A note on the Gelfand-Mazur theorem

In this note, we show that if (E,.p),0<p <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E,\left\| .\right\| _{p}),\ 0<p\le 1$$\end{document}, is a unital semi-simple complex advertible complete A-p-normed algebra such that, for every x is an element of Fr(G(E))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in Fr(G(E))$$\end{document}, SpE(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sp_{E}(y)$$\end{document} is star-shaped domain for every y is an element of Bx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in B_{x}$$\end{document}, then E similar or equal to C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\simeq \mathbb {C}$$\end{document}. In the involutive case, we obtain the same conclusion under the starry hypothesis on the spectrum of each normal element of Fr(G(E)). If the algebra is additionally hermitian, it suffices to assume that the spectrum of each unitary element of Fr(G(E)) is star-shaped domain. The case of algebras with an involution anti-morphism is also considered.