Characterizations of $w^{\ast }$-homomorphisms and expectations

If $\mathfrak {A}$ and $\mathfrak {B}$ are $\ast$-algebras a map $\phi :\mathfrak {A} \to \mathfrak {B}$ is called a Schwarz map if it is linear and satisfies the Cauchy-Schwarz inequality $\phi {(a)^ \ast }\phi (a) \leq \phi (a ^\ast a)$ for all $a \in \mathfrak {A}$. Under mild restrictions on $\mathfrak {A}$ and $\mathfrak {B}$, $\ast$-homomorphisms and expectations are characterized in terms of Schwarz maps $\phi :\mathfrak {A} \to \mathfrak {B}$. The proofs are based on an elementary result on the multiplicative properties of Schwarz maps.