On Coarse Spectral Geometry in Even Dimension

Let $\sigma$ be the involution of the Roe algebra $\Roe{\RR}$ which is induced from the reflection $\RR\to\RR; x\mapsto -x$. A graded Fredholm module over a separable $C^*$-algebra $A$ gives rise to a homomorphism $\tilde{\rho}:A\to\Roe{\RR}^\sigma$ 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 $K_0(\Roe{\RR}^\sigma)\cong\ZZ$, $K_1(\Roe{\RR})=0$, and that the induced map $\tilde{\rho}_*:K_0(A) \to \ZZ$ on $K$-theory gives the index pairing of $K$-homology with $K$-theory.