nLab
amplimorphism

Contents

Context

Algebra

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 concept of amplimorphism is a way to present bimodules in terms of linear maps.

For AA an associative algebra (not necessarily unital) over some ring RR (possibly with extra structure, in applications often a C*-algebra) then an amplimorphism from AA to AA is an algebra homomorphism of the form

α:AA RMat n(R)Mat n(A) \alpha \;\colon\; A \longrightarrow A \otimes_R Mat_n(R) \simeq Mat_n(A)

for nn \in \mathbb{N} and Mat n(R)Mat_n(R) the ring of matrices with coefficients in RR under matrix multiplication.

This map induces the RR-bimodules N αAR nN_\alpha \subset A \otimes R^n on elements N α={ψ|α(1)ψ=ψ}N_\alpha = \{\psi \;|\; \alpha(1)\psi = \psi\} with left AA-action given by α\alpha and right AA action given by componentwise multiplication with AA from the right.

(If R=R = \mathbb{C} and AA is a C*-algebra then this is canonically equipped with the structure of a Hilbert bimodule).

References

At least in the context of AQFT amplimorphisms were introduced

  • K. Szlachányi, K. Vecsernyes, Quantum symmetry and braid group statistics in GG-spin models, Commun. Math. Phys. 156:1 (1993), 127-168, euclid doi

The concept is recalled for instance in

  • Ezio Vasselli, page 6 of The C *C^\ast-algebra of a vector bundle and fields of Cuntz algebras, Journal of Functional Analysis 222(2) (2005), 491-502, arXiv:math/0404166

  • Fernando Lledó, Ezio Vasselli, section 3.3 of Realization of minimal C *C^\ast-dynamical systems in terms of Cuntz-Pimsner algebras (arXiv:math.OA/0702775)

Last revised on January 15, 2014 at 01:05:30. See the history of this page for a list of all contributions to it.