A Family of Semi-norms in $C^*$-algebras
Athul Augustine, Pintu Bhunia, P. Shankar
TL;DR
The paper introduces a family of interpolation semi-norms on a unital $C^*$-algebra $\mathcal{A}$ that lie between the algebraic numerical radius and the operator norm by defining $\|a\|_{\sigma_{\mu}}$ via normalized states and a symmetric-mean interpolation path $\sigma_{\mu}$. The authors establish fundamental properties, including $v(a)\le\|a\|_{\sigma_{\mu}}\le\|a\|$ and several operator-class inequalities, and they show that $\|\cdot\|_{\nabla}$ is a norm with a detailed equality characterization for the triangle inequality. They derive extensive upper and lower bounds for $\|a\|_{\sigma}$ in terms of $v(a)$, Crawford's $m(a)$, and moments of $|a|$, $|a^{*}|$, including a parameterized family using $\phi(t)=t^{\nu}$, and they provide a lower bound $\|a\|_{\nabla}\ge \frac{1}{\sqrt{2}}\|a\|$ along with Crawford-type relations. The results extend the semi-norm framework from $B(\mathcal{H})$ to general unital $C^*$-algebras and connect to the broader literature on operator means and numerical radii. An illustrative matrix example shows the sharp positioning of $v(a)$, $\|a\|_{\sigma}$, and $\|a\|$ in a concrete setting. Overall, the work furnishes a unified approach to estimating norms and numerical radii via interpolation semi-norms in operator algebras.
Abstract
We introduce a new family of non-negative real-valued functions on a $C^*$-algebra $\mathcal{A}$, i.e., for $0\leq μ\leq 1,$ $$\|a\|_{σ_μ}= \text{sup}\left\lbrace \sqrt{|f(a)|^2 σ_μ f(a^*a)}: f\in \mathcal{A}', \, f(1)=\|f\|=1 \right\rbrace, \quad $$ where $a\in \mathcal{A}$ and $σ_μ$ is an interpolation path of the symmetric mean $σ$. 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.