Order-zero condition for a real spectral triple

Definition

Let $(\mathcal A,H,D,J)$ be spectral data with antiunitary $J$ , and define the represented opposite-algebra action by

b^\circ=Jb^*J^{-1}.

The order-zero condition is

[a,b^\circ]=0 \qquad\text{for every }a,b\in\mathcal A.

Equivalently, the left representation of $\mathcal A$ and the right representation of $\mathcal A^{\mathrm{op}}$ have commuting ranges, so together they give a representation of $\mathcal A\otimes\mathcal A^{\mathrm{op}}$ on $H$ . This condition involves only the two algebra actions and $J$ ; it places no restriction on $D$ . The separate first-order condition controls commutators with $D$ .

Bimodule interpretation

Define $a\xi b=ab^\circ\xi$ . The equality

(a\xi)b=a(\xi b)

holds precisely because $a$ commutes with $b^\circ$ . Thus the order-zero axiom makes $H$ an $\mathcal A$ -bimodule in the represented sense. The involution in $b^\circ=Jb^*J^{-1}$ is needed for this right action to be complex-linear and multiplicative Connes, §2.

Example and near miss

For the spin spectral triple of a compact manifold, both $a$ and $b^\circ$ act by multiplication by functions, so the condition holds.

By contrast, if $J$ is chosen so that $J\mathcal AJ^{-1}$ does not lie in the commutant of $\mathcal A$ , then the displayed commutator can be nonzero. The data may still satisfy all spectral-triple axioms, but they do not define the required bimodule action and hence fail order zero.

References

A. Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2, especially the opposite action and order-zero relation.
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, 2008. DOI record. Relevant: Definition 1.124 and equations (1.471)–(1.472).