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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., 31 pages. Final version to appear in the Journal of Functional Analysis

In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group Γ in PGL 2 ( C ) that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for PGL 2 ( C ) . This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi.

We study the semigroup C*-algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P\cap P^{-1}=\{e\} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C*-algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group. Here we give a new definition of a controlled map and discuss examples from different sources. We apply our main result to establish nuclearity for semigroup C*-algebras of a class of one-relator semigroups, motivated by a recent work of Li, Omland and Spielberg. This includes all the Baumslag--Solitar semigroups. We also analyse semidirect products of weakly quasi-lattice ordered groups and use our theorem in examples to prove nuclearity of the semigroup C*-algebra. Moreover, we prove that the graph product of weak quasi-lattices is again a weak quasi-lattice, and show that the corresponding semigroup C*-algebra is nuclear when the underlying groups are amenable., 36 pages, to appear in Journal of Functional Analysis, minor changes from previous version

In this thesis, we define and study the free wreath product of a compact quantum group by a quantum automorphism group and, in this way, we generalize the previous notion of free wreath product by the quantum symmetric group introduced by Bichon.Our investigation is divided into two part. In the first, we define the free wreath product of a discrete group by a quantum automorphism group. We show how to describe its intertwiners by making use of decorated noncrossing partitions and from this, thanks to a result of Lemeux, we deduce the irreducible representations and the fusion rules. Then, we prove some properties of the operator algebras associated to this compact quantum group, such as the simplicity of the reduced C*-algebra and the Haagerup property of the von Neumann algebra.The second part is a generalization of the first one. We start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group. We generalize the description of the spaces of the intertwiners obtained in the discrete case and, by adapting a monoidal equivalence result of Lemeux and Tarrago, we find the irreducible representations and the fusion rules. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.

In this paper we find the fusion rules of the free wreath products $\widehat{\Gamma}\wr_*S_N^+$ for any (discrete) group $\Gamma$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced $C^*$-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type $II_1$-factor and that the dual of $\widehat{\Gamma}\wr_*S_N^+$ has the Haagerup approximation property for all finite groups $\Gamma$., Comment: 35 pages

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H. We prove Lp Poincare inequalities for the group measure space L∞(ΩH,γ)⋊G, where both the group action and the Gaussian measure space (ΩH,γ) are associated with the representation α. The idea of proof comes from Pisierʼs method on the boundedness of Riesz transform and Lust-Piquardʼs work on spin systems. Then we deduce a transportation type inequality from the Lp Poincare inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffelʼs compact quantum metric spaces.

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd. © 2009 Elsevier Inc. All rights reserved. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Paternostro, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

If $G$ is a countable, discrete group generated by two finite subgroups $H$ and $K$ and $P$ is a II$_1$ factor with an outer G-action, one can construct the group-type subfactor $P^H \subset P \rtimes K$ introduced in \cite{BH}. This construction was used in \cite{BH} to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra (in the sense of Jones \cite{J2}) of this subfactor and prove that any subfactor with an abstract planar algebra of "group type" arises from such a subfactor. The action of Jones' planar operad is determined explicitly., 25 pages, 18 figures, To appear in JFA, reviewer's suggestions incorporated

Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable.

Suppose T is a bounded self-adjoint operator on the Hilbert space L2(X,μ) and let T=∫SpL2TλdE(λ) be its spectral resolution. Let F be a Borel bounded function on [−a,a], SpL2T⊂[−a,a]. We say that F is a spectral Lp-multiplier for T, if F(T)=∫SpL2TF(λ)dE(λ) is a bounded operator on Lp(X,μ). The paper deals with l1-multipliers, where X=G is a discrete (countable) solvable group with ∀x∈G, x4=1, μ is the counting measure and TΦ:l2(G)∋ξ↦ξ∗Φ∈l2(G), where Φ=Φ∗ is a l1(G) function, suppΦ generates G. The main result of the paper states that there exists a Ψ on G such that all l1-multipliers for TΨ are real analytic at every interior point of Spl2(G)TΨ. We also exhibit self-adjoint Φ′s in l1(G) such that suppΦ generates G and F∈Cc2 are l1-multipliers for TΦ.

In this paper, we define an equivariant higher index map from K∗Γ(X) to K∗(C∗(X)Γ) if a torsion-free discrete group Γ acts on a manifold X properly, where C∗(X)Γ is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum–Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1–20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C∗-algebras, in: C∗-Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291]. When Γ is trivial, this equivariant higher index map is the coarse Baum–Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective.

We will study some permanence properties of C ∗ -unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C ∗ -unique group. Moreover, any amenable discrete group is a retract of a discrete C ∗ -unique group.

Sz.-Nagy's dilation theorem for a contraction on a Hilbert space has been extended by Mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group. We generalize to this case the intertwining lifting theorem of Sz.-Nagy and Foias. Some previous interpolation results on ordered groups are obtained as consequences.

