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

We introduce a new family of non-negative real-valued functions on a $C^{\ast }$-algebra $\mathcal{A}$, i.e., for $0\le \mu \le 1,$ $$\mathrm{\parallel }a{\mathrm{\parallel }}_{{\sigma }_{\mu }}=\text{sup}\{\sqrt{\mathrm{\mid }f(a){\mathrm{\mid }}^2{\sigma }_{\mu }f(a^{\ast }a)}:f\in {\mathcal{A}}^{\mathrm{\prime }},f(1)=\mathrm{\parallel }f\mathrm{\parallel }=1\},$$ 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.