Applications of Kazhdan’s property (T ) to automatic continuity

Let G be a compact group which has the strong Kazhdan’s property (T ) and let τ be an isometric representation of G on Lp(μ) for some measure space (Ω,Σ,μ) and 1 < p <∞. We characterize the continuity of all the linear operators from Lp(μ) which commute with translations and the property of · p being the unique norm (up to equivalence) on Lp(μ) with respect to which all translations are continuous in terms of the finite-dimensionality of the subspace of all invariant functions.  2004 Elsevier Inc. All rights reserved