Some permanence properties of C-unique groups

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.     

On the norms of group-invariant transition operators on graphs

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel ²-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.     

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.     

Transversals in non-discrete groups

The concept of ‘topological right transversal’ is introduced to study right transversals in topological groups. Given any right quasigroupS with a Tychonoff topologyT, it is proved that there exists a Hausdorff topological group in whichS can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal(S, T _S ,T ^S , o) is such thatT _S =T ^S is a locally compact Hausdorff topology onS, thenS can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.     

Quasi-uniformities on topological semigroups and bicompletion

We study the bicompletion of the quasi-uniformities that are induced in a natural way on a topological semigroup which has a neutral element. In particular, we show that if X is a topological semigroup, with neutral element, for which the left translations are open, then the bicompletion of the left quasi-uniformity of X can be considered a topological semigroup which contains the topological space X as a sup-dense subsemigroup. The bicompletion in the case that the left translations are not necessarily open is also discussed. In particular, both Abelian and left-cancellable topological semigroups are considered. For semigroups which are (left-)cancellable or which are locally totally bounded, theorems similar to those known from the classical theory of (para)topological groups are established. July 1, 1999     

Duality between compactness and discreteness beyond pontryagin duality

One of the most striking results of Pontryagin’s duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height 2 whose dual spaces are quasicompact and non-Hausdorff (they are T ₁ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a T ₁ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.     

Multiplicity formulas for finite dimensional and generalized principal series representations

The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.     

Covering groups of nonconnected topological groups and 2-groups

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are equal.     

Self-adjointness of perturbed bi-Laplacians on infinite graphs

We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

A Low Complexity Lie Group Method on the Stiefel Manifold

A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and used as a coordinate map. The GPC maps prove to adapt well to the Stiefel manifold. For the n×k matrix representation of the Stiefel manifold the arithmetic complexity of the method presented is of order nk ², and for nk this leads to huge savings in computation time compared to ordinary Lie group methods. Numerical experiments compare the method to a standard Lie group method using the matrix exponential, and conclude that on the examples presented, the methods perform equally on both accuracy and maintaining orthogonality.     

Complete hyperexpansivity, subnormality and inverted boundedness conditions

Athavale introduced in [3] the notion of a completely hyperexpansive operator. In this paper some results concerning powers of completely (alternatingly) hyperexpansive operators (not necessarily bounded) are extended tok-hyperexpansive ones. A semispectral measure is associated with a subnormal contraction as well as with a completely hyperexpansive operator, and an operator version of the Levy-Khinchin representation is obtained. Passing to the Naimark dilation of the semispectral measure, such an operator is related to a positive contraction in a natural way. New characterizations of a completely hyperexpansive operator and a subnormal contraction are given. The power bounded completely hyperexpansive operators are characterized. All these are illustrated using weighted shifts.     

ON VECTOR SPACES WITH LOCALLY BOUNDED CONVERGENCE STRUCTURES

ABSTRACT

It is known that a precompact set in a limit vector space is not necessarily bounded. In this paper it is shown that the topological vector space result that every precompact set is bounded can be extended to the large class of locally bounded prelimit vector spaces. This result is used to extend the well-known characterization of finite-dimensional separated topological vector spaces to locally bounded prelimit vector spaces.     

On Order Bounded Subsets of Locally Solid Riesz Spaces

In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Subshifts with slow complexity and simple groups with the Liouville property

We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson–Furstenberg boundary. We also get explicit upper bounds for the growth of Følner sets.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

Ramsey-Milman phenomenon,Urysohn metric spaces,and extremely amenable groups

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso of isometries of Urysohn’s universal complete separable metric space , equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.     

Completezza degli spaziL p per misure finitamente additive

Summary A criterion of completeness of metric (not necessarily normed) L^p spaces with respect to bounded finitely additive measures is established (L ^p spaces are defined differently that it is usually done in the setting of finitely additive measures). This result hinges on the Stone representation theorem for Boolean algebras.