Operator Algebras of Universal Quantum Homomorphisms

Given two unital C*-algebras $A$ and $B$, we study, when it exists, the universal unital $C^*$-algebra $\mathcal{U}(A,B)$ generated by the coefficients of a unital $*$-homomorphism $\rho\,:\, A\rightarrow B\otimes\mathcal{U}(A,B)$. When $B$ is finite dimensional, it is well known that $\mathcal{U}(A,B)$ exists and we study in this case properties LP, RFD, primitiveness and the UCT as well as $K$-theory. We also construct a reduced version of $\mathcal{U}(A,B)$ for which we study exactness, nuclearity, simplicity, absence of non-trivial projection and $K$-theory. Then, we consider the von Neumann algebra generated by the reduced version and study factoriality, amenability, fullness, primeness, absence of Cartan, Connes' invariants, Haagerup property and Connes' embeddability. Next, we consider the case when $B$ is infinite dimensional: we show that for any non-trivial separable unital $C^*$-algebra $A$, $\mathcal{U}(A,B)$ exists if and only if $B$ is finite dimensional. Nevertheless, we show that there exists a unique unital locally $C^*$-algebra generated by the coefficients of a unital continuous $*$-homomorphism $\rho\,:\, A\rightarrow B\otimes\mathcal{U}(A,B)$. Finally, we study a natural quantum semigroup structure on $\mathcal{U}(A,A)$.