Definition. Let U be a C∗-algebra. A representation of U is a pair (H,π), where H is a Hilbert space and π is a ∗-homomorphism of U into B(H). A representation (H,π) is said to be irreducible if π(U) is weakly dense in B(H). A representation (H,π) is said to be faithful if π is an isomorphism.
- unitarily equivalent if there is a unitary U : H → K such that Uπ(A) = φ(A)U for all A ∈ U.
- quasiequivalent if the von Neumann algebras π(U)′′ and φ(U)′′ are ∗-isomorphic.
- disjoint if they are not quasiequivalent.
We construct the Hilbert space H from equivalence classes of elements in U, and the representation π is given by the action of left multiplication. In particular, define a bounded sesquilinear form on U by setting
Let H be the quotient of A induced by the norm ∥A∥ω = ⟨A,A⟩ω1/2. Let H be the unique completion of the pre-Hilbert space H0. Thus there is an inclusion mapping j : U → H with j(U) dense in H. Define the operator π(A) on H by setting
One must verify that π(A) is well-defined, and extends uniquely to a bounded linear operator on H. One must also then verify that π is a ∗-homomorphism. Finally, if we let Ω = j(I), then Ω is obviously cyclic for π(U).