The Big Question about Limit Operators I

A recent paper on the ArXiv (see bibliography below) is entitled “An Affirmative Answer to the Big Question about Limit Operators”.   I want to do a series of posts about this paper.  In this one I will try to explain the background, at least in the most elementary (Hilbert space) case.  In this introduction I will begin by re-expressing matters in the language of coarse geometry, following my paper (also in the bibliography below).

The basic setting is a discrete group \(\Gamma\) with a (left-invariant) word metric; in the usual literature about limit operators this group is \(\mathbb Z\) or \({\mathbb Z}^n\), but there is no particular need for this restriction.  Let \(A\) denote the rough algebra of \(\Gamma\), that is the C*-algebra of operators on \(H=\ell^2(\Gamma)\) generated by \(\ell^\infty(\Gamma)\) together with the right translations by elements of \(\Gamma\); equivalently, the closure of the bounded finite propagation operators on \(H\).  If \(T\in A\), then all the translates \(L_\gamma T L_\gamma^*\) also belong to \(A\), and indeed it is easy to see that they form a precompact subset of \(A\) in the strong (or *-strong) operator topology.  The set of *-strong limit points of this subset is called the \emph{operator spectrum} of \(A\) and denoted \(\sigma_{op}(A) \).

Remark. In the limit operator literature the operator spectrum is defined slightly differently, in terms of the limits of convergent subsequences rather than limit points in the strong topology.  One should pause for a moment here, in view of the fact that sequential closures (in weak Banach space topologies) need not be sequentially closed.  However, all is well in this instance: bounded subsets of \({\mathfrak B}(H)\) for separable \(H\) are *-strongly metrizable, and this implies that sequential and topological closures agree.

One can envisage the construction of the operator spectrum in terms of the Stone-Cech compactification as follows: the map \(\gamma\mapsto L_\gamma T L_\gamma^* \) from \(\Gamma\) to a compact space extends to a *-strongly continuous map of \(\partial\Gamma\) (the Stone-Cech boundary) to \(A\), and the range of this map is precisely the operator spectrum of \(T\).  In the paper below I make a couple of observations:

  • Every *-strongly continuous map from a compact Hausdorff space like \(\partial\Gamma\) to \({\mathfrak B}(H)\) is norm bounded (by the uniform boundedness principle) and thus the collection of such maps forms a C*-algebra under the sup norm, which we denote \(C_s(\partial \Gamma;A)\)
  • The process of passing from an operator to its limit operators defines a *-homomorphism (the operator symbol) \(A \to C_s(\partial \Gamma;A)\);
  • The kernel of the operator symbol homomorphism is precisely G. Yu’s ideal of ghost operators.  Thus for an exact group, the kernel is the compacts and we find that an operator in \(A\) is Fredholm if its symbol is invertible.  This is the Fredholm theory for limit operators, now generalized to all exact groups.

The Fredholm theory focuses attention on the question: when is an element of \(C_s(\partial \Gamma;A)\) invertible (in that C*-algebra)?  It is tempting to assume, by analogy with C*-algebras of continuous functions, that such a function will be invertible if and only if it is invertible “pointwise”, which would yield the claim that \(T\in A\) is Fredholm if and only if all its limit operators are invertible.  But not so fast!  Since the relevant topology on \(A\) is the strong topology, not the norm topology, the best that can be said a priori is that an element of \(C_s(\partial \Gamma;A)\) is invertible if it is invertible pointwise and the pointwise norms of the inverses are uniformly bounded.  Whether the italicized condition can be removed in the special case of the operator spectrum is the Big Question to which Lindner and Seidel have now given an affirmative answer.  (Note that the elements of \(C_s(\partial \Gamma;A)\) arising as symbols have some extra equivariance properties not present for general functions; the question is how to exploit them.)

 

References

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.

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