Odd Fredholm module - Imported Knowlpedia Content

Definition

Let $A$ be a complex $C^*$ -algebra. An odd Fredholm module over $A$ is a Fredholm module $(H,\pi,F)$ regarded without a $\mathbb Z/2$ -grading: $H$ is a complex Hilbert space, $\pi:A\to\mathcal B(H)$ is a representation, and the commutator and local self-adjointness and involutivity defects of $F$ are compact. In the normalized convention this says

F=F^*,\qquad F^2=I,\qquad [F,\pi(a)]\ \text{is compact for all }a\in A.

“Odd” records K-homological degree, not that $F$ is odd relative to an omitted grading.

Projection picture

For a normalized odd module,

P=\frac{I+F}{2}

is an orthogonal projection. Compactness of $[F,\pi(a)]$ is equivalent to compactness of $[P,\pi(a)]$ . Consequently $P\pi(a)P$ is multiplicative modulo compact operators, which is the Toeplitz-type mechanism behind the odd index pairing. Connes, Chapter IV, Section 1, Proposition 2(b).

For an unnormalized module over a unital algebra represented unitally, the image of $(I+F)/2$ in the Calkin algebra is a projection. In the nonunital convention the corresponding assertion is only local relative to the represented algebra. One can avoid this distinction by first passing to a normalized representative.

Structure and consequences

Odd modules represent degree-one classes in analytic K-homology. Their direct sum is addition, and stable homotopy identifies cycles that carry the same class. The terminology differs from graded algebra: there is no operator whose degree is being measured inside the ungraded Hilbert space.

An odd spectral triple has no grading compatible with its algebra and Dirac operator. Its bounded transform is therefore an odd Fredholm module, provided the spectral-triple compactness and commutator hypotheses hold.

Examples and non-examples

On $H=L^2(S^1)$ , let $P$ project onto the Hardy space and put $F=2P-I$ . Multiplication by $C(S^1)$ has compact commutators with $P$ , so this gives the fundamental odd Fredholm module of the circle. Compressing the coordinate unitary produces the unilateral shift and detects a nonzero index.

A normalized Fredholm module equipped with a compatible grading that commutes with the representation and anticommutes with $F$ is an even Fredholm module, not an odd one under the standard parity convention.

References