A note on the Gelfand-Mazur theorem (Q6634289)

The authors find sufficient conditions for a complex unital semi-simple advertibly complete \(A\)-\(p\)-normed algebra \((E, \|\cdot\|_p)\), \(0<p\leqslant 1\), to be topologically isomorphic to the field \(\mathbb C\) of complex numbers.\N\NThe suitable choices for sufficient conditions found in this paper are the following:\N\begin{itemize}\N\item[1)] for every \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a star-shaped domain;\N\item[2)] for every \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a convex set;\N\item[3)] \((E, \|\cdot\|_p)\) has involution \(^*\) and for every normal \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a star-shaped domain;\N\item[4)] \((E, \|\cdot\|_p)\) has involution \(^*\) and for every normal \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a convex set;\N\item[5)] \((E, \|\cdot\|_p)\) has involution \(^*\), is a \(p\)-Banach algebra and for every unitary \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a star-shaped domain;\N\item[6)] \((E, \|\cdot\|_p)\) has involution \(^*\) which is an anti-morphism, and for every normal \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a star-shaped domain;\N\item[7)] \((E, \|\cdot\|_p)\) has involution \(^*\) which is an anti-morphism, and for every normal \(x\in\) Fr\((G(E))\), Sp\(_E(x)\) is a convex set.\N\end{itemize}