Simplicity of algebras associated to étale groupoids

We prove that the full C ^?-algebra of a second-countable, Hausdorff, étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex ^?-algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.     

Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.     

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \(\hat \otimes\)-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ?: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H ⁿ(?): H ⁿ (x) → H ⁿ (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \(\hat \otimes\)-algebras: the tensor algebra E \(\hat \otimes\) F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε^*(G) on a compact Lie group G.     

Derivations,automorphisms, and representations of complex ω-Lie algebras

Let (𝔤,ω) be a finite-dimensional non-Lie complex ω-Lie algebra. We study the derivation algebra Der(𝔤) and the automorphism group Aut(𝔤) of (𝔤,ω). We introduce the notions of ω-derivations and ω-automorphisms of (𝔤,ω) which naturally preserve the bilinear form ω. We show that the set Der_ω(𝔤) of all ω-derivations is a Lie subalgebra of Der(𝔤) and the set Aut_ω(𝔤) of all ω-automorphisms is a subgroup of Aut(𝔤). For any three-dimensional and four-dimensional nontrivial ω-Lie algebra 𝔤, we compute Der(𝔤) and Aut(𝔤) explicitly, and study some Lie group properties of Aut(𝔤). We also study representation theory of ω-Lie algebras. We show that all three-dimensional nontrivial ω-Lie algebras are multiplicative, as well as we provide a four-dimensional example of ω-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple ω-Lie algebra C_α(α≠0,?1) is one-dimensional.     

Annihilating polynomials, étale algebras, trace forms and the Galois number

Annihilating polynomials for quadratic forms in the Witt ring are obtained via an étale algebra interpretation of the Burnside ring together with a homomorphism to the Witt ring.     

Connectedness of the invertibles in certain nest algebras of order type ω

We solve the connectedness problem for a class of nests of order type ω with finite dimensional atoms.     

Étale descent for hochschild and cyclic homology

IfB is an étale extension of ak-algebraA, we prove for Hochschild homology thatHH _*(B)≅HH_*(A)⊗_AB. For Galois descent with groupG there is a similar result for cyclic homology:HC _*≅HC_*(B)^G if . In the process of proving these results we give a localization result for Hochschild homology without any flatness assumption. We then extend the definition of Hochschild homology to all schemes and show that Hochschild homology satisfies cohomological descent for the Zariski, Nisnevich and étale topologies. We extend the definition of cyclic homology to finite-dimensional noetherian schemes and show that cyclic homology satisfies cohomological descent for the Zariski and Nisnevich topologies, as well as for the étale topology overQ. Finally we apply these results to complete the computation of the algebraicK-theory of seminormal curves in characteristic zero. Partially supported by National Science Foundation grant DMS-8803497 Partially supported by National Security Agency grant MDA904-90-H-4019     

Local-global principle for étale cohomology

Letk be a field of characteristic different from 2 and ø be a galoisian cohomological class ofk, with values in /2. J. K. Arason proved that ø is killed by a cup-product power of (–1) if and only if the restriction of ø is zero in all the real closedk-extensions. In this paper, we extend such a local-global principle to semilocal rings with 2 a unit, étale cohomology replacing Galois cohomology.     

On Fitting Ideals of Certain étale K-groups

Let F be an Abelian number field and S the set of primes of F that are either ramified or over p, with p an odd prime. In this paper we compute the (first) Fitting ideal of K _2i–2 ^ét (O _F ^S () for i 2, where O _F ^S is the ring of S-integers of F and is a character of Gal(F/) of order prime to p different from the ith power of the Teichmüller character. This Fitting ideal proves to be principal and generated by a Stickelberger element.     

Cyclic Cohomology of étale Groupoids: The General Case

We give a general method for computing the cyclic cohomology of crossed products by étale groupoids, extending the Feigin–Tsygan–Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor, and Tsygan for the convolution algebra C _c (G) of an étale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered.     

On generators and relations for certain involutory subalgebras of kac-moody lie algebras ∗

We find generators and relations for those subalgebras of Kac-Moody Lie algebras that are the fixed point algebras of certain involutions. Specifically the involution must involve the Cartan involution which interchanges the positive and negative generators. We go on to apply these results to the G.I.M. algebras, which were introduced as natural generalizations of Kac-Moody algebras by P. Slodowy. We show such algebras are isomorphic to subalgebras of Kac-Moody algebras. From this we are able to derive someinteresting interrelations between certain Kac-Moody algebras.     

Theorems of Main Boundedness Type, Derivations and Module Derivations

The study of automatic continuity for Banach algebras rests heavily on threetechnical results, namely the main boundedness theorem (cf. [1] and [2]),stability of separating spaces and finiteness of discontinuous points (Lemma1. 6 and Theorem 2.3 in [3]. It has long been asked whether the last two results     

Representations of étale Lie groupoids and modules over Hopf algebroids

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.     

On some algebras diagonalized by M-bases of ℓ2

We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ². Examples are given whereA differs from ( being the rank one subalgebra ofA), and where together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in . We show that this holds provided that I is approximable by compact operators inA at any two points of ². We show that the spaceA+^* (where is the annihilator of ) is ultraweakly dense in (²), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.     

Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy—including all Deaconu–Renault groupoids associated to discrete abelian groups—M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.     

F-rationality of certain Rees algebras and counterexamples to “Boutot's Theorem” for F-rational rings

F-rational rings are defined for rings of characteristic p > 0 using the Frobenius endomorphism and corresponds to rational singularities in characteristic 0. We study F-rationality of certain Rees algebras and prove that every Cohen-Macaulay local ring with isolated singularity and negative a-invariant has a Rees algebra which is F-rational. As a consequence, we find that “Boutot's Theorem” asserting that a pure subring of a rational singularity is a rational singularity is not true for a F-rational ring.