Fuglede–Putnam Theorem in Algebras with Involutions

The validity of the analogs of the Fuglede–Putnam theorem in the algebra $(\mathcal B(H),\star)$ of bounded operators acting on a Hilbert space $H$ with an arbitrary involution $\star$ is considered, together with the same problem in certain $*$-subalgebras of these algebras and in related constructions. The results obtained in this way are used to solve stability problems for “Fuglede” classes with respect to extensions and to the operation of tensor products.