Algebras generated by a weighted shift

Techniques of Olin and Thomson and of Chevreau, Pearcy and Shields are used to prove the following: If T is an injective weighted shift on H with r(T)=?T? then for each L ? (a(T), weak-*)*, there exist f, g ? H so that L(A)=?Af, g? for all A ?a(T). Thus the map i(A)=A is a homeomorphism from (a(T), weak-*) onto (a(T), WOT).