Treffer: Quasi *-Algebras and Multiplication of Distributions: Quasi *-algebras and multiplication of distributions

Let \(A\) be a linear space and \(A_0\) a *-algebra contained in \(A\). The linear space \(A\) is said to be a quasi *-algebra over \(A_0\) if (i) the right and left multiplications of an element of \(A\) and an element of \(A_0\) are always defined and linear; and (ii) an involution * (which extends the involution of \(A_0\)) is defined in \(A\) with the property \((AB)^*= B^*A^*\) whenever the multiplication is defined. The authors investigate the space of distributions \({\mathcal S}_X'(\Omega)\) associated with a selfadjoint operator \(X\) in the space \(L^2(\Omega,\mu)\), where \(\Omega\) is a locally compact Hausdorff measure space with a positive Borel measure \(\mu\). The main purpose of their study is to find conditions on the operator \(X\) for \({\mathcal S}_X'(\Omega)\) to be a quasi *-algebra in the sense given above.