On rings with involution equipped with some new product (Q2714166)

Let \(R\) be a ring with involution \(*\). Introduce a new product \(\diamond\) in \(R\) by \(r\diamond s=rs-sr^*\). This product first appeared in work by \textit{P. Šemrl} [Stud. Math. 97, No. 3, 157-165 (1991; Zbl 0761.46047)], in the problem of representing quadratic functionals by sesquilinear functionals. Recently, \textit{L. Molnár} [Linear Algebra Appl. 235, 229-234 (1996; Zbl 0852.46021)] has initiated a systematic study of this product, showing that a subset of \({\mathcal B}(H)\), the algebra of all bounded linear operators on a real or complex Hilbert space \(H\) of dimension at least 2, is an ideal if and only if it is a left \(\diamond\)-ideal in \({\mathcal B}(H)\). The authors generalize this result in different ways and study algebraic properties of left and right \(\diamond\)-ideals. Moreover, several important examples are provided.