Z/2-graded Hilbert space - Imported Knowlpedia Content

Definition

A $Z/2$ -graded Hilbert space is a Hilbert space $H$ with an orthogonal decomposition

H=H^{0}\oplus H^{1}

into closed subspaces, called the even and odd parts. Equivalently, it is a Hilbert space equipped with a grading operator $\Gamma$ satisfying

\Gamma=\Gamma^*,\qquad \Gamma^2=1,

where $H^0=\ker(\Gamma-1)$ and $H^1=\ker(\Gamma+1)$ . This is the topological, inner-product-compatible version of a $Z/2$ -graded module: closedness and orthogonality ensure that the decomposition is a Hilbert direct sum.

Even and odd operators

A bounded operator $T$ is even when $T\Gamma=\Gamma T$ , equivalently when it preserves $H^0$ and $H^1$ . It is odd when $T\Gamma=-\Gamma T$ , equivalently when it exchanges the two parts. For an unbounded operator, either assertion also requires $\Gamma\operatorname{Dom}(T)\subseteq\operatorname{Dom}(T)$ , and the commutation relation is imposed on that domain. These are the parity conventions encoded by an even or odd operator.

Spectral-triple convention

An even spectral triple $(\mathcal A,H,D,\Gamma)$ requires the represented algebra to act evenly, $[\Gamma,\pi(a)]=0$ , and the Dirac operator to act oddly, $\Gamma D=-D\Gamma$ on $\operatorname{Dom}(D)$ . The latter statement includes invariance of the domain under $\Gamma$ . By contrast, an odd spectral triple normally means that no grading operator is part of the data; it does not mean that $D$ is even.

Examples and conventions

For any Hilbert spaces $K_0$ and $K_1$ , the direct sum $K_0\oplus K_1$ is graded by $\Gamma(\xi_0,\xi_1)=(\xi_0,-\xi_1)$ . Differential forms are graded by even and odd degree, with the parity operator acting by $(-1)^k$ on $k$ -forms. Authors also write $\mathbb Z_2$ , $\mathbb Z/2$ , or “super” Hilbert space; in analytic $K$ -homology these expressions refer to the same two-term orthogonal grading.

References

Nigel Higson and John Roe, Analytic K-Homology, Oxford University Press, 2000. Publisher record. Relevant: Chapter 8 on graded Hilbert spaces and cycles.
José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §3.2 on $Z/2$ -gradings and operator parity.