  We study an operator norm localization property and its applications to the
coarse Novikov conjecture in operator K-theory. A metric space X is said to
have operator norm localization property if there exists a positive number c
such that for every r>0, there is R>0 for which, if m is a positive locally
finite Borel measure on X, H is a separable infinite dimensional Hilbert space
and T is a bounded linear operator acting on L^2(X,m) with propagation r, then
there exists an unit vector v satisfying with support of diameter at most R and
such that |Tv| is larger or equal than c|T|. If X has finite asymptotic
dimension, then X has operator norm localization property. In this paper, we
introduce a sufficient geometric condition for the operator norm localization
property. This is used to give many examples of finitely generated groups with
infinite asymptotic dimension and the operator norm localization property. We
also show that any sequence of expanding graphs does not possess the operator
norm localization property.
