Normalizers of Irreducible Subfactors

We consider normalizers of an irreducible inclusion $N\subseteq M$ of $\mathrm{II}_1$ factors. In the infinite index setting an inclusion $uNu^*\subseteq N$ can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these normalizers of $N$ in $M$ to projections in the basic construction and show that every trace one projection in the relative commutant $N'\cap$ is of the form $u^*e_Nu$ for some unitary $u\in M$ with $uNu^*\subseteq N$. This enables us to identify the normalizers and the algebras they generate in several situations. In particular each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of irreducible subfactors arising from subgroup--group inclusions $H\subseteq G$. Here the normalizers are the normalizing group elements modulo a unitary from $L(H)$. We are also able to identify the finite trace $L(H)$-bimodules in $\ell^2(G)$ as double cosets which are also finite unions of left cosets.Comment: 33 Page

Kadison-Kastler stable factors

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For n≥3 and a free, ergodic, probability measure-preserving action of SL<sub>n</sub>(Z) on a standard nonatomic probability space (X,μ), write M=(L<sup>∞</sup>(X,μ)⋊SL<sub>n</sub>(Z))⊗¯¯¯R, where R is the hyperfinite II1-factor. We show that whenever M is represented as a von Neumann algebra on some Hilbert space H and N⊆B(H) is sufficiently close to M, then there is a unitary u on H close to the identity operator with uMu∗=N. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture. We also obtain stability results for crossed products L<sup>∞</sup>(X,μ)⋊Γ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L<sup>2</sup>(X,μ). In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when Γ is a free group

A remark on the similarity and perturbation problems

In this note we show that Kadison's similarity problem for C*-algebras is equivalent to a problem in perturbation theory: must close C*-algebras have close commutants?Comment: 6 Pages, minor typos fixed. C. R. Acad. Sci. Canada, to appea

Computer modelling techniques for industrial and marine cyclo-converter drives

Large cyclo-converter drives using either synchronous or induction machines are found in a number of low speed high torque applications: these include steel mills, gearless cement mills, mine winders and ice breakers. The cyclo-converter output frequency is synthesised from portions of the high voltage supply system. At low frequencies this produces an output current waveform which is reasonably sinusoidal, as the frequency increases the waveform become more distorted. There is concern for the level of current harmonics produced by the cyclo-converter in both the motor windings and the supply system. The waveform distortion is mainly due to the switching action of the cyclo-converter, this is modified by the effects of interbridge changeover delay, commutation overlap, the supply and decoupling reactances. Industrial drives are required to conform to the Electricity Council's regulations on input current harmonics and voltage waveform distortion. In marine drives the main generators feed the ship propulsion system and the ship's auxiliary supplies. The level of interference produced by the converter input line current on the supply network is of particular concern. High levels of voltage distortion may cause malfunction of the auxiliary equipment and produce large torque pulsations in the supply alternator. The cyclo-converter generates unwanted harmonics in the output current waveforms, these are responsible for torque pulsations on the motor drive. With naval marine drives the torque pulsation is of particular interest, since this level must be minimised to increase the sensitivity of detecting instruments such as passive sonar. To date, mathematical models and computer based models have not detailed all the aspects which are responsible for the distortion to the converter input and output waveforms. This thesis seeks to address this and identify the effects on the converter performance of, the converter interbridqe delay, overlap and the supply parameters supply and decoupling reactance. Various motor models will also be examined. The author has used the mathematical work by Pelly as a basis, and has identified the percentage errors in the level of harmonic currents which are likely to be introduced if his results are used in designing new systems. Comparisons will be made with experimentally produced results obtained from a mine winder at the Wearmouth Colliery. Results from a lower powered cyclo-converter produced experimentally and computationally at Newcastle University will also be examined

Computer modelling techniques for industrial and marine cyclo-converter drives

Large cyclo-converter drives using either synchronous or induction machines are found in a number of low speed high torque applications: these include steel mills, gearless cement mills, mine winders and ice breakers. The cyclo-converter output frequency is synthesised from portions of the high voltage supply system. At low frequencies this produces an output current waveform which is reasonably sinusoidal, as the frequency increases the waveform become more distorted. There is concern for the level of current harmonics produced by the cyclo-converter in both the motor windings and the supply system. The waveform distortion is mainly due to the switching action of the cyclo-converter, this is modified by the effects of interbridge changeover delay, commutation overlap, the supply and decoupling reactances. Industrial drives are required to conform to the Electricity Council's regulations on input current harmonics and voltage waveform distortion. In marine drives the main generators feed the ship propulsion system and the ship's auxiliary supplies. The level of interference produced by the converter input line current on the supply network is of particular concern. High levels of voltage distortion may cause malfunction of the auxiliary equipment and produce large torque pulsations in the supply alternator. The cyclo-converter generates unwanted harmonics in the output current waveforms, these are responsible for torque pulsations on the motor drive. With naval marine drives the torque pulsation is of particular interest, since this level must be minimised to increase the sensitivity of detecting instruments such as passive sonar. To date, mathematical models and computer based models have not detailed all the aspects which are responsible for the distortion to the converter input and output waveforms. This thesis seeks to address this and identify the effects on the converter performance of, the converter interbridqe delay, overlap and the supply parameters supply and decoupling reactance. Various motor models will also be examined. The author has used the mathematical work by Pelly as a basis, and has identified the percentage errors in the level of harmonic currents which are likely to be introduced if his results are used in designing new systems. Comparisons will be made with experimentally produced results obtained from a mine winder at the Wearmouth Colliery. Results from a lower powered cyclo-converter produced experimentally and computationally at Newcastle University will also be examined

