Groupoid normalisers of tensor products: infinite von Neumann algebras

The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$establishing the formula$$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$and normalisers$v_i\in\mathcal{N}_{M_i}(B_i)$). We obtain a positive result when one of the$M_i$is finite or both of the$B_i$are infinite. For the remaining case, we characterise the II$_1$factors$B_1$for which such factorisations always occur (for all$M_1, B_2$and$M_2$) as those with a trivial fundamental group.Comment: 22 page

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

Report on 1:100 000 Scale Geological and Metallogenic Maps Sheet 3163-19 Cosquín, Province of Córdoba

Fil: Stuart-Smith, Peter G. Australian Geological Survey Organisation; Australia.Fil: Skirrow, Roger G. Australian Geological Survey Organisation; Australia.Cubrimiento parcial de la hoja.The Cosquin 1:100.000 Sheet area lies within the Córdoba Province, between 31o00’-31o20’S and 64o00’-64o30’W. The area is part of the Córdoba (3163-III)1:250 000 sheet area. The region includes the central northern part of the Sierra Chica, one of several north-trending mountain ranges which traverse the northern part of the Córdoba Province. The Sierra Chica is drained by the easterly flowing Ríos La Granja, Ascochinga, and Santa Sabina. Access to the region, from Córdoba city, is via El Manzano and Ruta Provincial 9 in the east. An unsealed road traverses the eastern flank of the Sierra Chica from El Manzano to La Cumbre (Jesús María sheet area)

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

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

Normalisers of irreducible subfactors

We consider normalizers of an infinite index irreducible inclusion Nsubset of or equal toM of II1 factors. Unlike the finite index setting, an inclusion uNu*subset of or equal toN can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these one-sided normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N′∩left angle bracketM,eNright-pointing angle bracket is of the form u*eNu for some unitary uset membership, variantM with uNu*subset of or equal toN generalizing the finite index situation considered by Pimsner and Popa. We use this to show that each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of infinite index irreducible subfactors arising from subgroup–group inclusions Hsubset of or equal toG. Here the one-sided normalizers arise from appropriate group elements modulo a unitary from L(H). We are also able to identify the finite trace L(H)-bimodules in ℓ2(G) as double cosets which are also finite unions of left cosets

The Regulatory Repertoire of Pseudomonas aeruginosa AmpC ß-Lactamase Regulator AmpR Includes Virulence Genes

In Enterobacteriaceae, the transcriptional regulator AmpR, a member of the LysR family, regulates the expression of a chromosomal β-lactamase AmpC. The regulatory repertoire of AmpR is broader in Pseudomonas aeruginosa, an opportunistic pathogen responsible for numerous acute and chronic infections including cystic fibrosis. In addition to regulating ampC, P. aeruginosa AmpR regulates the sigma factor AlgT/U and production of some quorum sensing (QS)-regulated virulence factors. In order to better understand the ampR regulon, we compared the transcriptional profile generated using DNA microarrays of the prototypic P. aeruginosa PAO1 strain with its isogenic ampR deletion mutant, PAOΔampR. Transcriptome analysis demonstrates that the AmpR regulon is much more extensive than previously thought, with the deletion of ampR influencing the differential expression of over 500 genes. In addition to regulating resistance to β-lactam antibiotics via AmpC, AmpR also regulates non-β-lactam antibiotic resistance by modulating the MexEF-OprN efflux pump. Other virulence mechanisms including biofilm formation and QS-regulated acute virulence factors are AmpR-regulated. Real-time PCR and phenotypic assays confirmed the microarray data. Further, using a Caenorhabditis elegans model, we demonstrate that a functional AmpR is required for P. aeruginosa pathogenicity. AmpR, a member of the core genome, also regulates genes in the regions of genome plasticity that are acquired by horizontal gene transfer. Further, we show differential regulation of other transcriptional regulators and sigma factors by AmpR, accounting for the extensive AmpR regulon. The data demonstrates that AmpR functions as a global regulator in P. aeruginosa and is a positive regulator of acute virulence while negatively regulating biofilm formation, a chronic infection phenotype. Unraveling this complex regulatory circuit will provide a better understanding of the bacterial response to antibiotics and how the organism coordinately regulates a myriad of virulence factors in response to antibiotic exposure

C*-algebras nearly contained in type I algebras

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