Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

Gene-for-gene disease resistance without the hypersensitive response in Arabidopsis dnd1 mutant

The cell death response known as the hypersensitive response (HR) is a central feature of gene-for-gene plant disease resistance. A mutant line of Arabidopsis thaliana was identified in which effective gene-for-gene resistance occurs despite the virtual absence of HR cell death. Plants mutated at the DND1 locus are defective in HR cell death but retain characteristic responses to avirulent Pseudomonas syringae such as induction of pathogenesis-related gene expression and strong restriction of pathogen growth. Mutant dnd1 plants also exhibit enhanced resistance against a broad spectrum of virulent fungal, bacterial, and viral pathogens. The resistance against virulent pathogens in dnd1 plants is quantitatively less strong and is differentiable from the gene-for-gene resistance mediated by resistance genes RPS2 and RPM1. Levels of salicylic acid compounds and mRNAs for pathogenesis-related genes are elevated constitutively in dnd1 plants. This constitutive induction of systemic acquired resistance may substitute for HR cell death in potentiating the stronger gene-for-gene defense response. Although cell death may contribute to defense signal transduction in wild-type plants, the dnd1 mutant demonstrates that strong restriction of pathogen growth can occur in the absence of extensive HR cell death in the gene-for-gene resistance response of Arabidopsis against P. syringae.

Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.