### Abstract:

A Jordan homomorphism is a linear map Ø : A → satisfying Ø(x<sup>2</sup> = Ø(x<sup>2</sup> for every x € A, where A and B are algebras. A few decades ago, Kaplansky raised the question whether unital linear invertibility preserving maps between unital algebras are Jordan homomorphisms . This question is still unanswered, and the progress that has been made has mainly been in the context of Banach algebras, including C* -algebras and von Neumann algebras. Let M be a von Neumann algebra with a faithful semifinite normal trace τ, and M the algebra of τ-measurable operators (measurable for short) affiliated with M. The algebra M can be endowed with a topology γ<sub>cm</sub>, called the topology of convergence in measure, such that M becomes a complete metrizable topological <sup>*</sup>-algebra in which M is dense. We have found affirmative answers to Kaplansky's question in the context of algebras of measurable operators. We also prove, amongst other things, that every self-adjoint Jordan homomorphism between algebras of measurable operators is γ<sub>cm</sub> — γ<sub>cm</sub> continuous and can be expressed as a sum of a self-adjoint algebra homomorphism and a self-adjoint algebra anti-homomorphism. Derivations between algebras of measurable operators are also considered. Recall that a derivation on an algebra A is a linear map D : A → A satisfying D(xy) = xD(y) + D(x)y for every x,y € A. It is well known that every derivation on a C<sup>*</sup>-algebra is continuous and that every derivation on a von Neumann algebra is inner. We have found that these results don't generally carry over to algebras of measurable operators. Motivated by the Singer-Wermer theorem for commutative Banach algebras as well as the non-commutative Singer-Wermer conjecture for Banach algebras, we also ask whether primitive ideals of M are invariant under derivations on M. We have found affirmative answers if M is commutative or if the trace on M is finite. The thesis ends with finding answers to Kaplansky's question when the algebras in volved are locally C<sup>*</sup>-algebras and locally W<sup>*</sup>-algrebras We also investigate derivations on locally C<sup>*</sup>-algebras and locally W<sup>*</sup>-algebras. We give a partial answer to the question whether all derivations on a locally W<sup>*</sup>-algebra are inner.