On \({\mathcal O}^*\)-representable algebras (Q2784961)

A \(*\)-algebra \(A\) is called proper if \(xx^*=0\) for \(x\in A\) implies \(x=0\), and completely proper if all the algebras \(A\otimes M_n(\mathbb C)\), \(n=1,2,\ldots \), are proper. The author shows that if a \(*\)-algebra has a faithful \(*\)-representation by unbounded operators, then it is completely proper. Conditions are also found for the existence of a faithful \(\sigma\)-bounded representation \(\pi\). The latter means that its domain \(D(\pi)\) is a union of an increasing sequence of Hilbert spaces in each of which \(\pi\) induces a representation by bounded operators.