By Bruce Blackadar
This ebook is the main finished remedy to be had of the overall idea of C*-algebras and von Neumann algebras. starting with the fundamentals, the idea is built via such subject matters as tensor items, nuclearity and exactness, crossed items, type of injective components, K-theory, finiteness, good rank, and quasidiagonality.
The presentation concentrates on rigorously and accurately explaining the most beneficial properties of every a part of the idea of operator algebras; most crucial arguments are no less than defined, and lots of are awarded in complete element, so the quantity is far greater than a trifling survey.
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Additional resources for Operator algebras: theory of C*-algebras and von Neumann algebras
Sample text
The index is constant on connected components of C \ σe (T ) and vanishes on C \ σ(T ) and, in particular, on the unbounded component of C \ σe (T ). Thus, if C \ σe (T ) is connected, we have σe (T ) = ∩K∈K(H) σ(T + K). 13 Let G be the group of invertible elements of Q(H), and Go the connected component (path component) of the identity in G. 8 implies that index is a homomorphism from G/Go to the additive group Z. 9 imply that index is actually an isomorphism from G/Go ∼ = K0 (K(H)). 14 Many of the properties of Fredholm operators carry over to semiFredholm operators.
5 Examples. 3). Then S n → 0 weakly, but not strongly. (S ∗ )n → 0 strongly, but not in norm. 6 This example shows that the adjoint operation is not strongly continuous. ) One can define another, stronger topology, the strong-* operator topology, by the seminorms T → T ξ and T → T ∗ ξ for ξ ∈ H, in which the adjoint is continuous. There is similarly a σ-strong-* operator topology. Even though the adjoint is not strongly continuous, the set of self-adjoint operators is strongly closed (because it is weakly closed).
Thus the unilateral shift S is in the strong closure, but not the strong-* closure, and S ∗ is in the weak closure but not the strong closure. In fact, the weak closure of U(H) is the entire closed unit ball of L(H) [Hal67, Problem 224]. 10 Theorem. [DD63] Let H be an infinite-dimensional Hilbert space. Then U(H) is contractible in the weak/strong/strong-* topologies, and the unit sphere {ξ ∈ H : ξ = 1} of H is contractible in the norm topology. Proof: First suppose H is separable, and hence H can be identified with L2 [0, 1].