Definition. A C∗-algebra is a pair consisting of a ∗-algebra U and a norm

∥ · ∥ : A → C such that

∥AB∥ ≤ ∥A∥ · ∥B∥, ∥A∗A∥ = ∥A∥^{2},

∀ A, B ∈ A. We usually use A to denote the algebra and its norm.

Definition. A state ω on A is a linear functional such that ω(A∗A) ≥ 0 ∀ A ∈ U, and ω(I) = 1.

- 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.

Two representations are quasiequivalent iff they have unitarily equivalent subrepresentations.

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