nLab
Hilbert bimodule

Contents

Context

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The notion of Hilbert C *C^\ast-bimodule adapts the notion of bimodules over associative algebras to operator algebra/C-star-algebra theory.

Definition

Definition

For A,BA,B \in C*Alg two C-star algebras, an (A,B)(A,B)-Hilbert C *C^\ast-bimodule (or just Hilbert bimodule, for short) is

  • a right BB-Hilbert C*-module (N,,)(N, \langle -,-\rangle);

  • equipped with a further left AA-representation A(N)A \to \mathcal{B}(N) by adjointable operators, hence such that a,=,a *\langle a -,- \rangle = \langle -,a^\ast -\rangle for all aAa \in A.

A isomorphism between two (A,B)(A,B)-bimodules (N 1,,)(N 2,, 2)(N_1, \langle -,-\rangle) \to (N_2, \langle -,-\rangle_2) is a linear operator N 1N 2N_1\to N_2 which is unitary with respect to , 2\langle -,-\rangle_2.

Definition

Given an (A,B)(A,B)-Hilbert bimodule (N 1,, 1)(N_1, \langle -,-\rangle_1) and a (B,C)(B,C)-Hilbert bimodule (N 2,, 2)(N_2, \langle -,-\rangle_2), the tensor product of Hilbert bimodules N 1 BN 2N_1 \otimes_B N_2 is the (A,C)(A,C)-Hilbert bimodule obtained from the ordinary (algebraic) tensor product of modules over \mathbb{C} N 1 N 2N_1 \otimes_{\mathbb{C}} N_2 by

  1. equipping it with the CC-valued inner product defined by

    ξ 1η 1,ξ 2η 2η 1,ξ 1,ξ 2η 2. \left\langle \xi_1 \otimes \eta_1 , \xi_2 \otimes \eta_2\right\rangle \coloneqq \left\langle \eta_1, \left\langle \xi_1, \xi_2\right\rangle \cdot \eta_2 \right\rangle \,.
  2. forming the quotient by the submodule of elements vv for which v,v=0\langle v,v\rangle = 0;

  3. forming the completion of this quotient with respect to the induced norm.

Remark

Def. really does yield a kind of tensor product over BB: elements of the form

vbwvbw v \cdot b \otimes w - v \otimes b \cdot w

are in the submodule that it being divided out, because

vbwvbw,vbwvbw w,vb,vbw +bw,v,vbw w,vb,vbw bw,v,vbw =(1+111)w,vb,vbw =0, \begin{aligned} \langle v \cdot b \otimes w - v \otimes b \cdot w \;,\; v \cdot b \otimes w - v \otimes b \cdot w \rangle & \coloneqq \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & + \langle b \cdot w, \langle v,v\rangle \cdot b \cdot w\rangle \\ & - \langle w, \langle v\cdot b, v \rangle \cdot b \cdot w\rangle \\ & - \langle b \cdot w, \langle v, v \cdot b \rangle \cdot w\rangle \\ & = (1+1-1-1) \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & = 0 \end{aligned} \,,

where we use that by definition the left actions are required to have adjoints, so that for instance

bw,v,vbw =w,b *v,vbt =w,vb,vbw \begin{aligned} \langle b \cdot w , \langle v,v\rangle \cdot b \cdot w\rangle & = \langle w , b^\ast \cdot \langle v,v\rangle \cdot b \cdot t\rangle \\ & = \langle w , \langle v \cdot b,v \cdot b\rangle \cdot w\rangle \end{aligned}
Proposition

There is a (2,1)-category C *Alg bC^\ast Alg_b whose

(Buss-Zhu-Meyer 09)

Examples

An (,B)(\mathbb{C},B)-Hilbert C *C^\ast-bimodule is equialently just an BB-Hilbert C*-module.

An (A,)(A, \mathbb{C})-Hilbert C *C^\ast-bimodule is equivalently just as representation of a C-star-algebra.

Properties

(…)

References

Reviews are for instance in

The tensor product of Hilbert bimodules and the induced 2-category structure is discussed in

Last revised on March 4, 2014 at 02:21:16. See the history of this page for a list of all contributions to it.