We investigate a class of triangular UHF algebras which have the special property that there exists a sequence of unital multiplicative contractive finite-rank conditional expectations of the algebra into itself, which converges strongly to the identity, whose ranges form an increasing chain with dense union. A partial classification is given in terms of a local finiteness property, and a delimiting counterexample is obtained. Associated inverse limit algebras are considered, and a characterization of residually finite dimensional triangular AF algebras is obtained.

1991 Mathematics Subject Classification: 47D25, 46J35.

Let $G$ be an almost simple algebraic group defined over ${\Bbb F}_p$ for some prime $p$. Denote by $G_1$ the first Frobenius kernel in $G$ and let $T$ be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of $G_1T$. We have such filtrations on the baby Verma modules $Z(\lambda)$, where $\lambda$ is a character of $T$. They are obtained via a certain deformation of the natural homomorphism from $Z(\lambda)$ into its contravariant dual $Z(\lambda)^\tau$. Using the same deformation we construct for each projective $G_1T$-module $Q$ a filtration of the vector space $F_\lambda(Q)=\text{Hom}_{G_1T}(Z(\lambda)^\tau, Q)$. We then prove that this filtration may also be described in terms of the above-mentioned homomorphism $Z(\lambda) \rightarrow Z(\lambda)^\tau$ and this leads us to a sum formula for our filtrations. When $Q$ is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on $F_\lambda(Q)$ explicitly for all $\lambda$. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation between the dimensions of the layers in the filtrations of $F_\lambda (Q)$ for an arbitrary indecomposable projective $Q$ and the coefficients in the corresponding Kazhdan--Lusztig polynomials. We conclude the paper by proving that the above results in the $G_1T$ theory have some analogues in the representation theory of $G$ (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.

We develop some cohomological tools for the study of the local geometry of moduli and parameter spaces in complex Algebraic Geometry. Notably, we develop canonical formulae for the differential operators of arbitrary order and their natural action on suitable `natural' modules (for example, functions); in particular, we obtain a formula, in terms of the moduli problem, for the Lie bracket of vector fields on a moduli space. As an application, we obtain another construction and proof of flatness for the familiar KZW or Hitchin connection on moduli spaces of curves.

Let $f$ be an orientation-preserving homeomorphism of a compact orientable manifold. Sufficient conditions are given for the persistence of a collection of periodic points under isotopy of $f$ relative to a compact invariant set $A$. Two main applications are described. In the first,~$A$ is the closure of a single discrete orbit of~$f$, and~$f$ has a Smale horseshoe, all of whose periodic orbits persist; in the second,~$A$ is a minimal invariant Cantor set obtained as the limit of a sequence of nested periodic orbits, all of which are shown to persist under isotopy relative to~$A$.

1991 Mathematics Subject Classification: 58F20, 58F15.

Dedicated to the memory of Helmut Wielandt

In this paper a classification is obtained of all pairs G, H of finite primitive permutation groups such that G < H, the socles of G and H are different, and G and H share a common non-trivial orbital. An application is given to the study of automorphism groups of edge-transitive graphs.

2000 Mathematical Subject Classification: 20B15, 20B25.

A general mean value theorem for Dirichlet series, with a sharp error estimate near the boundary of the critical strip, is proved. Applications of this theorem to various automorphic $L$-functions are discussed. Moreover, sharp upper bounds of mean square values of $L$-functions are obtained when they are attached to lifted forms, such as Doi–Naganuma and Ikeda lifts in the theory of Siegel modular forms.

In this paper we obtain uniform estimates for the lattice pointproblem in the hyperbolic plane $\Bbb H$ under theassumption that the action is by a Fuchsian group $\Gamma$ which is co-finite. We fix a point $w$ from $\Bbb H$ and set $N_t(z,w)$ equal to the number of translates of $w$ by the group$\Gamma$ which lie in a geodesic ball of radius$t$ centred at a point $z$ of $\Bbb H$. The behaviour of$N_t(z,w)$ is then examined when $t$ is large and $z$ is allowed to vary over $\Bbb H$. We show that the finite quantity

$\limsup_{t\rightarrow\infty}\sup_{z\in\Bbb H} N_t (z,w) / \cosh ^2(t/2)$

depends crucially on the point $w$, and indeed can become arbitrarily large with $w$. On the other hand, for the average of this quotient we derive the estimate

\[\sup_{z\in\Bbb H} \ \frac{1}{t}\int_0^t \frac{N_{\tau}(z,w)}{\cosh ^2(\tau /2)} \, d\tau\ = \ \frac{4\pi}{| \Gamma _w | \mbox{vol}(F)} \,+ \, O\!\left(\frac{\log t}{t}\right)\]

as $t\rightarrow \infty$, where the implied constant is an explicit function of $w$. In this formula, $\mbox{vol}(F)$ is the hyperbolic volume of a Dirichletfundamental domain $F$ for $\Gamma$, and $|\Gamma _w |$ denotes the number of elements from $\Gamma$ fixing $w$. This estimate is then combined with a recent sampling theorem of K. Seip to obtain an inequality which decides whether or not the orbit $\Gamma . w$ forms a set of interpolation for a given weighted Bergman space in $\Bbb H$.

