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

Equivariant Fredholm modules for the full quantum flag manifold of SUq(3)

We introduce C∗-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct SLq(3, C)-equivariant Fredholm modules for the full quantum flag manifold Xq = SUq(3)/T of SUq(3), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold Xq satisfies Poincar´e duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to SUq(3)

On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus

Polyhomogeneous symbols, defined by Kohn-Nirenberg and H\"ormander in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of $\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$ which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz class functions. This result holds for arbitrary graded dilations on the vector space $\mathbb{R}^d$. As an application, using the generalisation of A.~Connes' tangent groupoid for a filtered manifold, we show that the Heisenberg calculus of Beals and Greiner on a contact manifold or a codimension 1 foliation coincides with the groupoid calculus of Van Erp and the second author

Equivariant Fredholm modules for the full quantum flag manifold of SUq(3)

We introduce C∗-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct SLq(3, C)-equivariant Fredholm modules for the full quantum flag manifold Xq = SUq(3)/T of SUq(3), based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold Xq satisfies Poincar´e duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to SUq(3)