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IMI/Publicaţii/BASM/Ediţii/BASM n.2 (75), 2014/

A semi-isometric isomorphism on a ring of matrices

Authors: Svetlana Aleschenko

Abstract

Let $(R,\xi)$ be a pseudonormed ring and $R_{n}$ be a ring of matrices over the ring $R$. We prove that if $1\leq\gamma, \sigma\leq\infty$ and $\frac{1}{\gamma} + \frac{1}{\sigma} \geq 1$, then the function $\eta_{\xi ,\gamma ,\sigma }$ is a pseudonorm on the ring $R_{n}$. Let now $\varphi:(R,\xi)\rightarrow(\bar{R},\overline{\xi})$ be a semi-isometric isomorphism of pseudonormed rings. We prove that $\Phi:(R_{n},\eta_{\xi,\gamma ,\sigma })\rightarrow(\bar{R}_{n},\eta_{\bar{\xi}, \gamma ,\sigma })$ is a semi-isometric isomorphism too for all $1\leq\gamma, \sigma\leq\infty$ such that $\frac{1}{\gamma} + \frac{1}{\sigma} \geq 1$.

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