A Family of Semi-norms in $C^*$-algebras
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910854641876992 |
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| author | Augustine, Athul Bhunia, Pintu Shankar, P. |
| author_facet | Augustine, Athul Bhunia, Pintu Shankar, P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Family of Semi-norms in $C^*$-algebras Augustine, Athul Bhunia, Pintu Shankar, P. Functional Analysis Operator Algebras 47A12, 47A30, 26E60, 46L05 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. |
| title | A Family of Semi-norms in $C^*$-algebras |
| topic | Functional Analysis Operator Algebras 47A12, 47A30, 26E60, 46L05 |
| url | https://arxiv.org/abs/2503.01331 |