For a positive normal linear functional \(\varphi \) on a von Neumann algebra \(\mathscr {A}\) , we prove that the following conditions are equivalent: (i) \(\varphi \) is tracial, (ii) \(|\varphi (\textrm{Re}(A^2)|\le \varphi (|A|^2)\) for all \(A \in \mathscr {A}\) , and (iii) \(|\varphi (A^2)|\le \varphi (|A|^2)\) for all \(A \in \mathscr {A}\) . Based on this result, we present some criteria for commutativity of a von Neumann algebra. For a trace \(\varphi \) on a \(C^*\) -algebra \(\mathscr {A}\) , we prove that \(-\varphi (A^2B^2)\le \varphi ((AB)^2)\le \varphi (A^2B^2)\) for certain elements of \(\mathscr {A}\) , and show that when \(\varphi \) is faithful, the equality in the second inequality is achieved if and only if \(AB=BA\) . Moreover, we partially generalize the Araki–Lieb–Thirring inequality to arbitrary traces on any \(C^*\) -algebras and to self-adjoint elements. Furthermore, we present a simple joint proof for \({{\,\mathrm{Tr\,\!}\,}}(AB) \pm {{\,\mathrm{Tr\,\!}\,}}(X^*X)\le {{\,\mathrm{Tr\,\!}\,}}(A) {{\,\mathrm{Tr\,\!}\,}}(B) \pm {{\,\mathrm{Tr\,\!}\,}}(X^*) {{\,\mathrm{Tr\,\!}\,}}(X)\) provided that \(\begin{bmatrix} A & X \\ X^* & B \end{bmatrix}\) is positive semidefinite, without using the fact that \(\Phi (X)= X + ({{\,\mathrm{Tr\,\!}\,}}X)I\) is completely copositive, and then present a characterization of the trace on the full matrix algebra \(\mathbb {M}_n\) .
