arXiv subjects

Ruy Exel

Publications and source records attributed to Ruy Exel.

At least 55 records · Page 3Linked to original sources

arXiv2009

Non-Hausdorff groupoids

We present examples of non-Hausdorff, etale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bissection.

math.OA↗

arXiv2009

Fell bundles over inverse semigroups and twisted etale groupoids

Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.

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arXiv2009

Reconstructing a totally disconnected groupoid from its ample semigroup

We show that a (not necessarily Hausdorff) etale, second countable groupoid G with totally disconnected unit space may be reconstructed solely from the algebraic structure of its ample semigroup S. We also show that C*(G) possesses a universal property related to tight representations of S.

math.OA↗

arXiv2008

Noncommutative Cartan sub-algebras of C*-algebras

J. Renault has recently found a generalization of the caracterization of C*-diagonals obtained by A. Kumjian in the eighties, which in turn is a C*-algebraic version of J. Feldman and C. Moore's well known Theorem on Cartan subalgebras of von Neumann algebras. Here we propose to give a version of Renault's result in which the Cartan subalgebra is not necessarily commutative [sic]. Instead of describing a Cartan pair as a twisted groupoid C*-algebra we use N. Sieben's notion of Fell bundles over inverse semigroups which we believe should be thought of as "twisted etale groupoids with noncommutative unit space". En passant we prove a theorem on uniqueness of conditional expectations.

math.OA↗

arXiv2008

Inverse Semigroups and Combinatorial C*-Algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term "tight". These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the "tight spectrum", which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from a semigroupoid, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

math.OA↗

arXiv2008

Partial Crossed Product of a group G vs Crossed Product of S(G)

In this work we present a new definition to the Partial Crossed Product by actions of inverse semigroups in a C^*-algebra, without using the covariant representations as Sieben did in [5]. Also we present an isomorphism between the partial crossed products by partial actions of a group G and the partial crossed product by actions of S(G), an inverse semigroup associated to G introduced by Exel in [2].

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arXiv2007

Semigroupoid C*-Algebras

A semigroupoid is a set equipped with a partially defined associative operation. Given a semigroupoid Λwe construct a C*-algebra C*(Λ) from it. We then present two main examples of semigroupoids, namely the Markov semigroupoid associated to an infinite 0-1 matrix, and the semigroupoid associated to a row-finite higher-rank graph without sources. In both cases the semigroupoid C*-algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object, namely the Cuntz-Krieger algebras and the higher-rank graph C*-algebras, respectively. In the case of a higher-rank graph (Λ,d), it follows that the dimension function d is superfluous for defining the corresponding C*-algebra.

math.OA↗

arXiv2007

Tight representations of semilattices and inverse semigroups

By a "Boolean inverse semigroup" we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup S in a Boolean inverse semigroup which are "tight" in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of "Booleannes" present in the semilattice of idempotents of S. After observing that the Vagner-Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the "regular representation". We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to "continuous" inverse semigroups, a class generalizing the E*-unitaries.

math.RT↗

arXiv2006

Semigroups of local homeomorphisms and interaction groups

Given a semigroup of local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. Under suitable hypotheses we may also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

math.OA↗

arXiv2006

A new look at the Crossed-Product of a C*-Algebra by a Semigroup of Endomorphisms

Let G be a group and let P be a subsemigroup of G. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of "interaction groups" which consists of an assignment of a positive operator V_g on A for each g in G, obeying a partial group law, and such that (V_g,V_{g^{-1}}) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism crossed product constructions.

math.OA↗

arXiv2005

Hecke algebras for protonormal groups

We introduce the term "protonormal" to refer to a subgroup H of a group G such that for every x in G the subgroups x^{-1}Hx and H commute as sets. If moreover (G,H) is a Hecke pair we show that the Hecke algebra H(G,H) is generated by the range of a canonical partial representation of G vanishing on H. As a consequence we show that there exists a maximum C*-norm on H(G,H), generalizing previous results by Brenken, Hall, Laca, Larsen, Kaliszewski, Landstad and Quigg. When there exists a normal subgroup N of G, containing H as a normal subgroup, we prove a new formula for the product of the generators and give a very clean description of H(G,H) in terms of generators and relations. We also give a description of H(G,H) as a crossed product relative to a twisted partial action of the group G/N on the group algebra of N/H. Based on our presentation of H(G,H) in terms of generators and relations we propose a generalized construction for Hecke algebras in case (G,H) does not satisfy the Hecke condition.