An early Harvard 'Memorandum' on anti-depression policies

[Introductory Note by David Laidler (University of Western Ontario) and Roger Sandilands (University of Strathclyde) ...] The Memorandum which this note introduces was completed by three young members of the Harvard economics department sometime in January 1932 Two of them, Lauchlin Currie and Harry Dexter White were soon to play key roles on the American, indeed the world-wide, policy scene. Both of them would go to Washington in 1934 as founding members of Jacob Viner’s 'Freshman Brains Trust'. In due course, first at the Federal Reserve Board, and later at the Treasury and the White House, Currie would become a highly visible and leading advocate of expansionary fiscal policy, while White, at the Treasury, was to be a co-architect, with Keynes, of the Bretton Woods system. Both would fall victim to anti-communist witch-hunts in the late 1940s, in White’s case perhaps at the cost of his life, since he died of a heart attack in 1948 three days after a strenuous hearing before the House Committee on Unamerican Activities (HUAC). The third author, P. T. Ellsworth, later a Professor of Economics at the University of Wisconsin, is perhaps best remembered nowadays as the author of a leading textbook in International Economics, though it is worth noting that he was also a very early (late 1936) but hitherto unrecognised discoverer of what came to be called the IS-LM model as a means of elucidating issues raised by Keynes' 'General Theory'. It is not known how widely this Memorandum was circulated, but the fact that it is a piece of policy advocacy, combined with its relatively polished style, makes it inconceivable that it was meant for the eyes and files of its authors alone. As readers will see, it sketches out an explanation of the then rapidly developing Great Contraction, as well as a comprehensive and radical policy programme for dealing with it. In keeping with its authors’ explanation of the Contraction as a consequence of a collapsing money supply, the main domestic components of that programme were to be vigorously expansionary open-market operations and substantial deficit spending that, particularly in its early stages, was to be financed by money creation; its international dimension involved a return to free trade and serious efforts to resolve the problems of international indebtedness that had originated in the Great War and in the Treaty of Versailles which had brought it to an uneasy end in 1919. [...

Magnetic Phase Diagram of Spin-1/2 Two-Leg Ladder with Four-Spin Ring Exchange

We study the spin-1/2 two-leg Heisenberg ladder with four-spin ring exchanges under a magnetic field. We introduce an exact duality transformation which is an extension of the spin-chirality duality developed previously and yields a new self-dual surface in the parameter space. We then determine the magnetic phase diagram using the numerical approaches of the density-matrix renormalization-group and exact diagonalization methods. We demonstrate the appearance of a magnetization plateau and the Tomonaga-Luttinger liquid with dominant vector-chirality quasi-long-range order for a wide parameter regime of strong ring exchange. A "nematic" phase, in which magnons form bound pairs and the magnon-pairing correlation functions dominate, is also identified.Comment: 18pages, 7 figure

Groupoid normalizers of tensor products

We consider an inclusion B [subset of or equal to] M of finite von Neumann algebras satisfying B′∩M [subset of or equal to] B. A partial isometry vset membership, variantM is called a groupoid normalizer if vBv*,v*Bv[subset of or equal to] B. Given two such inclusions B<sub>i</sub> [subset of or equal to] M<sub>i</sub>, i=1,2, we find approximations to the groupoid normalizers of [formula] in [formula], from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis [formula], i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries vset membership, variantM satisfying vBv*[subset of or equal to] B and v*v,vv*[set membership, variant] B

Ferromagnetism of 3^3He Films in the Low Field Limit

We provide evidence for a finite temperature ferromagnetic transition in 2-dimensions as $H \to 0$ in thin films of $^3$He on graphite, a model system for the study of two-dimensional magnetism. We perform pulsed and CW NMR experiments at fields of 0.03 - 0.48 mT on $^3$He at areal densities of 20.5 - 24.2 atoms/nm$^2$. At these densities, the second layer of $^3$He has a strongly ferromagnetic tendency. With decreasing temperature, we find a rapid onset of magnetization that becomes independent of the applied field at temperatures in the vicinity of 1 mK. Both the dipolar field and the NMR linewidth grow rapidly as well, which is consistent with a large (order unity) polarization of the $^3$He spins.Comment: 4 figure