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Operator Algebras: Theory of C*-Algebras and von Neumann by Bruce Blackadar

By Bruce Blackadar

This booklet bargains a entire creation to the overall concept of C*-algebras and von Neumann algebras. starting with the fundamentals, the speculation is built via such themes as tensor items, nuclearity and exactness, crossed items, K-theory, and quasidiagonality. The presentation conscientiously and accurately explains the most positive factors of every a part of the idea of operator algebras; most vital arguments are at the least defined and plenty of are awarded in complete aspect.

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Extra resources for Operator Algebras: Theory of C*-Algebras and von Neumann Algebras (Encyclopaedia of Mathematical Sciences)

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Proof: It suffices to show that if T is closed, then T is adjointable and (T ∗ )∗ = T . Let V be the unitary operator on H ⊕ H defined by V (ξ, η) = (η, −ξ). V 2 = −I, and Γ(S ∗ ) = [V Γ(S)]⊥ for any densely defined operator S on H, and in particular Γ(T ∗ ) = [V Γ(T )]⊥ . So, since V is unitary, V Γ(T ∗ ) = [V 2 Γ(T )]⊥ = Γ(T )⊥ 30 I Operators on Hilbert Space [V Γ(T ∗ )]⊥ = Γ(T )⊥⊥ = Γ(T ) since T is closed. If η ∈ D(T ∗ )⊥ , then (0, η) ∈ [V Γ(T ∗ )]⊥ = Γ(T ), so η = 0 and D(T ∗ ) is dense. Furthermore, setting S = T ∗ above, Γ((T ∗ )∗ ) = [V Γ(T ∗ )]⊥ = Γ(T ), (T ∗ )∗ = T .

Limω (Ti∗ ) = (limω Ti )∗ . limω Ti is a weak limit point of (Ti ). If Ti → T weakly, then T = limω Ti . Property (v) implies the weak compactness of the unit ball of L(H). 5 Proposition. e. Ti ≤ Tj for i ≤ j, and ∃K with Ti ≤ K for all i). Then there is a positive operator T on H with Ti → T strongly, and T = sup Ti . T is the least upper bound for {Ti } in L(H)+ . For the proof, an application of the CBS inequality yields that, for any ξ ∈ H, (Ti ξ) is a Cauchy net in H, which converges to a vector we call T ξ.

If the closure of Γ(T ) is the graph of a (densely defined) operator (called the closure of T ). An everywhere defined closed operator is bounded by the Closed Graph Theorem. Conversely, a bounded densely defined operator is closable, and its closure is its unique everywhere defined extension. We will only consider closed (or sometimes closable) operators. If T is closed and S ∈ L(H), then S + T is closed. If T is closed and oneto-one with dense range, then T −1 is also closed. If T is closed and bounded below on D(T ), then R(T ) is closed.

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