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The notion of a localizing algebra was introduced by Lomonosov, Radjavi, and Troitsky as a side condition to build invariant subspaces for operators on Banach spaces. The goal of this paper is to show that the unital algebra 
generated by a single diagonal operator on a separable Hilbert space is localizing if and only if there is a non-zero compact operator in the weak closure of the unit ball of the algebra 
.
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