by Lo Chui-sim. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 91-93). / Abstract --- p.i / Introcution --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Preliminaries on von-Neumann algebra --- p.3 / Chapter 1.2 --- Modular theory --- p.6 / Chapter 2 --- Abstract Lp Spaces --- p.10 / Chapter 2.1 --- "Preliminaries on dual action, dual weights and extended positive part" --- p.10 / Chapter 2.2 --- Abstract LP spaces associated with von-Neumann algebras --- p.20 / Chapter 2.3 --- "LP(M) is a Banach space for p E [1, ∞ ]" --- p.25 / Chapter 2.4 --- Independence of the choice of ψ --- p.32 / Chapter 3 --- Spatial Lp Spaces --- p.34 / Chapter 3.1 --- Definition and elementary properties of spatial derivative --- p.35 / Chapter 3.2 --- Modular properties of spatial derivatives --- p.47 / Chapter 3.3 --- Spatial Lp spaces --- p.51 / Chapter 4 --- LP Spaces constructed by using complex interpolation method --- p.60 / Chapter 4.1 --- The complex interpolation space --- p.60 / Chapter 4.2 --- LP space with respect to a faithful normal state --- p.71 / Chapter 4.3 --- LP spaces with respect to a normal faithful semifinite weight . . --- p.78 / Chapter 4.4 --- Equivalence to spatial LP spaces --- p.87 / Bibliography --- p.91

The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and in particular, type III factors. We replace traces with weights, and for a von Neumann algebra $\mathcal{M}$ which admits a weight $\phi$, we show the existence of an automorphic action $\sigma^\phi:\mathbb{R}\rightarrow\text{Aut}(\mathcal{M})$. After showing the existence of these actions we can discuss the crossed product construction, which will then allow us to study the structure of the algebra.

The purpose of this thesis is to provide an exposition of the \textit{modular theory} of von Neumann algebras. The motivation of the theory is to classify and describe von Neumann algebras which do not admit a trace, and in particular, type III factors. We replace traces with weights, and for a von Neumann algebra $\mathcal{M}$ which admits a weight $\phi$, we show the existence of an automorphic action $\sigma^\phi:\mathbb{R}\rightarrow\text{Aut}(\mathcal{M})$. After showing the existence of these actions we can discuss the crossed product construction, which will then allow us to study the structure of the algebra.

Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering. / Typescript bound. Includes bibliographical references: (leaves 174-180) and index. Also available online.

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))

Let A be a von Neumann algebra of linear operators on the Hilbert space H . A linear operator T (resp. a linear bounded. functional ϕ ) on A is said to be normal if for any increasing net [formula omitted] of positive elements in A with least upper bound B , T(B) is the least upper bound of [formula omitted]. Two linear positive functionals ψ1 and ψ2 on A are said to be equivalent if ψ1 (B) = 0 <=> ψ2 (B) = 0 for any positive element B in A. Let ϕ0 be a positive normal linear functional on A . Let S be a semigroup and, {T(s) : s ε S} an antirepresentation of S as normal positive linear contraction operators on A . We find in this thesis equivalent conditions for the existence of a positive normal linear functional ϕ on A which is equivalent to ϕ0 and invariant under the semigroup {T(s) : s ε S} (i.e. ϕ(T(s)B) = ϕ(B) for all B in A and s ε S ). We also extend the concept of weakly-wandering sets, which was first introduced by Hajian-Kakutani, to weakly-wandering projections in A. We give a relation between the non-existence of weakly-wandering projections in A and the existence of positive normal linear functionals on A, invariant with respect to an antirepresentation {T(s) : s ε S} of normal *-homomorphisms on A . Finally we investigate the existence of a complete set of positive normal linear functionals on A which are invariant under the semigroup {T(s) : s ε S}. / Science, Faculty of / Mathematics, Department of / Graduate

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.

The measure-multiplicity-invariant for masas in II_1 factors was introduced by Dykema, Smith and Sinclair to distinguish masas that have the same Pukanszky invariant. In this dissertation, the measure class (left-right-measure) in the measuremultiplicity- invariant is studied, which equivalent to studying the structure of the standard Hilbert space as an associated bimodule. The focal point of this analysis is: To what extent the associated bimodule remembers properties of the masa. The structure of normaliser of any masa is characterized depending on this measure class, by using Baire category methods (Selection principle of Jankov and von Neumann). Measure theoretic proofs of Chifan's normaliser formula and the equivalence of weak asymptotic homomorphism property (WAHP) and singularity is presented. Stronger notions of singularity is also investigated. Analytical conditions based on Fourier coefficients of certain measures are discussed, that partially characterize strongly mixing masas and masas with nontrivial centralizing sequences. The analysis also provide conditions in terms of operators and L2 vectors that characterize masas whose left-right-measure belongs to the class of product measure. An example of a simple masa in the hyperfinite II1 factor whose left-right-measure is the class of product measure is exhibited. An example of a masa in the hyperfinite II1 factor whose leftright- measure is singular to the product measure is also presented. Unitary conjugacy of masas is studied by providing examples of non unitary conjugate masas. Finally, it is shown that for k greater than/equal to 2 and for each subset S \subseteq N, there exist uncountably many non conjugate singular masas in L(Fk) whose Pukanszky invariant is S u {1}.

For an inclusion N \subseteq M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N \subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B \subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N \subseteq M is a regular inclusion of II_1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II_1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.

The algebra of matrices M with entries in an abelian von Neumann algebra is a C*-module. C*-modules were originally defined and studied by Kaplansky and we outline the foundations of the theory and particular properties of M. Furthermore, we prove a structure theorem for ultraweakly closed submodules of M, using techniques from the theory of type I finite von Neumann algebras. By analogy with the classical join in topology, the join for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith. Assuming that K is finite dimensional, Gilfeather and Smith calculated the Hochschild cohomology groups of the join. We assume that M is the algebra of matrices with entries in a maximal abelian von Neumann algebra U, A is an operator algebra acting on a Hilbert space K, and B is an ultraweakly closed subalgebra of M containing U. In this new context, we redefine the join, generalize the calculations of Gilfeather and Smith, and calculate the cohomology groups of the join.