On bounded part of an algebra of unbounded operators (Q909976)

The author has characterized the symmetry of an \(Op^*\)-algebra \({\mathcal A}\) by topological properties of the bounded part \({\mathcal A}_ b\) of \({\mathcal A}:\) Theorem 1. (a) If \({\mathcal A}\) is symmetric, then \({\mathcal A}_ b\) is sequentially dense in any *-algebra topology \(\tau\) in \({\mathcal A}\) such that \(B_ 0\equiv \{T\in {\mathcal A}_ b\); \(\| T\| \leq 1\}\) is \(\tau\)-bounded. (b) Let \(\tau\) be any *-algebra topology on \({\mathcal A}\) such that the multiplication in \({\mathcal A}\) is \(\tau\)-hypocontinuous, and \(B_ 0\) is \(\tau\)-bounded and \(\tau\)-sequentially complete. If (\({\mathcal A}_{b,\| \|})\) is sequentially \(\tau\)-dense in \({\mathcal A}\), then \({\mathcal A}\) is symmetric. Furthermore, he has obtained the following result: Theorem 2. Let \({\mathcal A}\) be an \(Op^*\)-algebra satisfying condition (I); that is, there exists a sequence \(\{A_ n\}\) in \({\mathcal A}^{\dag}\) such that \(A_ n\geq 1\) and \(A_ n^{-1}\in {\mathcal A}\) for all n, and for each \(x\in {\mathcal A}\) \(x^*x\leq kA_ n\) for some n and \(k>0\). Then, the \(\rho\)-topology, the \(\sigma\)-weak topology and the ordered topology on \({\mathcal A}\) coincide.