Eta cocycles, relative pairings and the Godbillon-Vey index theorem

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the present context takes the form $0\to J\to A\to B\to 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(\tau_{GV}^r,\sigma_{GV})$ for the pair $A\to B$; $\tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, \`a la Melrose, of the usual Godbillon-Vey cyclic cocycle $\tau_{GV}$; $\sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $\tau_{GV}$ and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call $\sigma_{GV}$ the eta cocycle associated to $\tau_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class \Ind (D,D^\partial)\in K_* (A,B) and establishing the equality =<\Ind (D,D^\partial), [\tau^r_{GV}, \sigma_{GV}]>$. The Godbillon-Vey eta invariant$\eta_{GV}$is obtained through the eta cocycle$\sigma_{GV}$.Comment: 86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173

A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings

Let $\Gamma$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $\Gamma$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (\Gamma;\mathbb{C})$ which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way

Relative pairings and the Atiyah-Patodi-Singer index formula for the Godbillon-Vey cocycle

We describe a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle (X, F) with boundary; in particular, we define a Godbillon-Vey eta invariant on (partial derivative X, F(partial derivative)), that is; a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the classic Atiyah-Patodi-Singer index formula for (X, F). Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairing of K-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form 0 -> J -> U -> B -> 0 with J dense and holomorphically closed in C*(X, F) and B depending only on boundary data

Carboxylic Acids with Certain Molecular Structures Decrease Osmotic Fragility against Osmotic Pressure in Cattle Erythrocytes In Vitro : Appearance of a Wedge-like Effect Similar to RBCs in Other Animal Species

Osmotic fragility (OF) in red blood cells (RBCs) is a useful tool for evaluating the actions of various chemicals on the cell membrane in vitro. The effects of monocarboxylic and dicarboxylic acids on OF were evaluated in cattle RBCs. Isolated cattle RBCs were immersed in various carboxylic acids at 0-100 mM in a buffer solution for 1 hr and the 50% hemolysis was then determined by soaking in 0.1-0.8% NaCl solution. Although n-caprylic acid at 100 mM induced hemolysis, the other monocarboxylic acids possessing straight hydrocarbons did not affect OF. The dicarboxylic acids possessing straight hydrocarbons, except for glutaric acid, decreased OF in a dose-dependent manner. Some monocarboxylic acids with branched hydrocarbons tended to decrease OF, but these changes were not statistically significant. Although cyclopentanecarboxylic and cyclohexanecarboxylic acids at 100 mM decreased OF, other monocarboxylic acids with cyclic hydrocarbons did not affect OF. Among the dicarboxylic acids with cyclic hydrocarbons tested, only 1,2-cyclohexanedicarboxylic acid and phthalic acid with a benzene ring significantly decreased OF. There is no clear correlation between the effect of monocarboxylic or dicarboxylic acids on OF, and their partition coefficients. Thus, the partition coefficient is not a suitable parameter for explaining the effect of both groups of carboxylic acids on OF in cattle RBCs. With regard to the effect of monocarboxylic acids on OF, although an increase in OF was demonstrated in rat RBCs, no effect or rather a decrease in OF was demonstrated in cattle RBCs, similar to the results observed for guinea pig and sheep RBCs. With regard to the effect of dicarboxylic acids, decreases in OF were already demonstrated in rat, guinea pig and sheep RBCs. We have proposed that dicarboxylic acids exhibit a common stabilizing effect on the RBC membrane in various animals, which we termed a “wedge-like effect”. We clarified that cattle RBCs also show a similar OF response to dicarboxylic acids in this experiment

Operator algebras and geometry

In the early 1980's topologists and geometers for the first time came across unfamiliar words like C^*-algebras and von Neumann algebras through the discovery of new knot invariants (by V. F. R. Jones) or through a remarkable result on the relationship between characteristic classes of foliations and the types of certain von Neumann algebras. During the following two decades, a great deal of progress was achieved in studying the interaction between geometry and analysis, in particular in noncommutative geometry and mathematical physics. The present book provides an overview of operator algebra theory and an introduction to basic tools used in noncommutative geometry. The book concludes with applications of operator algebras to Atiyah-Singer type index theorems. The purpose of the book is to convey an outline and general idea of operator algebra theory, to some extent focusing on examples. The book is aimed at researchers and graduate students working in differential topology, differential geometry, and global analysis who are interested in learning about operator algebras