How to solve an operator equation (Q2366847)

The author discusses the techniques for solving operator equations on \(C^*\)-algebras. The equations are of the form \(T_{\alpha, x_ 1,\dots,x_ n} =0\) where \(\alpha\) is a parameter and \(x_ 1,\dots, x_ n\) are unknowns in a \(C^*\)-algebra. There are examples involving derivations, completely positive operators, centralizing mappings and generators of dynamical semigroups. The main device for solving such equations is the so-called local multiplier algebra which provides a unified approach to the above examples.