Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.     

Equivariant sheaf cohomdlogy

In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)^G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology H_G(X;-).     

Fixed point property for Banach algebras associated to locally compact groups

In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak^∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier-Stieltjes algebra of G has the weak^∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T.     

A generalization of amenability and inner amenability of groups

Let G be a locally compact group. We continue our work [A. Ghaffari: Γ-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of Γ-amenability of a locally compact group G defined with respect to a closed subgroup Γ of G × G. In this paper, among other things, we introduce and study a closed subspace A _Γ ^p (G) of L ^∞(Γ) and then characterize the Γ-amenability of G using A _Γ ^p (G). Various necessary and sufficient conditions are found for a locally compact group to possess a Γ-invariant mean.     

Multipliers and Modulus on Banach Algebras Related to Locally Compact Groups

LetGbe a locally compact group. In this paper we study moduli of products of elements and of multipliers of Banach algebras which are related to locally compact groups and which admit lattice structure. As a consequence, we obtain a characterization of operators onL^∞(G) which commute with convolutions whenGis amenable as discrete.     

Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

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.     

The number of possibilities for random dating

Let G be a regular graph and H a subgraph on the same vertex set. We give surprisingly compact formulas for the number of copies of H one expects to find in a random subgraph of G.     

On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC₂(G×H) the space of uniformly continuous functionals in VN(G×H)=A^∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least ²b(G)² left ideals of dimensions at least ²b(G)² in A(G×H)^∗∗ and in UC₂^∗(G×H). We show that every nontrivial right ideal in A(G×H)^∗∗ and in UC₂^∗(G×H) has dimension at least ²b(G)².     

Ideals with bounded approximate identities in Fourier algebras

We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.     

Amenability properties of the centres of group algebras

Let G be a locally compact group, and ZL¹(G) be the centre of its group algebra. We show that when G is compact ZL¹(G) is not amenable when G is either non-abelian and connected, or is a product of infinitely many finite non-abelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL¹(G).     

Strong liftings with application to measurable cross sections in locally compact groups

There are two principal theorems. The adjustment theorem asserts that a lifting may be changed on a set of measure zero so as to become slightly stronger. In conjunction with the standard lifting theorem, it yields generalizations (with shorter proofs) of a number of known results in the theory of strong liftings. It also inspires a characterization of strong liftings, when the measure is regular, by the fact that they induce upon every open set an artificial “closure” of that set which differs from it by a set of measure zero. The projection theorem asserts that, in the presence of a strict disintegration, a strong lifting may be transferred or “projected” from one topological measure space onto another. In conjunction with Losert's example, it yields regular Borel, measures, carried on compact Hausdorff spaces of arbitrarily large weight, which everywhere fail to have the strong lifting property. It also provides the final link needed to obtained, with no separability assumptions, a measurable cross section (or right inverse) for the canonical map Ω:G→G/H, whereG is an arbitrary locally compact group, and whereH is an arbitrary closed subgroup ofG.     

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.     

Symmetry and nonsymmetry for locally compact groups

The class [S] of locally compact groups G is considered, for which the algebra L¹(G) is symmetric (=Hermitian). It is shown that [S] is stable under semidirect compact extensions, i.e., H ? [S] and K compact implies K ×_sH? [S]. For connected solvable Lie groups inductive conditions for symmetry are given. A construction for nonsymmetric Banach algebras is given which shows that there exists exactly one connected and simply solvable Lie group of dimension ?4 which is not in [S]. This example shows that $\text{G}\text{Z}\text{?}$ [S]. Z the center of G, in general does not imply G ?[S]. It is shown that nevertheless for discrete groups and a (possibly) stronger form of symmetry this implication holds, implying a new and shorter proof of the fact that [S] contains all discrete nilpotent groups.     

A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups

Let G be a locally compact Abelian group. In this paper we study in which way the qualitative uncertainty principle is modified when we consider only functions f∈L²(G) which generate a Gabor frame associated with a uniform lattice K in G. This provides us with sharp lower bounds for the measure of the support of such functions and their Plancherel transforms.