The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Let G be a discrete group and (G, G₊) be a quasi-ordered group.Set G₊(G₊)^–1 and G₁= (G₊\){e}. Let F^G₁(G) andF^G₊(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofF^G₊(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from F^G₁(G)to F^G₊(G). As an application, it is shown that when G = Z² andG₊ = Z₊ x Z, the K-groups K₀(F^G₁(G)) Z², K₁(F^G₁(G)) Z andall Fredholm operators in F^G₁(G) are of index zero.     

Interpolating Blaschke Products and Factorization Theorems

Let M(H) be the maximal ideal space of H the Banach algebraof bounded analytic functions on the open unit disk. Let G bethe set of nontrivial points in M(H). By Hoffman's work, G hasdeep connections with the zero sets of interpolating Blaschkeproducts. It is proved that for a closed -separated subset Eof M(H) with E G, there exists an interpolating Blaschke productwhose zero set contains E. This is a generalization of Lingenberg'stheorem. Let f be a continuous function on M(H). Suppose thatf is analytic on a nontrivial Gleason part P(x), f(x) = 0, andf 0 on P(x). It is proved that there is an interpolating Blaschkeproduct b with zeros {z_n}_n such that b(x) = 0 and f(z_n) = 0for every n. This fact can be used for factorization theoremsin Douglas algebras and in algebras of functions analytic onGleason parts.     

One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras

The following results are proved for a non-compact, locallycompact group G: the dimension of every non-trivial right idealin L¹(G)** (equipped with the first Arens product) is at least, where (G) is the minimalnumber of compact sets required to cover G; there exist left ideals in L¹(G)** and in LUC(G)* with trivialintersections, and the linear span of right-cancellable elementsis weak*-dense in the annihilator of C₀(G) in LUC(G)* and inthe annihilator of (theL-functions that vanish at infinity) in L(G)*. The same resultsare proved for weighted algebras when the weight function isdiagonally bounded.     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Relative Completions and the Cohomology of Linear Groups Over Local Rings

For a discrete group G there are two well known completions.The first is the Malcev (or unipotent) completion. This is aprounipotent group U, defined over Q, together with a homomorphism : G U that is universal among maps from G into prounipotentQ-groups. To construct U, it suffices for us to consider thecase where G is nilpotent; the general case is handled by takingthe inverse limit of the Malcev completions of the G/^rG, where^•G denotes the lower central series of G. If G is abelian,then U = G Q. We review this construction in Section 2.     

Continuity of Multiplication in the Largest Compactification of a Locally Compact Group

G^LUC is the largest semigroup compactification of the locallycompact group G. When G is not compact, given q G* = G^LUC \G, there is p G* such that x qx is discontinuous at p (Theorem2). If G is -compact, there is one p which will serve for allq (Theorem 1).     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.     

Essential Equivariant Maps and Borsuk-Ulam Theorems

A full characterization is given of those compact Lie groupsG with the property that every G-map XX on a finite-dimensionalG-complex X of finite orbit type, X^G = Ø, is (non-equivariantly)essential. For arbitrary G, conditions are given on the G-spaceX which guarantee this property. Finally, conditions are givenfor the non-existence of a G-map XY inducing a homotopy equivalenceX^GY^G on the fixed point sets. These results have applicationsto critical point theory of almost G-invariant functionals.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

Twistors and 3-symmetric spaces

We describe complex twistor spaces over inner 3-symmetric spacesG/H, such that H acts transitively on the fibre. As in the symmetriccase, the complex twistor spaces are flag manifolds G/K, whereK is the centralizer of a torus in G. Moreover, they carry analmost complex structure defined using the horizontal distributionof the normal connection on G/H that coincides with the complexstructure associated to a parabolic subgroup PG if it is integrable.Conversely, starting from a complex flag manifold G/P, thereexists a natural fibration with complex fibres on a 3-symmetricspace, called fibration of degree 3.     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

On a p-Local Geometry Associated with a Complex of Non-Trivial p-Subgroups

Let D be a complex of non-trivial p-subgroups of a finite groupG, closed under G-conjugation. In this paper, a p-local geometry^p(G; D) is introduced for G associated with the complex D. Thehomotopy equivalence between ^p(G; D) and D is also studied.2000 Mathematics Subject Classification 20E15, 51E24.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

The Consistency of Holt's Conjectures on Cohomological Dimension of Locally Finite Groups

Let G be a locally finite group of cardinality _n where n isa natural number. Let (G) be the set of primes p for which Ghas an element of order p. In [5], Holt conjectures that ifk is a finite field with char k (G) then (1) G has cohomological dimension n+1 over k; (2) Hⁿ⁺¹(G, kG) has cardinality 2^_n; (3) Hⁱ(G, kG) = 0 for 0 i n.     

An exact minimum degree condition for Hamilton cycles in oriented graphs

We show that every sufficiently large oriented graph G with⁺(G), ^–(G)(3n–4)/8 contains a Hamilton cycle. Thisis best possible and solves a problem of Thomassen from 1979.     

Relation Modules of Infinite Groups

Let F_n be the free group of rank n with basis x₁, x₂, ..., x_n,and let d(G) denote the minimal number of generators of thefinitely generated group G. Suppose that n d(G). There existsan exact sequence and wemay view the free abelian group as a right ZG-module by defining (rR')^g = r^g–1R' for allg G, where g^–1 is any preimage of g under , and = (g^–1)^–1 r(g^–1),the conjugate of r by g^–1. We call the relation module of G associated with the presentation(1), and say that has ambient rank n. Furthermore, we call the group F_n/R' the free abelianizedextension of G associated with (1). 1991 Mathematics SubjectClassification 20F05, 20C07.     

Exceptional Functions and Normality

Yang proved in [10] that if f and f^(k) have no fix-points forevery fF, where F is a family of meromorphic functions in adomain G and k a fixed integer, then F is normal in G. In thispaper we prove normality for families F for which every fF omits₁ and f^(k) omits ₂, where ₁ and ₂ are analytic functions with. 1991 Mathematics SubjectClassification 30D35, 30D45.