Tomita-takesaki Theory

نویسنده

BRENT NELSON

چکیده

Definition 1.1. Let A be an involutive algebra over C with involution ξ 7→ ξ (resp. ξ 7→ ξ). We say A is a left (resp. right) Hilbert algebra if A has a inner product (· | ·) satisfying: a. Multiplication on the left (resp. right) is a bounded operator; that is, ∀ξ ∈ A the map πl(ξ) : η 7→ ξη (resp. πr(ξ) : η 7→ ηξ) is bounded on A. b. (ξη | ζ) = (η | ξζ) (resp. (ξη | ζ) = (ξ | ζη)). c. The involution map ξ 7→ ξ (resp. ξ 7→ ξ) is closable. d. Denote by A the linear span of products ξη for ξ, η ∈ A (note this is a subalgebra). Then A ⊂ A is dense.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Tomita–Takesaki Modular Theory

We provide an brief overview of Tomita–Takesaki modular theory and some of its applications to mathematical physics. This is an article commissioned by the Encyclopedia of Mathematical Physics, edited by J.-P. Francoise, G. Naber and T.S. Tsun, to be published by the Elsevier publishing house.

متن کامل

Boundary conformal fields and Tomita–Takesaki theory

Motivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita–Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shown that, when normalisable the Ishibashi states are cyclic separ...

متن کامل

Algebraic Quantum Theory

We present the mathematical setting of an algebraic approach to quantum theory, including the Tomita–Takesaki theory, liouvilleans and quantum dynamical systems, and non-commutative integration up to the Falcone–Takesaki theory.

متن کامل

Application of Tomita-Takesaki theory in algebraic euclidean field theories

The construction of the known interacting quantum field theory models is mostly based on euclidean techniques. The expectation values of interesting quantities are usually given in terms of euclidean correlation functions from which one should be able to extract information about the behavior of the correlation functions of the Minkowskian counterpart. We think that the C*-algebraic approach to...

متن کامل

W ∗-algebras and integration

We present the mathematical setting of the theory of W ∗-algebras, including the Tomita– Takesaki modular theory, the Connes–Araki relative modular theory, the theory of W ∗dynamical systems (with derivations, liouvilleans, and crossed products), and noncommutative integration up to the Falcone–Takesaki theory of noncommutative Lp(N ) spaces. We pay special attention to abstract algebraic formu...

متن کامل