Normalizers of Operator Algebras and Reflexivity

The set of normalizers between von Neumann (or, more generally, reflexive) algebras A and B, (that is, the set of all operators x such that xAx* is a subset of B and x*Bx is a subset of A) possesses `local linear structure': it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces U for which UU*U is a subset of U. Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form U={x:xp=h(p)x, for all p in P}, where P is a set of projections and h a certain map defined on P. A normalizing space consists of normalizers between appropriate von Neumann algebras A and B. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian selfadjoint algebras consist of operators `supported' on sets of the form [f=g] where f and g are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.Comment: 20 pages; to appear in the Proceedings of the London Mathematical Societ

Operator algebras from the discrete Heisenberg semigroup

We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of H^{\infty}(\bb{T})\otimes\cl B(\cl H)

Ideals of the Fourier algebra, supports and harmonic operators

We examine the common null spaces of families of HerzSchurmultipliers and apply our results to study jointly harmonic operatorsand their relation with jointly harmonic functionals. We show howan annihilation formula obtained in [1] can be used to give a short proofas well as a generalisation of a result of Neufang and Runde concerningharmonic operators with respect to a normalised positive definite function.We compare the two notions of support of an operator that havebeen studied in the literature and show how one can be expressed interms of the other

Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.Comment: 10 pages, no figures To appear in the International Journal of Mathematic

Tensor products of subspace lattices and rank one density

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for \Alg M and any other reflexive operator algebra and give several further corollaries of these results.Comment: 15 page

Variant -and individual dependent nature of persistent Anaplasma phagocytophilum infection

<p>Abstract</p> <p>Background</p> <p><it>Anaplasma phagocytophilum </it>is the causative agent of tick-borne fever in ruminants and human granulocytotropic anaplasmosis (HGA). The bacterium is able to survive for several months in immune-competent sheep by modifying important cellular and humoral defence mechanisms. Little is known about how different strains of <it>A. phagocytophilum </it>propagate in their natural hosts during persistent infection.</p> <p>Methods</p> <p>Two groups of five lambs were infected with each of two <it>16S </it>rRNA gene variants of <it>A. phagocytophilum</it>, i.e. <it>16S </it>variant 1 which is identical to GenBank no <ext-link ext-link-id="M73220" ext-link-type="gen">M73220</ext-link> and <it>16S </it>variant 2 which is identical to GenBank no <ext-link ext-link-id="AF336220" ext-link-type="gen">AF336220</ext-link>, respectively. The lambs were infected intravenously and followed by blood sampling for six months. <it>A. phagocytophilum </it>infection in the peripheral blood was detected by absolute quantitative real-time PCR.</p> <p>Results</p> <p>Both <it>16S </it>rRNA gene variants of <it>A. phagocytophilum </it>established persistent infection for at least six months and showed cyclic bacteraemias, but variant 1 introduced more frequent periods of bacteraemia and higher number of organisms than <it>16S </it>rRNA gene variant 2 in the peripheral blood.</p> <p>Conclusion</p> <p>Organisms were available from blood more or less constantly during the persistent infection and there were individual differences in cyclic activity of <it>A. phagocytophilum </it>in the infected animals. Two <it>16S </it>rRNA gene variants of <it>A. phagocytophilum </it>show differences in cyclic activity during persistent infection in lambs.</p