On Coarse Spectral Geometry in Even Dimension
Résumé
Let $\sigma$ be the involution of the Roe algebra $C^*|R|$ which is induced from the reflection $R \to R; x \mapsto -x$. A graded Fredholm module over a separable C∗-algebra A gives rise to a homomorphism $\tilde{\rho}:A \to C^*|R|^\sigma$ to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the K-theory of this symmetric Roe algebra is $K_0(C^*|R|^\sigma)\cong Z$, $K_1(C^*|R|)=0$, and that the induced map $\tilde{\rho}_*:K_0(A) \to Z$ on K-theory gives the index pairing of K-homology with K-theory.