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For a Lie groupoid G over a smooth manifold M we construct the adjoint action of the etale Lie groupoid G# of germs of local bisections of G on the Lie algebroid g of G. With this action, we form the associated convolution…

We present a $C^*$-algebra which is naturally associated to the $ax+b$-semigroup over $\mathbb N$. It is simple and purely infinite and can be obtained from the algebra considered by Bost and Connes by adding one unitary generator which…

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to…

We start by reviewing the relation between toposes and Grothendieck quantales. We improve results of previous work on this relation by giving both a characterisation of the map from the tensor product of two internal sup-lattices to another…

In the present paper we prove a duality theory for compact groups in the case when the C*-algebra A, the fixed point algebra of the corresponding Hilbert C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies the…

A *Gerstenhaber-Schack (G-S) bialgebra* consists of a graded Hopf algebra $H$ together with multilinear operations $\{\omega^1_3,\omega^2_2,\omega^3_1\}\subset \{Hom^{-1}(H^{\otimes m},H^{\otimes n}): m+n=4\},$ whose sum is the degree $-1$…

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…

The Bost conjecture with C*-algebra coefficients for locally compact Hausdorff groups passes to open subgroups. We also prove that if a locally compact Hausdorff group acts on a tree, then the Bost conjecture with C*-coefficients is true…

In this paper we study some generalization of the notion of a formal group over ring, which may be called a formal group over Hopf algebra (FGoHA). The first example of FGoHA was found under the study of cobordism's ring of some $H$-space…

Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of all Cuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication $\Delta_{\phi}$ and a counit $\epsilon$. Let ${\rm BI}({\cal O}_{*})$ denote the set of…

Let A be a C*-algebra, h a Hilbert space and C the CAR algebra over h. We construct a twisted tensor product of A by C such that the two factors are not necessarily one in the relative commutant of the other. The resulting C*-algebra may be…

We construct dense, unconditional subalgebras of the reduced group $C^*$-algebra of a word-hyperbolic group, which are closed under holomorphic functional calculus and possess many bounded traces. Applications to the cyclic cohomology of…

For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete. In an earlier paper…

A host algebra of a topological group G is a C^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite…

A subgroup of an amenable group is amenable. The $C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear $C^*$-algebra $C^*_r(\ZZ_2*\ZZ_3)$ is a subalgebra of the nuclear Cuntz…

Given a continuous open surjective morphism $\pi :G\to H$ of \'etale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of…

In this paper we construct the notions of double Fell bundle and double C*-category for possible future use as tools to describe noncommutative spaces, in particular in finite dimensions. We identify the algebra of sections of a double Fell…

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways. The first…

The subspace generated by the eigenvectors pertaining to the peripheral spectrum of any stochastic matrix is naturally equipped with a structure of a (finite dimensional abelian) $C^*$-algebra, and the restriction of such a stochastic…

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account of the effects of the fundamental group of the spacetime. Using this notion of…