Let G be a discrete group with a proper length function l. In this paper, we prove that the Schwartz space S2l(G,A) is a spectral invariant dense subalgebra of the reduced crossed product C r ∗ (G,A) for all commutative C ∗ -algebra A with a G-action if and only if G has polynomial growth with respect to l.

Let G be a connected real semisimple Lie group which contains a compact Cartan subgroup such that it has non-empty discrete series. A holomorphic discrete model of G is a unitary G -representation consisting of all its holomorphic discrete series with multiplicity one. We perform geometric quantization to a class of G -invariant pseudo-Kahler manifolds and construct a holomorphic discrete model. The construction of discrete series which are not holomorphic is also discussed.

By the method of cyclic cohomology we prove that all tracial states on a twisted group C *( G ;σ), where G is a torsion free discrete group of polynomial growth and σ is a 2-cocycle on G with values in the unit circle group, induce the same map from K 0 ( C *( G ;σ)) into R .

This paper describes, for an induced representation π of a discrete group G , the structure of the center π( G )′ ∩ π( G )″. In particular, criteria for π being a factor representation are given. Also, the type of π is discussed, and an example of a type III quasi-regular factor representation is presented.

Let [F, be the free group on n generators and for each word w E [F, let 1 WI be its length. Haagerup [lo] showed that for each 0 < r < 1, the function H,(wp) = r”” is a positive definite function on [F,,. Consider the positive definite function q,(n) = rinl on h. Then the free product function cpI * cpr on ff, = Z * H coincides with H, and the result was extended this way by de Michele and Fig&Talamanca [7] and by Boiejko [4, 51. In [S] it is proved that the free product of the unital positive defined functions ui: Gi + Z(Z) is still positive defined on the free product group * Gi. The correspondence between the positive defined functions on a discrete group G and the completely positive maps on the full C*-algebra C*(G) and the isomorphism C*(G, * G2) N C*(G,) Z C*(G,) suggested we consider amalgamated products of unital linear maps on the amalgamated product of a family of unital C*-algebras over a C*-subalgebra. In Section 3 we prove that the amalgamated product of a family of unital completely positive B-bimodule maps Qi: A, + C is completely positive on the “biggest” amalgamated free product i .A i. As an application of our main result, in Section 2 we obtain some results concerning the dilation of noncommutative families of operators.

Let H be a hyperbolic plane, Γ be a co-finite discrete group acting on H. There is a well-known theorem of Selberg. If Γ⊂PSL(2, Z ) is a congruence subgroup then Γ is essentially cuspidal. That is, N Γ (λ)~ Vol (ΓβH) 4π λ, λ → ∞ . Here NΓ(λ) is the distribution function for the discrete spectrum of the automorphic Laplacian A(Γ). In a paper we give an infinite series of examples of noncongruence subgroups of PSL(2, Z ) which are essentially cuspidal.

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group G F are topological vector groups is investigated. It is shown that, if F is a real Frechet group and G F a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if G F is finite dimensional or if F is one dimensional and G F a separable Hilbert group.

Let G be a semi-simple Lie group of split rank 1 and Γ a discrete subgroup of G of cofinite volume. If P is a percuspidal parabolic of G with unipotent radical N and if χ is a non-trivial unitary character of N such that χ ( Γ ∩ N ) = 1 then a meromorphic family of functions M( v ) on gG / G that satisfy all of the conditions in the definition of automorphic form except for the condition of moderate growth is constructed. It is shown that the principal part of M( v ) at a pole v 0 with Re v 0 ⩾ 0 is square integrable and that “essentially” all square integrable automorphic forms with non-zero χ-Fourier coefficient can be constructed using the principal parts of the M -series. For square integrable automorphic forms that are fixed under a maximal compact subgroup the proviso “essentially” can be dropped. The Fourier coefficients of the M -series are computed. A specific term in the χ-Fourier coefficient is shown to determine the structure of the singularities of the M -series. This term is related to Selberg's “Kloosterman-Zeta function.” A functional equation for the M -series is derived. For the case of SL (2, R) the results are made more explicit and a complete family of square integrable automorphic forms is constructed. Also the paper introduces the conjecture that for semi-simple Lie groups of split rank > 1 and irreducible Γ the condition of moderate growth in the definition of automorphic form is redundant. Evidence for this conjecture is given for SO ( n , 1) over a number field.

Osborne and Warner have given a formula for the multiplicity of an integrable discrete series representation in L2(GΓ) when G is real rank one semi-simple Lie group and Γ is a discrete subgroup of co-finite volume. We simplify and evaluate this formula to show that for most G (as above) the multiplicity is the formal degree of the representation times the volume of GΓ. When it is not we give a simple interpretation of the difference.

Let Γ be a discrete subgroup of a semisimple Lie group G such that ΓβG has a finite volume. Using a theorem of Moscovici we express the multiplicity of discrete series representations of G in the discrete spectrum of L2(ΓβG) as the L2-index of a twisted Dirac operator. This result, which extends a result of Moscovici and of the author, holds for all integrable discrete series and for infinitely many nonintegrable discrete series. In particular, up to computing L2-indices in the special rank one case, it implies the Osborne-Warner formula.