math.OA↗

arXiv2004

Interactions

Given a C*-algebra B, a closed *-subalgebra A contained in B, and a partial isometry S in B which "interacts" with A in the sense that S*aS = H(a)S*S and SaS* = V(a)SS*, where V and H are positive linear operators on A, we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an "interaction" as being a pair of maps (V,H) satisfying the derived properties. Starting with an abstract interaction (V,H) over a C*-algebra A we construct a C*-algebra B containing A and a partial isometry S whose "interaction" with A follows the above rules. We then discuss the possibility of constructing a "covariance algebra" from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a "generalized correspondence". Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz-Pimsner algebras.

math.OA↗

arXiv2003

On embeddings of full amalgamated free product C*-algebras

We examine the question of when the *-homomorphism of full amalgamated free product C*-algebras λ: A *_D B --> A' *_{D'} B', arising from compatible inclusions of C*-algebras A in A', B in B' and D in D', is an embedding. Results giving sufficient conditions for λto be injective, as well of classes of examples where λfails to be injective, are obtained. As an application, we give necessary and sufficient condition for the full amalgamated free product of finite dimensional C*-algebras to be residually finite dimensional.

math.OA↗

arXiv2001

KMS states for generalized gauge actions on Cuntz-Krieger algebras (An application of the Ruelle-Perron-Frobenius Theorem)

Given a zero-one matrix A we consider certain one-parameter groups of automorphisms of the Cuntz-Krieger algebra O_A, generalizing the usual gauge group, and depending on a positive continuous function H defined on the Markov space Σ_A. The main result consists of an application of Ruelle's Perron-Frobenius Theorem to show that these automorphism groups admit a single KMS state.

math.OA↗

arXiv2001

Crossed-Products by Finite Index Endomorphisms and KMS states

Given a unital C*-algebra A, an injective endomorphism α:A --> A preserving the unit, and a conditional expectation E from A to the range of αwe consider the crossed-product of A by αrelative to the transfer operator L=α^{-1}E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a "gauge action" on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b in A (e.g. when A is commutative) then the KMS_βstates are precisely those of the form ψ= ϕG, where ϕis a trace on A satisfying the identity ϕ(a) = ϕ(L(h^{-β}ind(E)a)), where ind(E) is the Jones-Kosaki-Watatani index of E.

math.OA↗

arXiv2000

A New Look at The Crossed-Product of a C*-algebra by an Endomorphism

We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism α, which depends not only on the pair (A,α) but also on the choice of a transfer operator (defined in the paper). With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz-Krieger algebra O_A arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transfer operator.

math.OA↗

arXiv2000

Exact Groups, Induced Ideals, and Fell Bundles

Given a C*-algebra B which is graded over a discrete group G we consider ideals of B which are invariant under the projections onto each of the grading subspaces. If G is exact and the standard conditional expectation of B is faithful we show that all such ideals are induced, i.e. are generated by their intersection with the unit fiber algebra. This result is derived from a generalization to the context of Fell bundles of a Theorem by Kirchberg and Wasserman according to which a group is exact if and only if its reduced C*-algebra is exact.

math.OA↗

arXiv2000

Partial Dynamical Systems and the KMS Condition

Given a countably infinite 0-1 matrix A without identically zero rows, let O_A be the Cuntz-Krieger algebra recently introduced by the authors and T_A be the Toeplitz extension of O_A, once the latter is seen as a Cuntz-Pimsner algebra, as recently shown by Szymanski. We study the KMS equilibrium states of C*-dynamical systems based on O_A and T_A, with dynamics satisfying σ_t(s_x) = N_x^{it} s_x for the canonical generating partial isometries s_x and arbitrary real numbers N_x > 1. The KMS_βstates on both O_A and T_A are completely characterized for certain values of the inverse temperature β, according to the position of βrelative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for O_A are derived from those for T_A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for O_n and of Enomoto, Fujii and Watatani for O_A as particular cases.

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