Nonunital spectral triple - Imported Knowlpedia Content

Definition

In the locally compact convention, a nonunital spectral triple $(\mathcal A,H,D)$ consists of a nonunital involutive algebra $\mathcal A$ whose $C^*$ -completion acts faithfully by a nondegenerate representation on a Hilbert space $H$ , together with a densely defined self-adjoint operator $D$ . For every $a\in\mathcal A$ , one requires $a\operatorname{Dom}(D)\subseteq\operatorname{Dom}(D)$ , a bounded extension of $[D,a]$ , and the localizing operator below must be compact:

a(1+D^2)^{-1/2}\in K(H).

The last condition is local compactness; it replaces the global compact-resolvent axiom of a compact spectral triple.

Why local compactness is the right replacement

When $\mathcal A$ is unital, substituting $a=1$ recovers compactness of $(1+D^2)^{-1/2}$ , hence compact resolvent. In a genuinely nonunital example, multiplication by $a$ localizes the resolvent before compactness is tested. The operator $(1+D^2)^{-1/2}$ itself may therefore be noncompact, as expected for a Dirac operator on a noncompact manifold.

Standard examples and conventions

For a complete noncompact spin manifold, a suitable dense smooth subalgebra of $C_0(M)$ acts on $L^2(M,S)$ , and the Dirac operator gives the model example; local compactness follows from local elliptic compactness. Moyal planes provide genuinely noncommutative examples Gayral et al., Sections 2–4.

References

A. L. Carey, V. Gayral, A. Rennie, and F. A. Sukochev, Index Theory for Locally Compact Noncommutative Geometries, Memoirs of the American Mathematical Society 231, no. 1085, 2014. AMS record. Relevant: the nonunital locally compact framework and local index formula.
V. Gayral, J. M. Gracia-Bondía, B. Iochum, T. Schücker, and J. C. Várilly, “Moyal Planes Are Spectral Triples,” Communications in Mathematical Physics 246 (2004), 569–623. DOI record. Relevant: axioms and Moyal-plane examples.