Operator Algebras Homework 4
Deadline: 25th of November
Exercise 1
During the lectures we proved the following:
Theorem Let H be a Hilbert space, let x ∈ H be a unit vector and let A ⊂ B(H) be a C∗-subalgebra with1 A ⊃ K(H). Then the (vector) state a 7→ hax, xi is a pure2 state on A.
In this exercise we will give a more direct proof of this theorem, without using the GNS construction.
Let H be a Hilbert space. In parts (a), (b), (c) and (d) of this exercise, x ∈ H will denote a fixed unit vector, ex∈ B(H) will denote the orthogonal projection onto Cx, i.e. exh = hh, xix for all h ∈ H, and A ⊂ B(H) will denote a C∗-subalgebra such that ex∈ A.
(a) If a ∈ B(H), show that exaex= hax, xiex.
We define τx: B(H) → C to be the (vector) state on B(H) given by τx(a) = hax, xi for a ∈ B(H).
(b) Show that the state τx on B(H) restricts to a state on A (in what follows, this restriction of τx to A will also be denoted by τx).
Suppose that ρ : A → C is a positive linear functional with ρ ≤ τx. (c) Prove that for all a ∈ A we have ρ(a) = ρ(exaex).
Hint: First use the Cauchy-Schwarz inequality to prove that for all c, b ∈ A we have both ρ(cb − cbex) = 0 and ρ(bc−exbc) = 0 and use these equalities to prove that both ρ(b−bex) = 0 and ρ(b−exb) = 0 for all b ∈ A.
(d) Use the results of (a) and (c) to prove that τxis a pure state on A.
(e) Now let A ⊂ B(H) be a C∗-subalgebra with A ⊃ K(H) and let x ∈ H be an arbitrary unit vec- tor. Prove that the vector state τxon A is pure, i.e. prove the theorem above.
(f ) Explain in a few sentences why this proof of the theorem fails for C∗-subalgebras A ⊂ B(H) that do not contain K(H). Remark: You do not have to prove any standard results from functional analysis concerning compact operators, but you should be clear about what results you are using.
1Here K(H) denotes the C∗-algebra of compact operators on H.
2Recall that we have seen two equivalent definitions of a pure state on a C∗-algebra. In this exercise we will use the one on page 144 of Murphy: a state τ on a C∗-algebra A is called pure if it has the property that whenever ρ is a positive linear functional on A such that ρ ≤ τ , necessarily there is a number t ∈ [0, 1] such that ρ = t · τ .
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Exercise 2 (Schur’s lemma for C
∗-algebras)
Let A be a C∗-algebra and let ϕ1: A → B(H1) and ϕ2: A → B(H2) be two irreducible representations of A. Furthermore, let T : H1→ H2 be a bounded operator such that
T ϕ1(a) = ϕ2(a)T for all a ∈ A.
(a) Prove that there are c1, c2 ∈ C such that T∗T = c11H1 and T T∗ = c21H2. Also show that if T is non-zero, then c1 = c2∈ R. Warning: Because T ∈ B(H1, H2) and T∗ ∈ B(H2, H1) are not elements of a C∗-algebra, you have to be a bit careful if you want to prove that T∗T and T T∗ are positive operators in the C∗-algebras B(H1) and B(H2), respectively.
(b) Prove that if ϕ1 and ϕ2 are not unitarily equivalent, then we must have T = 0.
Exercise 3
Let A be a C∗-algebra and let ϕ : A → B(H) be a representation of A. Let x ∈ H be a unit vector and define the state τ : A → C by
τ (a) = hϕ(a)x, xi.
We write (Hτ, ϕτ) to denote the corresponding GNS representation. Construct an isometry V : Hτ → H such that
ϕ(a)V = V ϕτ(a) for all a ∈ A.
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