Unitarily covariant maps in approximately finite-dimensional $C^*$-algebras

We consider maps defined on a real space $A_\mathrm{sa}$ of all self-adjoint elements of a $C^*$-algebra $A$ commuting with the conjugation by unitaries: $F(u^*au)=u^*F(a)u$ for any $a\in A_\mathrm{sa}$, $u\in\mathcal U(A)$. In the case where $A$ is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized to the class of uniformly hyperfinite $C^*$-algebras and to the algebra of all compact operators in a Hilbert space.