1 Spectral triples
This chapter records the basic definitions of (odd and even) spectral triples and the first consequences of the axioms. It corresponds to Phase 1 of the project plan.
1.1 Odd spectral triples
Let \(\mathcal{H}\) be a Hilbert space over \(\Bbbk \in \{ \mathbb {R}, \mathbb {C}\} \), let \(A\) be a \(*\)-algebra over \(\Bbbk \), and let \(\pi \colon A \to \mathcal{B}(\mathcal{H})\) be a \(*\)-algebra homomorphism. A possibly unbounded linear operator \(D \colon \operatorname {dom}(D) \to \mathcal{H}\), with \(\operatorname {dom}(D) \subseteq \mathcal{H}\) a linear subspace, together with the data \((A, \mathcal{H}, \pi , D)\), is an odd spectral triple if:
\(D\) is self-adjoint (in particular, \(\operatorname {dom}(D)\) is dense and \(D\) is closed);
for every \(a \in A\), \(\pi (a)\) maps \(\operatorname {dom}(D)\) into \(\operatorname {dom}(D)\);
for every \(a \in A\), the commutator \([D, \pi (a)]\), viewed as an operator on \(\operatorname {dom}(D)\), is bounded on the closed unit ball of \(\operatorname {dom}(D)\).
If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then \(\operatorname {dom}(D)\) is dense in \(\mathcal{H}\).
If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then \(D\) is a closed operator.
If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then for every \(a \in A\) there is a real constant \(C \geq 0\) such that
for every \(\xi \in \operatorname {dom}(D)\) with \(\| \xi \| \leq 1\).
1.2 Even spectral triples
An even spectral triple \((A, \mathcal{H}, \pi , D, \gamma )\) consists of an odd spectral triple \((A, \mathcal{H}, \pi , D)\) together with a bounded operator \(\gamma \in \mathcal{B}(\mathcal{H})\) (the grading operator) such that:
\(\gamma \) is self-adjoint: \(\gamma ^* = \gamma \);
\(\gamma \) is an involution: \(\gamma ^2 = 1\);
\(\gamma \) commutes with \(\pi (a)\) for every \(a \in A\);
\(\gamma \) maps \(\operatorname {dom}(D)\) into \(\operatorname {dom}(D)\);
\(D \gamma \xi = -\gamma D \xi \) for every \(\xi \in \operatorname {dom}(D)\).
1.3 Resolvents and finitely summable spectral triples
This section introduces the resolvent set and resolvent of the (partially defined) Dirac operator \(D\), and uses them to define finitely summable spectral triples: those whose Dirac operator has compact resolvent.
Let \(D \colon \operatorname {dom}(D) \to \mathcal{H}\) be a (partially defined) linear operator, where \(\operatorname {dom}(D) \subseteq \mathcal{H}\) is a linear subspace. For \(z \in \Bbbk \), write \(z - D\) for the partially defined linear map \(\xi \mapsto z\xi - D\xi \) with domain \(\operatorname {dom}(D)\). The resolvent set \(\rho (D) \subseteq \Bbbk \) of \(D\) is the set of \(z \in \Bbbk \) for which \(z - D \colon \operatorname {dom}(D) \to \mathcal{H}\) is bijective.
For \(z \in \rho (D)\), the resolvent \(R(z, D) \colon \mathcal{H}\to \mathcal{H}\) is the (everywhere-defined, linear) inverse of \(z - D\).
For \(z \in \rho (D)\), the range of \(R(z, D)\) equals \(\operatorname {dom}(D)\).
If \(D\) is self-adjoint and \(z \in \Bbbk \), then for every \(\xi \in \operatorname {dom}(D)\),
If \(D\) is self-adjoint and \(\operatorname {Im}(z) \neq 0\), then \(z - D \colon \operatorname {dom}(D) \to \mathcal{H}\) is injective.
If \(D\) is self-adjoint and \(\operatorname {Im}(z) \neq 0\), then \(z - D\) is in fact bijective, i.e. \(z \in \rho (D)\).
This strengthens Lemma 10 from injectivity to bijectivity: the range of \(z - D\) is closed (the bounded-below estimate makes preimage sequences Cauchy, and the graph of \(D\) is closed) and dense (its orthogonal complement is trivial, via the adjoint), hence all of \(\mathcal{H}\). In particular \(i \in \rho (D)\), so the resolvent-set hypothesis of a finitely summable spectral triple holds automatically at \(z = i\).
An odd spectral triple \((A, \mathcal{H}, \pi , D)\) is finitely summable if the resolvent \(R(i, D)\) at the imaginary unit \(i \in \Bbbk \) is a compact operator on \(\mathcal{H}\). By Theorem 11, the hypothesis \(i \in \rho (D)\) holds automatically whenever \(\operatorname {Im}(i) \neq 0\) (e.g. when \(\Bbbk = \mathbb {C}\)), and is not required as part of the definition.