For a positive normal linear functional $\\\\\\\\varphi$ on a von Neumann algebra $\\\\\\\\mathscr{A}$, we prove that the following conditions are equivalent: {\\\\\\\\rm (i)} $\\\\\\\\varphi$ is tracial, {\\\\\\\\rm (ii)} $|\\\\\\\\varphi(\\\\\\\\mathrm{Re}(A^2)|\\\\\\\\leq \\\\\\\\varphi(|A|^2)$ for all $A \\\\\\\\in \\\\\\\\mathscr{A}$, and (iii) $|\\\\\\\\varphi(A^2)|\\\\\\\\leq \\\\\\\\varphi(|A|^2)$ 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 $-\\\\\\\\varphi(A^2B^2)\\\\\\\\leq\\\\\\\\varphi((AB)^2)\\\\\\\\leq\\\\\\\\varphi(A^2B^2)$ 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 $\\\\\\\\tr(AB) \\\\\\\\pm \\\\\\\\tr(X^*X)\\\\\\\\leq \\\\\\\\tr(A) \\\\\\\\tr(B) \\\\\\\\pm \\\\\\\\tr(X^*) \\\\\\\\tr(X)$ provided that $\\\\\\\\begin{bmatrix} A & X \\\\\\\\\\\\\\\\ X^* & B \\\\\\\\end{bmatrix}$ is positive semidefinite, without using the fact that $\\\\\\\\Phi(X)= X + (\\\\\\\\tr X)I$ is completely copositive, and then present a characterization of the trace on the full matrix algebra $\\\\\\\\mathbb{M}_n$.

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