On irreducible operator *-algebras on Banach spaces (Q796080)

Let L(X) be the algebra of all bounded linear operators on a Banach space X. A subalgebra \({\mathcal B}\subset L(X)\) is called irreducible if for each pair x,\(y\in X\), \(x\neq 0\) there exists \(A\in {\mathcal B}\) such that \(Ax=y.\) A subalgebra \({\mathcal B}\subset L(X)\) is called strongly irreducible if for each \(y\in X\) there exists a constant \(\alpha_ y\) with the property: If \(x\in X\), \(\| x\| =1,\) then there exists \(A\in {\mathcal B}\) such that \(Ax=y,\) and \(\| A\| \leq \alpha_ y.\) Let \({\mathcal A}\) be a real or complex Banach *-algebra with the identity element e. \({\mathcal A}\) is called symmetric if \((e+a^*a)^{-1}\) exists for each \(a\in {\mathcal A}\). The main purpose of the paper is to prove the result below which can be considered as a characterization of Hilbert spaces among all Banach spaces. Theorem 1. Let X be a real or complex Banach space. Suppose there exists a strongly irreducible symmetric Banach *-algebra \({\mathcal B}\subset L(X)\) which contains the identity operator. In this case there exists an inner product on X such that the corresponding norm is equivalent to the given norm, and that for each \(A\in {\mathcal B}\), \(A^*\) is the adjoint of A relative to the inner product. Using the well known Kadison's result concerning representations of \(B^*\)-algebras the following result is proved. Theorem 2: Let X be a complex Banach space, and suppose that there exists an irreducible \(B^*\)-algebra \({\mathcal B}\subset L(X)\) which contains the identity operator. In this case there exists an inner product on X such that the corresponding norm is equivalent to the given norm, and that for each \(A\in {\mathcal B}\), \(A^*\) is the adjoint of A relative to the inner product.