Unitary equivalence of spectral triples - Imported Knowlpedia Content

Definition

Two spectral triples $(\mathcal A_1,H_1,D_1)$ and $(\mathcal A_2,H_2,D_2)$ , with representations $\pi_1,\pi_2$ , are unitarily equivalent if there are a star-isomorphism $\alpha:\mathcal A_1\to\mathcal A_2$ and a unitary, hence a surjective linear isometry, $U:H_1\to H_2$ such that

U\pi_1(a)U^*=\pi_2(\alpha(a)),\qquad UD_1U^*=D_2

for every $a\in\mathcal A_1$ . The second equality includes $U\operatorname{Dom}(D_1)=\operatorname{Dom}(D_2)$ . If the algebras are already identified, one usually takes $\alpha$ to be the identity.

Preserved structure

Unitary conjugation preserves self-adjointness, spectrum with multiplicity, compactness of the resolvent, and norms of commutators:

[D_2,\pi_2(\alpha(a))] =U[D_1,\pi_1(a)]U^*.

It therefore preserves summability and the Connes metric after states are transported by $\alpha$ . This is the basic sameness relation for changing a Hilbert-space realization without changing the represented metric data van Suijlekom, §2.2.1, Definition 2.24.

Additional structures

For even triples, equivalence also requires $U\Gamma_1U^*=\Gamma_2$ . For real triples it requires $UJ_1U^*=J_2$ , with the algebra isomorphism understood in the represented left and right actions. Omitting these equations identifies the underlying ungraded complex triples but need not identify their parity or real structures.

Examples and distinctions

Changing an orthonormal basis of $H$ conjugates all represented data and gives a unitarily equivalent triple. Likewise, a spin-preserving isometry of closed spin manifolds induces a unitary equivalence of their canonical spectral triples.

Equality of spectra of $D_1$ and $D_2$ alone is not enough: an isospectral unitary need not intertwine the algebra representations. Morita equivalence is also weaker and more flexible; it may change the Hilbert-space module rather than identify it by one unitary.

References

Walter D. van Suijlekom, Noncommutative Geometry and Particle Physics, 2nd ed., Springer, 2025. Publisher chapter. Relevant: §2.2.1, Definition 2.24 on unitary equivalence of finite spectral triples.
Alain Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted book. Relevant: Part VI on spectral triples, their represented data, and unitary changes of realization.