# Uniform Roe algebra of virtually abelian group is type I C*-algebra?

Let $$G$$ be an arbitrary (discrete) group. It acts by left translation on $$\ell^\infty(G)$$. The uniform Roe algebra of $$G$$ is defined as the crossed product $$\ell^\infty (G) \rtimes_{\mathrm{red}}G$$.

Elmar Thoma has shown (Thoma, E., Eine Charakterisierung diskreter Gruppen vom Typ I, Invent. Math. 6, 190-196 (1968). ZBL0169.03802.) that the reduced group $$C^*$$-algebra $$C_{\mathrm{red}}^*(G)$$ of a group $$G$$ is a type I $$C^*$$-algebra (i.e. there exists some $$n\in \mathbb{N}$$ such that for every irreducible representation $$\pi$$ of $$C_{\mathrm{red}}^*(G)$$ the dimension of the corresponding Hilbert space is less or equal to $$n$$) if and only if $$G$$ is virtually abelian.

I'm wondering if for virtually abelian groups the corresponding uniform Roe algebra must also be of type I. Does a result into that direction exist?

It is not Type I in general. Probably it is not Type I whenever $$G$$ is infinite. Here is an argument when $$G=\mathbb{Z}.$$
Consider the projections in $$\ell^\infty(\mathbb{Z})$$ defined by characteristic functions for the following sets $$\{ 2^n\mathbb{Z}+k: n\geq 1,0\leq k <2^n \}.$$ Let $$A$$ be the C*-algebra generated by these projections. Then $$A=C(X)$$ where $$X=\{ 0,1 \}^\mathbb{N}$$. Moreover the $$\mathbb{Z}$$-action leaves $$A$$ invariant and defines the odometer action on $$\{ 0,1 \}^\mathbb{N}.$$ The odometer action is minimal and therefore $$A\rtimes \mathbb{Z}\subseteq \ell^\infty(\mathbb{Z})\rtimes \mathbb{Z}$$ is a simple, unital, infinite dimensional C*-algebra and therefore not Type I (hence it can't be contained in any Type I C*-algebra).
Side Note: Zeller-Meier (1968) gave a dynamical characterization of those $$\mathbb{Z}$$-actions on compact spaces whose crossed product is Type 1--$$C(X)\rtimes \mathbb{Z}$$ is Type I precisely when every orbit is discrete in the relative topology. Having a minimal subsystem (as above) is then about as far away as you can get from Zeller-Meier's condition
• Every infinite finitely generated group G contains $\mathbb{Z}$ isometrically. Hence, uniform Roe algebra of G contains uniform Roe algebra of $\mathbb{Z}$ as a C*-subalgebra. Moreover, type I passes to C*-subalgebras. – m07kl Oct 13 '19 at 12:23