Idéaux primitifs induits dans les produits croisés

Let an amenable second countable locally compact group G act continuously on a separable C¹-algebra A: the structure of the prime ideal space of the crossed product $\text{C}{}^1\text{(G, A)}$ is investigated without any restricting hypothesis on the action of G.     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.     

Maximal Abelian Subalgebras of the Hyperfinite Factor,Entropy and Ergodic Theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, μ), the crossed product L ^∞ (X, μ)⋊ G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra. Received February 24, 2002, Accepted August 5, 2002     

Applications of the Connes Spectrum to C1-dynamical systems,III

For a C¹-dynamical system (A, G, α) we show that the crossed product C¹-algebra is induced from a simple C¹-algebra equipped with an action of the Connes Spectrum, provided that A is G-simple and all isotropy subgroups of G under the action on the primitive ideal space of A are discrete. We then study the Borchers Spectrum of α and characterize its annihilator in G as the group of locally derivable automorphisms, under the assumption that the Arveson Spectrum of α is compact modulo the Borchers Spectrum. Finally a properly outer automorphism α is characterized by a series of equivalent conditions, one of which says that α is not close to the inner automorphisms on any ideal, another that α is not universally weakly inner on any ideal, and a third that the Borchers Spectrum of α on any invariant hereditary C¹-subalgebra is non-zero. This characterization leads to the conclusion that α is aperiodic (i.e., every non-zero power is properly outer) precisely when the Connes Spectrum of α is the full circle group.     

Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G ⁿ , the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G ⁿ , generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre??s notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.     

On traceable factor representations of crossed products

In this paper we give a complete classification of the traceable factor representations of the C¹-algebra which is the crossed product of the gauge group of automorphisms with the fermion algebra. Besides the type I representations, this algebra has an uncountable family of type II_∞ traceable factor representations. Unlike the fermion algebra, it has no finite factor representations. We present a similar analysis of the crossed product of SU(2) with the fermion algebra, where the action is the natural product action.     

Skew-products of higher-rank graphs and crossed products by semigroups

We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the C ^?-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the C ^?-algebra of a quotient graph. Our main tool is Laca’s dilation theory for endomorphic actions of Ore semigroups on C ^?-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger C ^?-algebra.     

Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions

Let G⊂Aut(A) be a discrete group which is exact, that is, admits an amenable action on some compact space. Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product A×G. This is shown for the topological entropy of an exact C∗-algebra A and for the dynamical entropy of an AFD von Neumann algebra A. These have applications to the case of transformations on Lebesgue spaces.     

On a duality for crossed products of C1-algebras

Let $\text{U}$ be a C¹-algebra, and G be a locally compact abelian group. Suppose α is a continuous action of G on $\text{U}$. Then there exists a continuous action $\text{}\text{?}$ga of the dual group $\text{G}\text{?}$ of G on the C¹-crossed product by α such that the C¹-crossed product is isomorphic to the tensor product and the C¹-algebra of all compact operators on L²(G).     

Crossed products of operator systems

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under G-equivariant maps, tensor products, and the canonical $C^{?}$-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects $C^{?}$-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.     

Coactions and Skew Products of Topological Graphs

We show that the C*-algebra of a skew-product topological graph E ×_κ G is isomorphic to the crossed product of C*(E) by a coaction of the locally compact group G.     

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T. Rieffel proved that if α is an action of G on a C ^*-algebra A and there is an equivariant embedding of C ₀(T) in M(A), then the action α of G on A is proper, and the crossed product \(A\rtimes_{\alpha,r}G\) is Morita equivalent to a generalised fixed-point algebra \({\operatorname{\mathtt{Fix}}}(A,\alpha)\) in M(A) ^α . We show that the assignment \((A,\alpha)\mapsto{\operatorname{\mathtt{Fix}}}(A,\alpha)\) extends to a functor \({\operatorname{\mathtt{Fix}}}\) on a category of C ^*-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel’s Morita equivalence implements a natural isomorphism between a crossed-product functor and \({\operatorname{\mathtt{Fix}}}\). From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg-Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.     

Crossed products of type I af algebras by abelian groups

Let (G, A, α) be a separableC*-dynamical system, withG abelian, and let Γ denote the dual group ofG. We characterize the Γ-invariant ideals of the crossed product algebraG×∩A, and use this characterization to prove that if in additionG is compact andA is type I AF, thenG×∩A is AF also. Finally, assumingG is discrete abelian and bothA andG×∩A are type I. we determine necessary and sufficient conditions, in terms ofA and the isotropy subgroups for the action ofG onÂ, forG×∩A to be AF.     

Module derivation problem for inverse semigroups

The derivation problem for a locally compact group G asserts that each bounded derivation from L ¹(G) to L ¹(G) is implemented by an element of M(G). Recently a simple proof of this result was announced. We show that basically the same argument with some extra manipulations with idempotents solves the module derivation problem for inverse semigroups, asserting that for an inverse semigroup S with set of idempotents E and maximal group homomorphic image G _S , if E acts on S trivially from the left and by multiplication from the right, any bounded module derivation from ? ¹(S) to ? ¹(G _S ) is inner.     

On some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).     

On the Regularity of Crossed Products

We study some generalizations of the notion of regular crossed products K * G. For the case when K is an algebraically closed field, we give necessary and sufficient conditions for the twisted group ring K * G to be an n-weakly regular ring, a ξ^* N-ring or a ring without nilpotent elements.     

Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C₀(X). We consider a category in which the objects are C^∗-dynamical systems (A,G,α) for which there is an equivariant homomorphism of (C₀(X),γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A_×α,rG. We show that the assignment (A,α)?Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.     

External derivations of internal groupoids

If H is a G-crossed module, the set of derivations of G in H is a monoid under the Whitehead product of derivations. We interpret the Whitehead product using the correspondence between crossed modules and internal groupoids in the category of groups. Working in the general context of internal groupoids in a finitely complete category, we relate derivations to holomorphisms, translations, affine transformations, and to the embedding category of a groupoid.     

Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T(N?N^×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1?β?2, the unique KMSβ-state is of type III₁. We prove this by reducing the type classification from T(N?N^×) to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N?N^× in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N?N^× on the Nica spectrum, we can also recover all the KMS-states of T(N?N^×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβ-states for β>2 that does not ostensibly come from an action of T on the C^?-algebra.