Schur multipliers and ideals in the translation algebra

Writing the Ghostbusting paper sent me back to the literature on “ideals in the Roe algebra” and in particular to this paper

Chen, Xiaoman, and Qin Wang. “Ideal Structure of Uniform Roe Algebras of Coarse Spaces.” Journal of Functional Analysis 216, no. 1 (November 1, 2004): 191–211. doi:10.1016/j.jfa.2003.11.015.

which contains (among other things) the following pretty theorem: Let \(X\) be a (bounded geometry discrete) coarse space, and let \(\phi\in\ell^\infty(X\times X)\) be a function with controlled support.  Then the Schur multiplier

\[ S_\phi\colon C^*_u(X) \to C^*_u(X) \]

maps any (closed, two-sided) ideal of \(C^*_u(X)\) into itself.

(Note that the usual “decompose into partial translations” argument shows that such a multiplier must be bounded, but the norm may be large.  If \(\phi\) is a positive kernel on \(X\times X\) then there is uniform control on the norm of the corresponding multiplier, but we don’t assume that here.)

To prove the theorem it seems best to start with a lemma: let \(I\) be an ideal, and let \(R\in I\) be a selfadjoint element.  Suppose that, for some subset \(Z\subseteq X\), we have \( R(x,x)\ge 1 \) for all \(x\in Z\).   Then the orthogonal projection \(P_Z\) (from \(\ell^2(X)\) onto \(\ell^2(Z)\)) belongs to the ideal \(I\).

Proof of the lemma: Choose a finite propagation, selfadjoint operator \(S\) such that \( \|R-S\| < \frac13\). Let \(E\) be the support of \(S\).  There is no loss of generality in assuming that \(Z\) is an \(E\)-separated set, meaning that \( E \cap (Z\times Z) \) is contained in the diagonal; because, for a fixed entourage \(E\), any \(Z\) can be partitioned into finitely many \(E\)-separated subsets.  But now \(P_ZRP_Z\) lies within \(\frac13\) of \(P_ZSP_Z\), which is invertible (with the norm of its inverse being at most\(\frac32\)) on \(Z\); it follows that \(P_ZRP_Z\in I\) is invertible on \(Z\), and therefore that \(P_Z\in I\).

To see how the theorem follows from this, let \(T\in I\), and note that it suffices to consider the case where \(\phi\) is the (characteristic function of the) graph of a partial translation, \(F\). Moreover, a simple approximation argument shows that we may assume that there is some \(\epsilon>0\) such that \(|T(x,y)|>\epsilon \) for all \( (x,y)\in F\).  (In fact, if \(F_\epsilon\) denotes the subset of \(F\) consisting of pairs \( (x,y)\) having this property, then \(S_{F_\epsilon}(T)\to S_F(T) \) as \(\epsilon\to 0\). )  Making this assumption, apply the previous lemma to the set \(Z\) and operator \(R\) defined by

\[ Z = \{ x: (x,y)\in F\}, \qquad R = \epsilon^{-2} T^*T. \]

We find that \(P_Z\in I\) and therefore that \(S_F(T) = P_ZS_F(T) \) belongs to \(I\) as well, QED.

It need not be the case that the truncations \( S_\phi(T),\ T\in I\) are dense in \(I\).  The above reasoning (together with basic facts about property A) shows that this is true if \(X\) has property A.  Our ghostbusting theorem shows that the implication is two-way: \(X\) has property A if and only if, for every ideal \(I \triangleleft C^*_u(X) \), the controlled truncations of elements of \(I\) are dense in \(I\).

Final remark: the conjecture in section 6 of the paper is now known to be false, see
Wang, Qin. “Remarks on Ghost Projections and Ideals in the Roe Algebras of Expander Sequences.” Archiv Der Mathematik 89, no. 5 (November 1, 2007): 459–465. doi:10.1007/s00013-007-2260-x.

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