Abstract
In the p-adic local Langlands correspondence for GL2 (ℚp), the following theorem of Berger and Breuil has played an important role: the locally algebraic representations of GL2(ℚp) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither (ϕ, Γ)-module techniques nor global methods. When F is a finite extension of ℚp, we also get a simpler proof of a theorem of Vigneras for the existence of integral structures for (locally algebraic) special series and for (smooth) tamely ramified principal series.
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