Geodesic
Seminorms Associated with Subadditive Weights on $C^*$-Algebras
Matematičeskie zametki, Tome 107 (2020) no. 3, pp. 341-350.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Hilbert space, bounded linear operator, Cayley transform, projection, von Neumann algebra, $C^*$-algebra, subadditive weight, seminorm
Mots-clés : matrix norm. 
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A. M. Bikchentaev. Seminorms Associated with Subadditive Weights on $C^*$-Algebras. Matematičeskie zametki, Tome 107 (2020) no. 3, pp. 341-350. http://geodesic.mathdoc.fr/item/MZM_2020_107_3_a1/

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