Functional Analysis

We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its R -transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.

Covariant stochastic partial differential equations are studied in any dimension. A special class of such equations is selected and it is proven that the solutions can be analytically continued to Minkowski space-time yielding tempered Wightman distributions which are covariant, obey the locality axiom and a weak form of the spectral axiom. Key words: stochastic partial differential equations, white noise, covariant Markov generalized random fields, Euclidean QFT, Schwinger functions, Wightman distributions

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.

Mansfield showed how to induce representations of crossed products of C*-algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable way of inducing representations of crossed products of C*-algebras by dual coactions. The construction works for homogeneous spaces as well as quotient groups, and we prove an imprimitivity theorem for these induced representations.

For $\MvN$ a separable, purely infinite von Neumann algebra with almost periodic weight ϕ , a decomposition of $\MvN$ as a crossed product of a semifinite von Neumann algebra by a trace--scaling action of a countable abelian group is given. Then Takasaki's continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki's notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one--parameter trace--scaling action of $\Real_+$ on the injective II ∞ factor. Finally, another model of the same action, due to work of Aubert and explained by Jones, is described.

For partial automorphisms of C ∗ -algebras, Takai-Takesaki crossed product duality tends to fail, in proportion to the extent to which the partial automorphism is not an automorphism.

Suppose that G has a representation group H , that G ab :=G/ [G,G] ¯ is compactly generated, and that A is a \cs-algebra for which the complete regularization of $\Prim(A)$ is a locally compact Hausdorff space X . In a previous article, we showed that there is a bijection α↦( Z α , f α ) between the collection of exterior equivalence classes of locally inner actions $\alpha:G\to\Aut(A)$, and the collection of principal $\hgab$-bundles Z α together with continuous functions $f_\alpha:X\to H^2(G,\T)$. In this paper, we compute the crossed products A ⋊ α G in terms of the data Z α , f α , and~$\cs(H)$.

We characterize when the primitive ideal space of a crossed product $\acg$ of a \cs-algebra A by a locally compact abelian group G is a σ -trivial $\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that $\Prim(\acg)$ is σ -trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup $\ttg(\alpha^\w)$ of the Connes spectrum of the system $(A_\w,G,\alpha^\w)$ is continuous, and there is a generalized Green twisting map for (A,G,α) . Our proof requires a substantial generalization of a theorem of Olesen and Pedersen in which we show that there is a generalized Green twisting map for (A,G,α) if and only if $\acg$ is isomorphic to a generalized induced algebra.

Given an arbitrary infinite 0--1 matrix A having no identically zero rows, we define an algebra OA as the universal C*-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize OA as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. OA is always nuclear and we obtain conditions for it to be unital and purely infinite.

We give a general method for computing the cyclic cohomology of crossed products by etale groupoids, extending the Feigin-Tsygan-Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea,Connes and Nistor for the convolution algebra -of compactly supported smooth functions- of an etale groupoid, removing the Hausdorffness assumption and including the computation of the hyperbolic components. Examples like group actions on manifolds and foliations are considered.