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Inequivalence and equivalence of certain kinds of non-normal operators Tam, Ping Kwan

Abstract

This thesis is concerned with the problem of unitary equivalence of certain kinds of non-normal operators. Suppose [m, K, G, g ↦ U [subscript g]] is an ergodic and free C-system, with G abelian. Let m = m [symbol omitted] 1, n = R(U[subscript g] [symbol omitted] V[subscript g] : g є G), and let a = R(m, n) = R[m, K, G, g ↦ U[subscript g]] be the von Neumann algebra constructed from [m, K, G, g ↦U[subscript g]] according to von Neumann. We compute: (1) the group A(α; m, n) of all automorphisms of α which keep m pointwise fixed and keep n invariant, and (2) the group A(m, α; n) (resp. G(m, α; n)) of all automorphisms of m which extend to automorphisms (resp. inner automorphisms) of α keeping n pointwise fixed. These calculations lead us to compute G' [symbol omitted] [G] and G' (where [symbol omitted] is the full group generated by G). We show that for an abelian and ergodic G on an abelian m G' [symbol omitted] [G] = G . In the course of this investigation we obtain several interesting results. For example we see that such [symbol omitted] G is automatically free on m. For a large class of tensor algebras (and in particular for a large class of multiplication algebras) we succeed in determining G'. (For the particular cases of multiplication algebras we only use measure-theoretical arguments.) These results are applied to solve the problem of unitary equivalence of certain kinds of non-normal operators. Finally for most of the interesting thick subalgebras E in the literature, we construct numerous unitarily non-equivalent operators A ,with R(Re A) = E.

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