Let Δ be a thick building of type Ã2, and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.

We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU(2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained.

For unbounded operators A1, …, Ad, Gevrey spaces Sλ1, …, λd (A1, …, Ad) of order (λ1, …, λd) are introduced, where the orders λ1, …, λd need not be equal. These extend the notion of Gevrey space defined by Goodman and Wallach where λ1 = … = λd. Several mild conditions on the operators A1, … Ad and the orders λ1, …, λd are presented such that the equality is valid. Examples are included.

In this paper, we generalize the classical F. and M. Riesz theorem to compact groups and compact commutative hypergroups. The group SU(2) of unitary matrices is also studied.

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

W. Rudin has proved that the union of the Riesz set N ⊆ R with a Λ(l)-subset of Z is again a Riesz set. In this note we generalize his result to compact groups whose contains a circle group, thereby extending an earlier F. and M. Riesz theorem for such groups by the author. We also investigate the possibility of constructing Λ(p)-sets for these groups, departing from Λ(p)-sets for the circle group in center.

Introduction. Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

We analyse the structure of a regular extension ℳ ⋊ γ, υQ of a von Neumann algebra ℳ by an action (modulo inner automorphisms) γ of a discrete group Q, and a nonabelian 2-cycle υ for γ, under the assumption that the “action” γ of Q is cocycle conjugate to an “action”, α which leaves globally invariant a cartan subalgebra of ℳ. we show that ℳ ⋊ γ, υQ is isomorphic with the algebra of the left regular projective representation of a certain discrete, non-principal groupoid ℜ V Q determined by the action of Q on the given cartan subalgebrs, where ℜ is the Takesaki relation associated to the pair (ℳ, ) we apply this description to give a decomposition of the regular representation of a group G into irreducibles, where G is a split extension of a type I group K by an abelian group Q, and work out the details of the author's earlier abstract plancherel theorem in the case when K is abelian.

Let G be a locally compact group, and let D(G) be a dense subalgebra of the convolution algebra L1(G). Suppose that π is a unitary representation of G and that, for each u in D(G), π(u)) is a trace-class operator. Then the linear functional u → tr(π(u)) (the trace of π(u)) is called the D-character of π. We give a simple proof that the D-character of such a representation determines the representation up to unitary equivalence. As an application, we give an easy proof of the result of Harish-Chandra that the K-finite characters of unitary representations of semisimple Lie groups determine the representations.

Recently M. Benedicks showed that if a function f ∈ L2(Rd) and its Fourier transform both have supports of finite measure, then f = 0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres.

Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

Consider a compact zero dimensional (profinite) monoid. While the group of units must be open, a regular D-class need not be open in the ideal it generates. This is the case if and only if the semigroup contains infinitely many copies of a certain semilattice composed of an increasing sequence of idempotents converging to an upper bound.

Using compactifications of free products, two generator compact monoids with these properties are constructed.

We prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on Rn belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.

In this paper it is proved that the principal series of representations of Γ = Z2*…*Z2 may be analytically continued to give uniformly bounded representations on the same Hilbert space, and that these representations are irreducible. Further, the reducibility of the restrictions to Γ ⊂ SL(2, Qp) of the irreducible unitary representations of SL(2, Qp) is examined.

The paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

For any group G, we introduce the subset S(G) of elements g which are conjugate to for some positive integer k. We show that, for any bounded representation π of G any g in S(G), either π(g) = 1 or the spectrum of π(g) is the full unit circle in C. As a corollary, S(G) is in the kernel of any homomorphism from G to the unitary group of a post-liminal C*-algebra with finite composition series.

Next, for a topological group G, we consider the subset of elements approximately conjugate to 1, and we prove that it is contained in the kernel of any uniformly continuous bounded representation of G, and of any strongly continuous unitary representation in a finite von Neumann algebra.

We apply these results to prove triviality for a number of representations of isotropic simple algebraic groups defined over various fields.

Let G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

This paper calculates the central Borel 2 cocycles for certain 2-step nilpotent Lie groups G with values in the injectives A of the category of 2nd countable locally compact abelian groups. The G's include, among others, all groups locally isomorphic to a Heisenberg group. The A's are direct sums of vector groups and (possibly infinite dimensional) tori, and in particular include R, T, and Cx. The main results are as follows.

(4.1) Every symmetric central 2 cocycle is trivial.

(4.2) Every central 2 cocycle is cohomologous with a skew symmetric bimultiplicative one (which is necessarily jointly continuous by [7]).

(4.3) The corresponding cohomology group H2cent (G, A) is calculated as the skew symmetric jointly continuous bimultiplicative maps modulo Homcont ([G, G]–, A).

These results generalize the case when G is a connected abelian Lie group and A = T, due to Kleppner [3]. Using standard facts of the cohomology of groups they can be interpreted as classifying all continuous central extensions (1) → A → E → G → (1) of the group G by the abelian group A. Finally some counterexamples are given to extending these results.

It is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.