R. Gow proved that the order of a solvable rational group is divisible only by the primes 2, 3 and 5. In this paper it is proved that in a solvable rational group the Sylow 5-subgroup is always normal and elementary Abelian. Moreover, the structure of rational {2, 5}-groups is described in detail.

Let $G$ be a $\sigma$-compact, locally compact group and $\mathcal I$ be a closed 2-sided ideal with finite codimension in $L^1(G)$. It is shown that there are a closed left ideal ${\mathcal L}$ having a right bounded approximate identity and a closed right ideal ${\mathcal R}$ having a left bounded approximate identity such that ${\mathcal I} = {\mathcal L} + {\mathcal R}$. The proof uses ideas from the theory of boundaries of random walks on groups. 2000 Mathematics Subject Classification: primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.

A graph $H$ is said to divide a graph $G$ if there exists a set $S$ of subgraphs of $G$, all isomorphic to $H$, such that the edge set of $G$ is partitioned by the edge sets of the subgraphs in $S$. Thus, a graph $G$ is a common multiple of two graphs if each of the two graphs divides $G$.

This paper considers common multiples of a complete graph of order $m$ and a complete graph of order $n$. The complete graph of order $n$ is denoted $K_n$. In particular, for all positive integers $n$, the set of integers $q$ for which there exists a common multiple of $K_3$ and $K_n$ having precisely $q$ edges is determined.

It is shown that there exists a common multiple of $K_3$ and $K_n$ having $q$ edges if and only if $q \equiv 0 \, ({\rm mod}\, 3)$, $q \equiv 0 \, ({\rm mod}\, \binom n2)$ and

(1) $q \neq 3 \binom n2$ when $n \equiv 5 \, ({\rm mod}\, 6)$;

(2) $q \geq (n + 1) \binom n2$ when $n$ is even;

(3) $q \notin \{36, 42, 48\}$ when $n = 4$.

The proof of this result uses a variety of techniques including the use of Johnson graphs, Skolem and Langford sequences, and equitable partial Steiner triple systems.

2000 Mathematical Subject Classification: 05C70, 05B30, 05B07.

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

The ideal space $\mbox{Id}(A)$ of a Banach algebra $A$ is studied as a bitopological space $(\mbox{Id}(A),\tau_u,\tau_n)$, where $\tau_u$ is the weakest topology for which all the norm functions $I\to\Vert a+I\Vert$ (with $a\in A$ and $I\in \mbox{Id}(A)$) are upper semi-continuous, and $\tau_n$ is the de Groot dual of $\tau_u$. When $A$ is separable, $\tau_n\vee\tau_u$ is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF-algebras are shown to exhibit the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI-Banach algebras, are given.

The principal 3-block of a Chevalley group $G_2(q)$ with $q$ a power of 2 satisfying $q \equiv 2$ or 5 mod 9 and the principal 3-block of $G_2(2)$ are Morita equivalent.

2000 Mathematical Subject Classification: 20C05, 20C20, 20C33.

Let $M$ be a compact, smooth, $2$-connected, $2m$-dimensional manifold with $\partial M$ simply connected. If $M$ has the homotopytype of an $m$-dimensional $CW$-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on $\partial M$ with at most $\mbox{cat}(M)+2$ critical points, all of which are of a certain ‘reasonable’ type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a ‘reasonable’ critical point.

1991 Mathematics Subject Classification: primary 55M30, 57R70; secondary 58E05, 55P50, 55P62.

In the paper we study finite covers of countable $\omega$-categorical structures by means of $\omega$-categorical expansions of bases. It is shown that the cohomology approach of P. Cameron to two-graphs can be applied to covers of some wide class of structures. For example, it works for some reducts of orderings and hypergraphs. The finite covers of the countable highly homogeneous structures and the reducts of the countable random graph are classified.

Let $A$ be any one of the three elliptic curves over $\mathbb{Q}$ with conductor 11. We show that $A$ has Mordell–Weil rank zero over its field of 5-division points. In each case we also compute the 5-primary part of the Tate–Shafarevich group. Our calculations make use of the Galois equivariance of the Cassels–Tate pairing.

It is shown that the theta functions of level $n$ on the principally polarised Prym varieties of an algebraic curve are dual to the sections of the orthogonal theta line bundle on the moduli space of Spin($n$)-bundles over the curve. As a by-product of our computations, we also note that when $n$ is odd, the Pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.

A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extends a result of A. Gardiner on graphs and a result of A. Devillers and J. Doyen on linear spaces.

2000 Mathematical Subject Classification: 05B25, 51E14, 20B25.

We characterise the 1-unconditional subsets $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices in the Schatten–von-Neumann class $\mathrm{S}^p$. The set of couples $I$ must be the set of edges of a bipartite graph without cycles of even length $4 \lel \le p$ if $p$ is an even integer, and without cycles at all if $p$ is a positive real number that is not an even integer. In the latter case, $I$ is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space $\mathrm{S}^p_I$ spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.