Introduction von Neumann Algebras and II_1 Factors

Abstract

Denote by B.(H) the algebra of bounded linear operators on the Hilbert space H. Recall that B(H) is naturally endowed with an involution x 7 ⟼ x^* associating with an operator x its adjoint x^*. The operator norm ∥x∥ of an element of B(H) is defined by ∥x∥ = sup {∥xξ∥ : ξ є H, ∥ξ∥ ≤ 1}. Endowed with this norm, B(H) is a Banach algebra with involution satisfying the identity ∥x^*x∥ = ∥x∥^2 (C^*-identity) i.e. a C^*-algebra. The weak operator topology on B(H) is the weakest topology making the map x ⟼ (xξ,η)

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