Let p and q be odd prime numbers. It is shown that all quasitriangular Hopf algebras of dimension pq over an algebraically closed field k of characteristic zero are semisimple and therefore isomorphic to a group algebra.

A special Pisot number is a Pisot number α such that α/(α−1) is also a Pisot number. Lagarias, Porta and Stolarsky found 11 special Pisot numbers. Here it is proved that there are no more. Also, the one known Salem number τ with the property that τ/(τ−1) is a Pisot number is shown to be the only such number.

The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

A theorem of Maranda [1, Section 30] states that if F is a finite group, p is a prime and pe exactly divides [mid ]F[mid ], then a ℤpF-lattice M is determined up to isomorphism by its finite quotient M/pe+1M. If M is a free ℤp-module of rank d, this is equivalent to saying that representations of F in GLd(ℤp) are determined up to equivalence by their images modulo pe+1.

Ali Fröhlich was one of the truly great mathematicians of his generation. In addition to making fundamental contributions, he is one of the very few of whom it can be said, “he founded a new subject”: namely that area now known as Galois module theory. In addition to establishing a theoretical basis for the subject, he also formulated some very deep and wonderful conjectures that would thrust him onto the world's stage. He also built up a school of researchers that played a central part in establishing British algebraic number theory as a world force. The outstanding nature of his contributions to mathematics was recognised in 1992 when the London Mathematical Society awarded him the De Morgan Medal: their highest and most distinguished prize, which is awarded triennially.

1. Introduction

Let k be an algebraically closed field of any characteristic. A toric variety over k is a normal variety X containing the algebraic group T=(k*)n as an open dense subset, with a group action T × X→X extending the group law of T.

On any smooth variety X over a field k, we can define the sheaf of differential operators [Dscr ], which carries a natural structure as an [Oscr ]X-bisubalgebra of Endk([Oscr ]X). A [Dscr ]-module on X is a sheaf [Fscr ] of abelian groups having a structure as a left [Dscr ]-module, such that [Fscr ] is quasi-coherent as an [Oscr ]X-module. A smooth variety X is called [Dscr ]-affine if for every [Dscr ]-module [Fscr ] we have

[bull ] [Fscr ] is generated by global sections over [Dscr ],

[bull ] Hi(X, [Fscr ])=0, i>0.

Beilinson and Bernstein have shown [1] that every flag variety over a field of characteristic zero is [Dscr ]-affine, from which they deduced a conjecture of Kazhdan and Lusztig. In fact, flag varieties are the only known examples of [Dscr ]-affine projective varieties. In this paper we prove that the [Dscr ]-affinity of a smooth complete toric variety implies that it is a product of projective spaces. Part of the method will be to translate a proof of the non [Dscr ]-affinity of a 2-dimensional Schubert variety, given by Haastert in [4], into the language of toric varieties.

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.

We prove that if A is a prime non-commutative JB*-algebra which is neither quadratic nor commutative, then there exist a prime C*-algebra B and a real number λ with ½ < λ [les ] 1 such that A = B as involutive Banach spaces, and the product of A is related to that of B (denoted by ∘, say) by means of the equality xy = λx ∘ y + (1 − λ)y ∘ x.

In a previous paper, the second author introduced a compact topology τr on the space of closed ideals of a unital Banach algebra A. If A is separable, then τr is either metrizable or else neither Hausdorff nor first countable. Here it is shown that τr is Hausdorff if A is C1[0, 1], but that if A is a uniform algebra, then τr is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.

In this paper, it is proved that for $n\geq 2$, any horizontally homothetic submersion $\varphi :\mathbb{R}^{n+1}\longrightarrow (N^{n}, h)$ is a Riemannian submersion up to a homothety. It is also shown that if $\varphi :\mathbb{S}^{n+1}\longrightarrow (N^{n}, h)$ is a horizontally homothetic submersion, then $n=2m$, $(N^{n}, h)$ is isometric to $\mathbb{C}P^{m}$ and, up to a homothety, $\varphi$ is a standard Hopf fibration $\mathbb{S}^{2m+1} \longrightarrow \mathbb{C}P^{m}$.

A norm on a Banach space $X$ is called maximal when no equivalent norm has a larger group of isometries. If, besides this, there is no equivalent norm with the same isometries (apart from its scalar multiples), the norm is said to be uniquely maximal, which is equivalent to the convex-transitivity of $X$: the convex hull of the orbits under the action of the isometry group on the unit sphere is dense in the unit ball of $X$. The main result of the paper is that the complex $C_0(\Omega)$ is convex-transitive in its natural supremum norm if $\Omega$ is a connected manifold (without boundary). As a complement, it is shown that if $\Omega$ is a connected manifold of dimension at least two, then the diameter norm is convex transitive on the corresponding space of real functions.

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrial and Applied Mathematics, died on 28 January 1995, aged 63.

