Abstract
We characterize the tracial functionals on the full matrix algebra \(\mathbb{M}_{n}(\mathbb{C})\) via the inequality \(16\,\varphi(BA^{2}B)\leq\varphi((A+B)^{4})\) for all \(A,B\in\mathbb{M}_{n}(\mathbb{C})^{+}\) . We proved that for a positive normal linear functional \(\varphi\) on a von Neumann algebra \(\mathcal{M}\) the following conditions are equivalent: (i) \(\varphi\) is tracial; (ii) \(\varphi(A^{p}+B^{p})\leq\varphi((A+B)^{p})\) for some \(p>1\) and for all \(A,B\in\mathcal{M}^{+}\) ; (iii) \(\varphi(A^{p}+B^{p})\geq\varphi((A+B)^{p})\) for some \(0<p<1\) and for all \(A,B\in\mathcal{M}^{+}\) ; (iv) \(8\,\varphi(PQP)\leq\varphi((P+Q)^{3})\) for all \(P,Q\in\mathcal{M}^{\textrm{pr}}\) .
