On Operator Norm Localization Property
by
Xiaoman Chen
Fudan University
Coauthors: Xianjin Wang

A metric space X is said to have operator norm localization property if there exists c > 0 such that for every r > 0, there is R > 0 for which, if n 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 L²(X, n)⊗H with propagation r, then there exists an unit vector x ∈ L²(X, n)⊗H satisfying the diameter(Supp(x)) ≤ R and ||T|| ≤ c||T x||.

We prove that a finitely generated group G which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁, ..., H_n} has operator norm localization property if and only if each H_i, i=1, 2, ..., n has operator norm localization property. Furthermore we prove the following result. Let p be the fundamental group of a connected finite graph of groups with finitely generated vertex groups G_P. If G_P has operator norm localization property for all vertices P then p has operator norm localization property.

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Date received: February 12, 2008