Seminorms Associated with Subadditive Weights on $C^*$-Algebras

Let $\varphi$ be a subadditive weight on a $C^*$-algebra $\mathscr A$, and let $\mathfrak M_\varphi^+$ be the set of all elements $x$ in $\mathscr A^+$ with $\varphi(x)<+\infty$. A seminorm ${\|\cdot\|}_\varphi$ is introduced on the lineal $\mathfrak M_\varphi^{\mathrm{sa}}=\operatorname{lin}_{\mathbb R}\mathfrak M_\varphi^+$, and a sufficient condition for the seminorm to be a norm is given. Let $I$ be the unit of the algebra $\mathscr A$, and let $\varphi(I)=1$. Then, for every element $x$ of $\mathscr A^{\mathrm{sa}}$, the limit $\rho_\varphi (x)=\lim_{t\to 0+}(\varphi(I+tx)-1)/t$ exists and is finite. Properties of $\rho_\varphi$ are investigated, and examples of subadditive weights on $C^*$-algebras are considered. On the basis of Lozinskii's 1958 results, specific subadditive weights on $\mathbb M_n(\mathbb C)$ are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained.