For finite-dimensional B, the universal quantum homomorphism algebra U(A,B) inherits LP, RFD, primitiveness, UCT, and K-theory from A, while for infinite-dimensional B a universal object exists only as a locally C*-algebra, and U(A,A) is a quantum semigroup.
Dykema, Exactness of reduced amalgamated free product C^* -algebras , Forum Math
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Operator Algebras of Universal Quantum Homomorphisms
For finite-dimensional B, the universal quantum homomorphism algebra U(A,B) inherits LP, RFD, primitiveness, UCT, and K-theory from A, while for infinite-dimensional B a universal object exists only as a locally C*-algebra, and U(A,A) is a quantum semigroup.