Cohomology of Banach algebras of operators and geometry of Banach spaces.

In the paper we give an exposition of the major results concerning the relation between first order cohomology of Banach algebras of operators on a Banach space with coefficients in specified modules and the geometry of the underlying Banach space. In particular we shall compare the properties weak amenability and amenability for Banach algebras A(X), the approximable operators on a Banach space X. Whereas amenability is a local property of the Banach space X, weak amenability is often the consequence of properties of large scale geometry.

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson’s space.

On the weak amenability of β(X)

We investigate the weak amenability of the Banach algebra ß(X) of all bounded linear operators on a Banach space X. Sufficient conditions are given for weak amenability of this and other Banach operator algebras with bounded one-sided approximate identities.

Common Breast Cancer Susceptibility Alleles and the Risk of Breast Cancer for BRCA1 and BRCA2 Mutation Carriers: Implications for Risk Prediction

The known breast cancer susceptibility polymorphisms in FGFR2, TNRC9/TOX3, MAP3K1, LSP1, and 2q35 confer increased risks of breast cancer for BRCA1 or BRCA2 mutation carriers. We evaluated the associations of 3 additional single nucleotide polymorphisms (SNPs), rs4973768 in SLC4A7/NEK10, rs6504950 in STXBP4/COX11, and rs10941679 at 5p12, and reanalyzed the previous associations using additional carriers in a sample of 12,525 BRCA1 and 7,409 BRCA2 carriers. Additionally, we investigated potential interactions between SNPs and assessed the implications for risk prediction. The minor alleles of rs4973768 and rs10941679 were associated with increased breast cancer risk for BRCA2 carriers (per-allele HR - 1.10, 95% CI: 1.03-1.18, P - 0.006 and HR - 1.09, 95% CI: 1.01-1.19, P = 0.03, respectively). Neither SNP was associated with breast cancer risk for BRCA1 carriers, and rs6504950 was not associated with breast cancer for either BRCA1 or BRCA2 carriers. Of the 9 polymorphisms investigated, 7 were associated with breast cancer for BRCA2 carriers (FGFR2, TOX3, MAP3K1, LSP1, 2q35, SLC4A7, 5p12, P 7 = 10 x (11) - 0.03), but only TOX3 and 2q35 were associated with the risk for BRCA1 carriers (P = 0.0049, 0.03, respectively). All risk-associated polymorphisms appear to interact multiplicatively on breast cancer risk for mutation carriers. Based on the joint genotype distribution of the 7 risk-associated SNPs in BRCA2 mutation carriers, the 5% of BRCA2 carriers at highest risk (i.e., between 95th and 100th percentiles) were predicted to have a probability between 80% and 96% of developing breast cancer by age 80, compared with 42%

On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.