# Operator Algebras. Theory of Casterisk-Algebras and von by Bruce Blackadar

This ebook deals a entire advent to the overall thought of C*-algebras and von Neumann algebras. starting with the fundamentals, the idea is constructed via such issues as tensor items, nuclearity and exactness, crossed items, K-theory, and quasidiagonality. The presentation conscientiously and accurately explains the most positive aspects of every a part of the idea of operator algebras; most vital arguments are not less than defined and plenty of are awarded in complete element.

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Example text

E. 5 Theorem. Let T be a normal operator on a separable Hilbert space H. Then, for 1 ≤ n ≤ ∞, there are disjoint Borel subsets Xn of σ(T ), with ∪Xn = σ(T ), and ﬁnite regular Borel measures µn on Xn , such that T is 2 unitarily equivalent to n Mz on n L (Xn , µn , Hn ), where Hn is an ndimensional Hilbert space and Mz is multiplication by f (z) = z. The Xn are uniquely determined up to sets of T -measure zero, and the µn are unique up to equivalence. In connection with functional calculus, the following fact is signiﬁcant.

Let H be a Hilbert space. (i) If S, T ∈ F(H), then index(ST ) = index(S) + index(T ). (ii) The index is locally constant on F(H), and hence constant on connected components (path components) of F(H). (iii) If T ∈ F(H) and K ∈ K(H), then index(T + K) = index(T ). The proof of (i) is a straightforward but moderately involved calculation, and (ii) is a simple consequence of (i) and the fact that the invertible operators form an open set. For (iii), note that T and T + K are connected by the path {T + tK : 0 ≤ t ≤ 1} in F(H).

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 . 4 Proposition. Let T be a closed operator on H which is one-to-one with dense range, and let T −1 be its “inverse,” with domain R(T ) (so T −1 is closed).