nLab amplimorphism

Idea

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

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

$$\alpha \;\colon\; A \longrightarrow A \otimes_R Mat_n(R) \simeq Mat_n(A)$$

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

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

(If $R = \mathbb{C}$ and $A$ 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 $G$-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 $C^\ast$-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 $C^\ast$-dynamical systems in terms of Cuntz-Pimsner algebras (arXiv:math.OA/0702775)