A Family of Semi-norms in $C^{\ast }$-algebras

Abstract

We introduce a new family of non-negative real-valued functions on a $C^{\ast }$-algebra $\mathcal{A}$, i.e., for $0\le \mu \le 1,$ $$\Vert a{\Vert }_{{\sigma }_{\mu }}=\text{sup}\left\{\sqrt{|f(a){|}^2{\sigma }_{\mu }f(a^{\ast }a)}:f\in {\mathcal{A}}^{\prime },f(1)=\Vert f\Vert =1\right\},$$ where $a\in \mathcal{A}$ and ${\sigma }_{\mu }$ is an interpolation path of the symmetric mean $\sigma$. These functions are semi-norms as they satisfy the norm axioms, except for the triangle inequality. Special cases satisfying triangle inequality, and a complete equality characterization is also discussed. Various bounds and relationships will be established for this new family, with a connection to the existing literature in the algebra of all bounded linear operators on a Hilbert space.