well-known when [Formula: see text]. In addition, we characterize the trace on [Formula: see text] among all positive linear functionals [Formula: see text] with [Formula: see text] through an inequality for determinant. We also establish that such a functional is equal to the usual trace if and only if [Formula: see text] for all positive integers [Formula: see text] and all [Formula: see text]. Furthermore, we show that there is no state [Formula: see text] on [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Finally, we establish that for a positive linear functional [Formula: see text] on a [Formula: see text]-algebra [Formula: see text], the following conditions are equivalent: (i) [Formula: see text] is tracial; (ii) [Formula: see text] for all [Formula: see text]. A new criterion for the commutativity of [Formula: see text]-algebras is also provided.