In the first post in this series, I gave some background to the “Big Question” on limit operators which it appears that Lindner and Seidel have solved for the case of free abelian groups. In the next couple of posts I want to sketch some of the key ideas of their proof and to explore to what extent it can also be generalized to all exact groups (in the same way that I generalized the basic theory of limit operators to all exact groups in my 2005 paper).

There are two components to the L-S argument, it seems to me.

- a localization property for the “lower norm” of a finite propagation operators, and
- a “condensation of singularities” argument.

In this post we’ll look at the first of those.

**Definition** Let \(A\) be an operator on a Hilbert space (or a Banach space more generally, but my attention is on the \(\ell^2\) case). The *lower norm* \(\nu(A)\) is defined to be

\[ \inf \{ \|A\xi\| : \|\xi\|=1\} \]

Thus, if \(A\) has a bounded inverse, the lower norm is just \(\nu(A) = \|A^{-1}\|^{-1}\).

For an operator \(A\in C^*_u(\Gamma)\), let \(A_\omega\) denote the limit operator corresponding to \(\omega\in\partial\Gamma\). The Big Question is then: if every \(A_\omega\) is invertible, must the function \(\omega\mapsto \nu(A_\omega) \) be bounded away from zero?

**Definition** Fix \(D>0\). We define \(\nu_D(A)\), for \(A\in C^*_u(\Gamma)\), to be the infimum

\[ \inf \{ \|A\xi\| : \|\xi\|=1,\quad\mbox{diam supp \((\xi)\)} < D\} \]

Clearly \(\nu_D(A)\ge \nu(A)\). The main estimate in the L-S proof is the following *localization principle for the lower norm*:

**Theorem** (Proposition 6 in L-S) Let \(\Gamma\) be free abelian. Given \(r>0\) and \(\delta>0\) there is \(D>0\) such that, for all \(A\in C^*_u(\Gamma)\) having both propagation and norm less than \(r\),

\[ \nu_D(A)\le \nu(A) +\delta. \]

This localization principle is an analog, for the lower norm, of the *operator norm localization principle* for the ordinary norm, which by the work of Sako turns out to be one of the many equivalent formulations of property A. Recall that a (bounded geometry uniformly discrete) space \(X\) has ONL if the following is true: there exists a constant \(c \in (0,1)\) such that, for every \(R>0\), there exists \(S>0\) such that for every operator \(B\) on \(\ell^2(X)\) with propagation \(\le R\), there exists \(\xi\in \ell^2(X)\) with diameter of support at most \(S\), such that

\[ \|B\xi\| \ge c \|B\| \|\xi\|. \]

As mentioned above, it is shown by Sako that ONL is equivalent to property A for uniformly discrete bounded geometry spaces – in particular, it is equivalent to exactness for groups.

Using ONL, I believe that one should be able to prove the following slightly weaker version of the localization principle for the lower norm.

**Proposed Theorem** Let \(\Gamma\) be exact. There is \(c\in (0,1)\) such that, given \(r>0\) and \(\delta>0\) there is \(D>0\) such that, for all \(A\in C^*_u(\Gamma)\) having both propagation and norm less than \(r\),

\[ \nu_D(A)\le c^{-1}(\nu(A) +\delta). \]

It appears that this result will suffice to execute the L-S proof. An outline of the proof of the proposed theorem is the following: first reduce to the case \(A\) positive, and then apply ONL to \( (A + (\delta/2)I)^{-1} \), or more precisely to a suitable polynomial approximation to this function of \(A\) (in the spirit of the *Ghostbusting* paper).

### References

Chen, Xiaoman, Romain Tessera, Xianjin Wang, and Guoliang Yu. “Metric Sparsification and Operator Norm Localization.” *Advances in Mathematics* 218, no. 5 (2008): 1496–1511. doi:10.1016/j.aim.2008.03.016.

Sako, Hiroki. “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” *arXiv:1203.5496* (March 25, 2012). http://arxiv.org/abs/1203.5496.

Lindner, Marko, and Markus Seidel. “An Affirmative Answer to the Big Question on Limit Operators.” *arXiv:1401.1300 [math]* (January 7, 2014). http://arxiv.org/abs/1401.1300.

Rabinovich, Vladimir, Steffen Roch, and Bernd Silbermann. *Limit Operators and Their Applications in Operator Theory*. Birkhäuser, 2004.

Roe, John. “Band-Dominated Fredholm Operators on Discrete Groups.” *Integral Equations and Operator Theory* 51, no. 3 (March 1, 2005): 411–416. doi:10.1007/s00020-004-1326-4.

Roe, John, and Rufus Willett. “Ghostbusting and Property A.” *Journal of Functional Analysis* 266 (2014): 1674–1684. doi:10.1016/j.jfa.2013.07.004.