Crash course: the hyperfinite II1 factor and the standard form

Masterclass “Topological quantum field theories, quantum groups and 3-manifold invariants”) This is a follow-up notes for the introductory talks (45min × 2) on the hyperfinite II1 factor and the standard form prepared for Prof. Ryszard Nest’s talks Oct.9-10. There may be typos or errors. 1 Talk 1 (Oct. 6 2014, 10:30-11:15) We define the notion of II1 factors and see one example, namely the hyperfinite II1 factor. We use the symbol H for complex Hilbert space (all Hilbert spaces in this notes are separable, and infinitedimensional). The inner product ⟨ · , · ⟩ : H × H → C is assumed to be linear in the first variable and conjugate-linear in the second variable. We use Greek letters ξ, η, · · · for vectors in H and Italic letters x, y, · · · for bounded linear operators on H. A linear opeartor x : H → H is bounded, if it is continuous in Hilbert space norm. In this case, there exists c > 0 such that ∥xξ∥ ≤ c∥ξ∥ (ξ ∈ H). The infimum of such c is called the (operator) norm of x, denoted as ∥x∥: ∥x∥ = sup ξ∈H, ∥ξ∥≤1 ∥xξ∥. Definition 1.1. We denote by B(H) the set of all bounded linear maps from H to itself. B(H) equipped with the operator norm is a Banach space. For each x ∈ B(H) there exists a unique x∗ ∈ B(H) satisfying ⟨xξ, η⟩ = ⟨ξ, x∗η⟩ (ξ, η ∈ H). x∗ is called the adjoint of x. The map x 7→ x∗ has the following properties (x, y ∈ B(H), λ, μ ∈ C): (1) (λx+ μy)∗ = λx∗ + μy∗. (2) (x∗)∗ = x. (3) (xy)∗ = y∗x∗. (4) ∥x∗x∥ = ∥x∥. B(H) has the *-algebra structure with respect to the operator sum, multiplication, etc. A vector subspace A of B(H) is called a *-subalgebra if A is closed under operator product and *. For simplicity we only consider *-subalgebras of B(H) containing the unit 1 = 1H . Definition 1.2. A *-subalgebra A of B(H) with unit 1H is called a C∗-algebra, if A is closed in operator norm topology. A is called a von Neumann algbera on H, if A is closed with respect to the strong operator topology (SOT). Here, a net {xi}i∈I ⊂ B(H) converges to x ∈ B(H) with respect to SOT, if limi ∥xiξ − xξ∥ = 0 for every ξ ∈ H. Below we always assume that all C∗-algebras on H contain the unit 1 = 1H . Next lemma is only relevant for Corollary 1.5, which is not important either for understanding the rest of the topics. So they can safely be ignored.