nLab amplimorphism




Operator algebra

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



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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).


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.