On the quasidiagonality of Roe algebras
It is shown as the main result of this paper that the following conditions are equivalent: {\parindent4.5mm \begin{itemize}\item[1.] The uniform Roe algebra \(B^*(X)\) of all locally compact operators on \(l^2(X)\) with finite propagation, associated to a noncompact discrete metric space \(X\) with bounded geometry is a quasidiagonal \(C^*\)-algebra (or set); \item[2.] \(B^*(X)\) is stably finite; \item[3.] All Fredholm operators in \(B^*(X) \otimes M_n(\mathbb C)\) \((n\geq 1)\) have Fredholm index zero; \item[4.] The space \(X\) does not have a coarse connected subspace containing infinitely many points; \item[5.] There is a sequence of mutually disjoint finite sets \(X_n\) of \(X\) whose union is \(X\) and with the distance between \(X_n\) and \(\cup_{i=1}^{n-1} X_i\) going to infinity as \(n \rightarrow \infty\). \end{itemize}} It is also shown that the similar conditions as above for the coarse Roe algebra \(C^*(X)\) of all locally compact operators on \(l^2(X) \otimes H\) with finite propagation and with \(H\) an infinite dimensional, separable Hilbert space are equivalent. Moreover, under an assumption on a discrete metric space called property A, which is invariant under coarse isometries, the quotient \(C^*\)-algebra of \(B^*(X)\) by the \(C^*\)-algebra of all compact operators on \(l^2(X)\) is quasidiagonal, but is not if \(X\) is an expander arising from a residually finite group with property T.
scientific article; zbMATH DE number 5908402