Alan went up to Oxford in 1951 to read Mathematics at Brasenose College. He obtained a first, and after a brief excursion to the Bristol Aircraft Company, he returned to work for a DPhil with George Temple. His thesis, completed in 1959 and entitled ‘Problems in compressible flow’, contained a mixture of analytic, approximate and numerical solutions which foreshadowed the new practical applied mathematics that he embraced later. He became University Lecturer and Tutorial Fellow of St Catherine's Society in 1959.

During the next twenty-five years, Alan Tayler brought a new ethos to applied mathematics. This change came about through his recognition that the status quo in the 1960s, which comprised a delicate balance between theory and practice in the area of applied mechanics, was capable of far-reaching generalisation; indeed, he saw that such a development was essential since the following decades were to be dominated by computers and an ever-increasing need for mathematical modelling. In 1967, with Leslie Fox, he initiated the mathematical Study Groups with Industry, wherein academic and industrial researchers interact in week-long workshops. These were an immediate success: (1) with industry, who found new insights into their problems and new recruiting possibilities; (2) with students, whose enthusiasm to use their theoretical knowledge soon led to the highly popular MSc in Mathematical Modelling and Numerical Analysis; and (3) with faculty, both pure and applied, who found an undreamed of source of fascinating new theoretical problems. For example, one intellectual consequence was the use of industrial case studies to uncover the new field of ‘free boundary problems’, on which several thousand learned articles have appeared since 1970.

The behaviour of the Ricci curvature along rays in a complete open manifold is examined.

A Radon measure $\mu$ on ${mathbb R}^n$ is said to be $k$-monotone if $r\mapsto{\mu(B(x,r))}/{r^k}$ is a non-decreasing function on $(0,\infty)$ for every $x\in {\mathbb R}^n$. (If $\mu$ is the $k$-dimensional Hausdorff measure restricted to a $k$-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1 -monotone measure $\mu$ in ${\mathbb R}^2$ with non-unique and non-conical tangent measures at a point. Furthermore, we show that $\mu$ can be the one-dimensional Hausdorff measure restricted to a closed set $A\subset {\mathbb R}^2$.

Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set [Hscr ]P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set [Hscr ]G of all G-harmonic functions. Then a necessary and sufficient condition for the equality [Hscr ]P(k)=[Hscr ]G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.

Let [Ascr ] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr ]0, and let {Tt, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr ]. The infinitesimal generator [Lscr ] of {Tt} is a bounded linear operator on the Banach space [Ascr ]. For any Hilbert space [Kscr ], denote by [Bscr ]([Kscr ]) the von Neumann algebra of all bounded operators on [Kscr ]. Christensen and Evans [3] have shown that [Lscr ] has the form

formula here

where π is a representation of [Ascr ] in [Bscr ]([Kscr ]) for some Hilbert space [Kscr ], R: [Hscr ]0 → [Kscr ] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr ]0, X∈[Ascr ]} is total in [Kscr ], and K0 is a fixed element of [Ascr ]. The unitality of {Tt} implies that [Lscr ](1) = 0, and consequently K0 = iH−½R*R, where H is a hermitian element of [Ascr ]. Thus (1.1) can be expressed as

formula here

We say that the quadruple ([Kscr ], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr ] of the semigroup {Tt}. It is quite possible that another set ([Kscr ]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr ]. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.

Extending a theorem of S. Mori [6], we characterize two-dimensional factorial algebras which are graded by a finitely generated abelian group of rank one and which are affine over an algebraically closed field. The result is related to the classification of weighted projective lines [2, 5] and to the topic of canonical algebras [7].

Let ${\cal A}$ be a unital maximal full algebra of operator fields with base space $[0, 1]^k$ and fibre algebras $\{{\cal A}_t\}_{t\in[0, 1]}^{k}$. It is shown in this paper that the stable rank of ${\cal A}$ is bounded above by the quantity sup$_{t\in[0, 1]^k}\,{\rm sr}(C([0, 1]^k)\,{\otimes}\,{\cal A}_t)$, where ‘sr’ means stable rank. Using the above estimate, the stable ranks of the C$^*$-algebras of the (possibly higher rank) discrete Heisenberg groups are computed.

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let π(G, A)[ratio ]IrrA(G)→Irr (CG(A)) be the Glauberman–Isaacs correspondence. Let B[les ]A and χ∈IrrA(G). We exhibit a counterexample to the conjecture that χπ(G, A) is an irreducible constituent of the restriction of χπ(G, B) to CG(A).

It is well known that there exist infinite closed subsets E of [ ] such that A(E) = C(E) (see, for example, [3]). Such sets are called Helson sets. Let E be a closed subset of [ ], let 0 < α < 1, and let Aα(E) be the restriction of the Beurling algebra Aα([ ]). Then Aα(E) ⊂ lipαE. We shall show that Aα(E) = lipαE if and only if E is finite. This answers a question raised by Pedersen [5], where partial results were obtained.