\(p\)-Banach algebras with generalized involution and \(C^*\)-algebra structure (Q2730861)

Let \((A,\|\|_p)\), \(0<p\leq 1\), be a complex \(p\)-Banach algebra with a generalized involution \(x\mapsto x^*\), that is, \(A\) is a complex algebra with a generalized involution \(x\mapsto x^*(x\mapsto x^*\) is a vector space involution satisfying \((xy)^*= y^*x^*\) or \(x^*y^*\), for all \(x,y\in A)\) and \(\|\cdot \|_p\) is a \(p\)-homogeneous norm on \(A\) satisfying \(\|xy \|_p\leq \|x\|_p\|y\|_p\), and \(A\) is complete. The author proves that a complex \(p\)-Banach algebra \((A,\|\|_p)\), \(0<p\leq 1\) with a generalized involution \(x\mapsto x^*\) is a \(C^*\)-algebra for a norm equivalent to \(\|\cdot \|_p\) if one of the following conditions (i)--(iv) holds:NEWLINENEWLINENEWLINE(i) \(\exists c>0\); \(\|x\|^2_p\leq c\|x^*x\|_p\), \(\forall x\in A\). NEWLINENEWLINENEWLINE(ii) \(\exists c>0\); \(\|h\|^2_p\leq c\rho(h)^p\), \(\forall h^*=h\in A (\rho(h)\) is the spectral radius of \(h)\).NEWLINENEWLINENEWLINE(iii) \(\exists c>0\); \(\|x \|_p\|x^*\|_p\leq c\|x^*x\|_p\), \(\forall x\in A\) with \(x^*x= xx^*\).NEWLINENEWLINENEWLINE(iv) \(\exists c<0\); \(\|x\|^2_p\leq c\rho (x^*x)^p\), \(\forall x \in A\) with \(x^*x=x x^*\).NEWLINENEWLINENEWLINE(v) \(A\) is unitary and \(\exists c>0\); \(\|u \|_p\leq c\), \(\forall u\in A\) with \(u^*u-uu^*=e\).