In the introduction we present several known results about preservers on matrix and operator spaces. In the second chapter we prove the fundamental theorem of affine geometry and the fundamental theorem of projective geometry. If ▫$V$▫ is a finite-dimensional real or complex vector space of dimension at least ▫$2$▫, then the fundamental theorem of affine geometry characterizes bijective maps ▫$V \to V$▫ which map lines into lines (they preserve colinearity). If ▫$V$▫ is a real or complex vector space of dimension at least ▫$3$▫, then the fundamental theorem of projective geometry characterizes bijective maps on the projective space of ▫$V$▫ which preserve complanarity. As a corollary we also prove Uhlhorn's theorem. These three theorems are our main tools when solving problems in later chapters. The problems of the following type are considered in the next chapter. Let ▫${\mathcal V}$▫ be the set of ▫$n \times n$▫ hermitian matrices or the set of ▫$n \times n$▫ real symmetric matrices or the set of all effects, or the set of all projections of rank one. Let ▫$c$▫ be a real number. We characterize bijective maps ▫$\phi : {\mathcal V} \to {\mathcal V}$ satisfying ${\rm sl} \left(AB\right) = c \iff {\rm sl} \left(\phi \left(A\right) \phi \left(B\right)\right) = c$▫ with some additional restrictions on ▫$c$▫, depending on the underlying set of matrices. Let ▫$\mathcal H$▫ be an infinite-dimensional real or complex Hilbert space and ▫$\mathcal{I_\infty (H)}$▫ the set of all bounded linear idempotent operators on ▫$\mathcal{H}$▫ with an infinite-dimensional image and an infinite-dimensional kernel. In the fourth chapter we characterize three types of maps on ▫$\mathcal{I_\infty (H)}$▫, namely poset automorphisms, bijective maps preserving orthogonality in both directions, and bijective maps preserving commutativity in both directions. In the last chapter we additionaly assume that ▫$\mathcal{H}$▫ is separable and and we denote the lattice of its closed subspaces by ▫${\rm Lat}\mathcal{H}$▫. We describe the general form of pairs of bijective maps ▫$\phi$▫, ▫$\psi$▫ on ▫${\rm Lat}\mathcal{H}$▫ having the following property: subspaces ▫$U,V \in {\rm Lat}\mathcal{H}$▫ are complemented if and only if the same holds for ▫$\phi \left( U \right)$▫ and▫$\psi \left( V \right)$▫. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.
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