Normed algebras with involution
Let \(A\) be a (complex) normed algebra with an identity element 1 and an involution \(*\), which may not be continuous. Denote by \(\widetilde {A}\) the completion of \(A\). Then \(\widetilde {A}\) is not a Banach \(*\)-algebra in general. For each \(x\) in \(A\), let \(r(x)\) be the spectral radius and \(p(x)= r(x^* x)^{1/2}\) the Pták function. Then \(A\) is said to be an \(H\)-algebra if \(r(x+y)\leq p(x)+p(y)\) for all \(x\), \(y\) in \(A\). The main result of this paper is: Let \(A\) be a normed \(*\)-algebra and \(H(A)\) the set of all selfadjoint elements of \(A\). Then the following conditions are equivalent: (i) \(A\) is an \(H\)-algebra. (ii) \(r(x)\leq p(x)\) and \(x+y\geq 0\) whenever \(x,y\in A\), \(x\geq 0\), \(y\geq 0\). (iii) \(r(x)\leq p(x)\) and \(p(x+y)\leq p(x)+p(y)\). (iv) There exists a Hilbert space \(H\) and a \(*\)-representation \(\pi: A\to B(H)\) such that \(r(\pi(x))=r(x)\) for every \(x\in A\). (v) \(1+a+b+ic\) is invertible in \(\widetilde A\) for every \(a,b,c\in H(A)\), \(a\geq 0\), \(b\geq 0\). (iv) \(1+a+b+ic\) is not a left (right) topological divisior of zero in \(A\) for all \(a,b,c\in H(A)\), \(a\geq 0\), \(b\geq 0\). This generalizes the known results for Hermitian Banach algebras.
scientific article; zbMATH DE number 91458
scientific article