nLab
C-star-correspondence

-correspondences

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

Index theory

C *C^*-correspondences

Idea

Roughly speaking, a C *C^*-correspondence between two C *C^*-algebras AA and BB is a generalised C *C^*-homomorphism from AA to BB.

Definition

Definition

Let AA and BB be C *C^*-algebras. An A,BA,B-correspondence is a pair (E,φ)(E,\varphi) consisting of a right Hilbert BB-module EE and a non-degerate C *C^*-homomorphism φ:A B(E)\varphi \colon A\to \mathcal{L}_B(E); i.e., the image φ(A)E\varphi(A)E is dense in EE (as Hilbert BB-modules).

We write

A(E,φ)B, A\stackrel{(E,\varphi)}{\to} B,

or just AEBA\stackrel{E}{\to}B, and call this a C *C^*-correspondence.

Properties

Composition of correspondences

Definition

Let A(E,φ)BA\stackrel{(E,\varphi)}{\longrightarrow}B and B(F,ψ)CB\stackrel{(F,\psi)}{\longrightarrow}C be C *C^*-correspondences. Then the internal tensor product E ψFE\otimes_{\psi}F is a Hilbert right CC-module. The composition

A(E,φ)(F,ψ)C A \stackrel{(E,\varphi)\circ(F,\psi)}{\longrightarrow} C

is defined by the the pair (E ψF,φψ)(E\otimes_{\psi}F,\varphi\otimes \psi), where

φψ: A C(E ψF) a φ(a)() ψ() \begin{array}{lccc} \varphi\otimes\psi \colon & A & \to & \mathcal{L}_C(E\otimes_{\psi}F) \\ & a & \mapsto & \varphi(a)(\cdot)\otimes_{\psi}(\cdot) \end{array}

Unitary intertwiners

Definition

Let A(E,φ)BA\stackrel{(E,\varphi)}{\longrightarrow}B and A(F,ψ)BA\stackrel{(F,\psi)}{\longrightarrow}B be C *C^*-correspondences. A unitary intertwiner

u:EF u\colon E\Rightarrow F

is a unitary u B(E,F)u\in \mathcal{L}_B(E,F) such that for all aAa\in A the following diagram commutes

E φ(a) E u u F ψ(a) F \array{E & \stackrel{\varphi(a)}{\longrightarrow} & E\\ ^u\downarrow && \downarrow^u\\ F & \stackrel{\psi(a)}{\longrightarrow} & F }

The 22-category ℭ𝔬𝔯𝔯\mathfrak{Corr}

References

  • Alcides Buss, Chenchang Zhu, Ralf Meyer, A higher category approach to twisted actions on C *C^*-algebras, Proceedings of the Edinburgh Mathematical Society, 56 (2013), pp 387-426, doi:10.1017/S0013091512000259

    https://doi.org/10.1017/S0013091512000259), arXiv:0908.0455

Relation to KK-theory:

  • El-kaioum Moutuou, Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory, Journal of K-theory, 13 (2014) pp83-113, doi:10.1017/is013010018jkt244

    https://doi.org/10.1017/is013010018jkt244), arXiv:1305.2495

Last revised on March 27, 2018 at 12:01:00. See the history of this page for a list of all contributions to it.