Hiroki Sato’s paper on the equivalence of property A and operator norm localization was recently published in *Crelle* ( “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” *Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)* 2014 (690): 207–16. doi:10.1515/crelle-2012-0065.) and I wanted to write up my understanding of this result. It completes a circle of proofs that various forms of “coarse amenability” are equivalent to one another, thus underlining the significance and naturalness of the “property A” idea that Guoliang came up with twenty years ago.

Let’s review the basic definitions. We work with a bounded geometry, uniformly discrete metric space \(X\).

**Definition** \(X\) *has property A* if there exists a sequence \( \{k_n\} \) of unital positive definite kernels on \(X\times X\) which have controlled support and such that \(k_n(x,y)\to 1\) (as \(n\to\infty\) ) uniformly on controlled sets.

This is not the original definition of property A but the equivalence was proved early on and is now a standard result.

Now consider a bounded operator \(T\) on a Hilbert space \(H\). For a constant \(c\in (0,1)\), a *\(c\)-witness to \( \|T\| \)* is a unit vector \(\xi\in H\) such that \( \|T\xi \| > c\|T\| \). Of course, such “witnesses” exist for every \(c\), that is just the definition of the norm. But things change if we demand that witnesses have specific geometric properties.

**Definition** \(X\) has the *operator norm localization property* if, for every \(c\in (0,1)\) and every \(R>0\), there is \(S>0\) such that every operator on \(\ell^2(X)\) of propagation \(\le R\) has a \(c\)-witness to its norm that has support diameter \(\le S\).

Once again, this is not the original definition, which just required this property for *one* \(c\) and which also allowed more general \(X\)-modules in place of the basic one \(\ell^2(X)\). Part of Sako’s paper is the (fairly standard) proof that the above version of the definition is equivalent to the original one.

**Sako’s Theorem** \(X\) has property A if and only if it has operator norm localization.

By chasing through the previously existing literature one could obtain a proof that A implies ONL (though Sako also gives a direct proof of that); so I’ll concentrate on the reverse implication, ONL implies A. First we need another reformulation of ONL.

Fix \(S>0\). For \(x\in X\) let \( M_S(x)\) denote the matrix algebra of all operators on the (finite-dimensional) Hilbert space \(\ell^2(B(x;S))\). Let \(M_S(X)\) denote the C*-algebra direct product \(\prod_{x\in X} M_S(x) \), and let \(\psi_S\) denote the obvious ucp cut-down map from \({\mathfrak B}(\ell^2(X)) \) to \(M_S(X)\). Let \(E_R \subseteq {\mathfrak B}(\ell^2(X)) \) denote the collection of all operators having propagation \(\le R\); it is an operator system. Notice that if \(R<S\) then the map

\[ \psi_S \colon E_R \to M_S(X) \]

is *injective*. By \(\phi_{R,S} \) we will denote the * inverse* of this map, considered as mapping \(\psi_S(E_R) \to E_R\). The following is then another reformulation of the idea that, given ONL, the norm of a finite propagation operator is “witnessed” by its behavior on vectors with not-too-large supports.

**Lemma** If \(X\) has ONL then the following is true: for each \(\epsilon>0\) and each \(R>0\), there is \(S>0\) such that \( \|\phi_{R,S}\| < 1 +\epsilon\). Moreover, we obtain the same condition if we replace the ordinary norm \( \|\phi_{R,S}\| \) by the completely bounded norm \( \|\phi_{R,S}\|_{cb} \) in this inequality.

**Proof that ONL implies property A:** Given \(\epsilon>0\) and \(R>0\), choose |(S>0\) such that \( \|\phi_{R,S}\| < 1 +\epsilon\), as in the lemma above. This says that the unital completely bounded map \(\phi_{R,S} \) is “almost” completely contractive. Now, a unital map which is honest-to-goodness (completely) contractive is automatically (completely) positive, because of the standard characterization of positivity in unital C*-algebras: \(a\in A\) is positive iff \( \|t1-a\|\le t \) for all sufficiently large \(t\). In Appendix B of Nate and Taka’s book (Brown, Nathanial P., and Narutaka Ozawa. 2008. *C*-Algebras and Finite-Dimensional Approximations*. American Mathematical Society) there is an “approximate” version of this statement: a ucb map like \(\phi_{R,S} \) which is within \(\epsilon\) of being completely contractive, is within \(2\epsilon\) of a ucp map; let \(\Phi_{R,S} \) be such a ucp map. Finally, define a kernel \(k\) on \(X\times X\) by

\[ k(x,y) = \langle \Phi_{R,S}\circ \psi_S(e_{y,z})\delta_z,\delta_y\rangle\]

where \(e_{y,z}\in {\mathfrak B}(\ell^2(X))\) is the matrix unit that maps \(\delta_z\) to \(\delta_y\). The complete positivity of \( \Phi\) and \(\psi\) implies that \(k\) is a positive definite kernel, and it clearly has propagation at most \(S\). If the distance \(d(y,z)<R\) then \(\phi_{R,S}\circ\psi_S(e_{y,z})=e_{y,z}\) and therefore

\[ \| \Phi_{R,S}\circ\psi_S(e_{y,z})-e_{y,z}\| < 2\epsilon.\]

Taking a sequence of such kernels \(k_n\) for \(R=n\), \(\epsilon=1/n\) now provides an approximate unit of completely positive multipliers as required by property A.