Even Fredholm module - Imported Knowlpedia Content

Definition

Let $A$ be a trivially graded complex $C^*$ -algebra. An even Fredholm module over $A$ is a Fredholm module $(H,\pi,F)$ together with a $\mathbb Z/2$ -grading operator $\Gamma$ such that

\Gamma=\Gamma^*,\qquad \Gamma^2=I,\qquad \Gamma\pi(a)=\pi(a)\Gamma,\qquad \Gamma F=-F\Gamma

for every $a\in A$ . Thus $\pi$ is even and $F$ is an odd operator. In the unnormalized convention, the compact-defect conditions for $F$ remain those of a Fredholm module. If $A$ is itself graded, the commutation rule is replaced by the graded representation rule.

Block form

Writing $H=H^+\oplus H^-$ for the $\pm1$ eigenspaces of $\Gamma$ , the operators have block form

\pi(a)= \begin{pmatrix} \pi^+(a)&0\\ 0&\pi^-(a) \end{pmatrix}, \qquad F= \begin{pmatrix} 0&F^-\\ F^+&0 \end{pmatrix}.

For a normalized module, $F^-=(F^+)^*$ and the two off-diagonal operators are mutual inverses. Before normalization they are inverses modulo the locally compact defects specified in the Fredholm-module axioms. This is the form used to construct the even index pairing. Connes, Chapter IV, Section 1, Definition 1 and Proposition 2(a).

Structure and consequences

Even modules represent degree-zero classes in analytic K-homology. Direct sum gives addition. Reversing the grading changes the sign of the represented class, subject to the accompanying standard convention for $F$ .

If an unbounded cycle has a grading that commutes with the algebra and anticommutes with its unbounded operator, its bounded transform is even. Thus the parity of an even spectral triple survives passage to bounded K-homology.

Examples and non-examples

Let $M$ be a closed even-dimensional spin manifold. The spinor Hilbert space splits into positive and negative chirality, multiplication by $C(M)$ preserves the split, and the bounded transform of the Dirac operator interchanges the two summands. It therefore defines an even Fredholm module. Connes, Chapter IV, Section 1 and Appendix A, Theorem 15.

An ungraded Fredholm module is not automatically even: a grading $\Gamma$ must exist and satisfy both displayed compatibility rules. In particular, a grading commuting with $F$ fails the oddness axiom.

References