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Paper

von Neumann algebras in JT gravity

Von Neumann Algebras, Subfactor and Knots - II. Subfactors - GAG_210312_Clare.pdf
https://prclare.people.wm.edu/GAG/GAG_210312_Clare.pdf

reference [40] in von Neumann algebras in JT gravity by David K. Kolchmeyer

Table of Contents

von Neumann algebra

adjoint

Setup:

\mathcal{H} : a Hilbert space

\mathfrak{B}(\mathcal{H}) : the space of bounded operators ( \mathcal{H} \to \mathcal{H} )

Theorem

\forall T \in \mathfrak{B}(\mathcal{H}),\, \exists!\, S \colon \mathcal{H} \to \mathcal{H} s.t.

\langle T x, y \rangle =\langle x, T^{*} y \rangle \quad \forall x,y \in \mathcal{H}.

This is called the adjoint of an operator T .

C*-algebra

Definition

A C*-algebra satisfies the following.

  1. subalgebra of \mathfrak{B}(\mathcal{H}) from a algebraic point of view
  2. closed from a topological point of view
  3. stable (invariant) under adjoint

Ex.

  • The matrix algebra \mathrm{M}(n,\mathbb{C}) equals the set of boundary operators \mathfrak{B}(\mathbb{C}^{n}) , which is a C*-algebra.

commutants

Def.

The commutant of a subset A \subseteq \mathfrak{B}(\mathcal{H}) is defined as

A' := \lbrace y \in \mathfrak{B}(\mathcal{H}) \mid \forall x \in A, xy = yx \rbrace

Def.

The bicommutant of A is the commutant of the commutant of A : A'' = (A')' .

Ex.

Let A = \mathrm{M}(n,\mathbb{C}) \cong \mathfrak{B}(\mathbb{C}^{n}) , then A' = \mathbb{C}. 1_N = \lbrace c 1_N \mid c \in \mathbb{C} \rbrace and A'' = A .

Definition of the von Neumann algebra

Def.

A von Neumann algebra is a (C)*-(sub)algebra such that the bicommutant equals itself.