Let N be a connected nilpotent Lie group and Γ be a discrete subgroup for which M = Γ \ N is compact. Let R be the regular representation of N in L 2 ( M ). Projections onto primary (irreducible) subspaces of R are given by convolution against distributions (the spherical distributions). In this paper we give formulas and several characterizations for these distributions. We apply these results in a specific case to the study of theta functions.

A C∗-algebra is called nuclear if there is a unique way of forming its tensor product with any other C∗-algebra. Takesaki [17] showed that all C∗-algebras of type I and all inductive limits of such algebras are nuclear, but that the C∗-algebra C r ∗ (G) generated by the left regular representation of G on l2(G) is nonnuclear, where G is the free group on two generators. In this paper an extension property for tensor products of C∗-algebras is introduced, and is characterized in terms of the existence of a certain family of weak expectations on the algebra. Nuclearity implies the extension property, and this is used to show that for a discrete group G, C r ∗ (G) is nuclear if and only if G is amenable. An approximation property in the dual of a C∗-algebra is introduced, and shown to imply nuclearity. It is not clear whether the converse implication holds, but it is proved that the known nuclear C∗-algebras satisfy the approximation property.

Let G be a noncompact simple Lie group, and let Γ be a discrete subgroup of G such that G Γ has finite volume. The main result of this paper is that the left ring of Γ is a Type II1 von Neumann algebra. This result in turn is applied to solve, in some generality, a problem in dynamical systems posed by I. M. Gelfand.

We consider the Laplace operator on quotients of hyperbolic n-dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non-maximal rank. Using methods developed by the first two authors, we prove a “Mourre estimate” and commutator estimates on the Laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates. These estimates will be used elsewhere to study the scattering matrix and Eisenstein series and their meromorphic continuations, and should be useful in studying trace formulas for these discrete groups.

Let G be a nilpotent discrete group and Prim ( C * ( G ) ) the primitive ideal space of the group C * -algebra C * ( G ) . If G is either finitely generated or has absolutely idempotent characters, we are able to describe the hull-kernel topology on Prim ( C * ( G ) ) in terms of a topology on a parametrizing space of subgroup-character pairs. For that purpose, we introduce and study induced traces and develop a Mackey machine for characters. We heavily exploit the fact that the groups under consideration have the property that every faithful character vanishes outside the finite conjugacy class subgroup.

Given a II 1 -factor M with separable predual and α, a free action of a countable amenable discrete group G on M , we show that the crossed product M × α G has property Γ (resp. is McDuff) when M itself has property Γ (resp. is McDuff).

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C ∗ -algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C ∗ -algebras. We show that our theory has a Mayer–Vietoris sequence. We prove that for a torsion-free countable discrete group Γ, the direct limit of the uniform K-homology of the Rips complexes of Γ, lim d → ∞ K * u ( P d Γ ) , is isomorphic to K * top ( Γ , l ∞ Γ ) , the left-hand side of the Baum–Connes conjecture with coefficients in l ∞ Γ . In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras.

Let G be a semisimple Lie group which has a compact Cartan subgroup H , let K be a maximal compact subgroup of G containing H , and let n be the sum of the negative root spaces of G corresponding to an arbitrary choice of a positive root system of ( G , H ). We compute the n -cohomology of the K -finite vectors in a limit of a discrete series representation π of G . In the special case when π is a discrete series representation our result reduces to the result of W. Schmid. In the special case when π is a holomorphic limit we interpret its multiplicity in L 2 ( ΓβG ) cohomologically, where Γ is a discrete subgroup of G . In the general case we present a conjecture for this multiplicity.

For a nondiscrete σ-compact locally compact Hausdorff group G, L∞(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {Nα} such that Nα1 + Nα2 = L∞(G) whenever α1 ≠ α2. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM⧹TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L∞(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.

The notion of a partial crossed product of a C*-algebra by the group of integers introduced by R. Exel is generalized to a partial crossed product by any discrete group. A reduced partial crossed product is defined and some of the standard facts relating amenability and K-amenability to ordinary crossed products are extended to this case. The exact sequence of Pimsner and Voiculescu for reduced crossed products by the free group on n generators is generalized to this setting. This result is used to calculate the K-groups of an amalgamated product of two finite dimensional C*-algebras over a common maximal commutative subalgebra.

Let G be a noncompact connected simple Lie group of split-rank 1. Assume that G possesses a compact Cartan subgroup so that the discrete series for G is not empty. Let Γ be a nonuniform lattice in G. In this paper, we give an explicit formula for the multiplicity with which an integrable discrete series representation of G occurs in the space of cusp forms in L2(GΓ).

Using techniques of stationary scattering theory for the Schrödinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds Mn arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds Mn of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.