Tracial joint spectral measures

Given two Hermitian matrices, $A$ $A$ and $B$ $B$ , we introduce a new type of spectral measure, a tracial joint spectral measure ${\mu }_{A,B}$ $\mu _{A, B}$ on the plane. Existence of this measure implies the following two results: 1) any two-dimensional subspace of the Schatten- $p$ $p$ class is isometric to a subspace of $L_p$ $L_{p}$ , and 2) if $f:\mathbb{R}\to \mathbb{R}$ $f : \mathbb{R}\to \mathbb{R}$ has non-negative $k$ $k$ th derivative and $A$ $A$ and $B$ $B$ are Hermitian matrices with $A$ $A$ positive semidefinite, then $t\mapsto trf(tA+B)$ $t \mapsto \operatorname{tr}f(t A + B)$ has non-negative $k$ $k$ th derivative. We also give an explicit expression for the measure ${\mu }_{A,B}$ $\mu _{A, B}$ .