We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a $C^*$-enveloping algebra.

Let $M$ be a subset of $\Bbb R$ with the following two invariance properties: (1) $M+k\subseteq M$ for all integers $k$, and (2) there exists a positive integer $l\ge 2$ such that $\frac{1}{l}M\subseteq M$. (For example, the set of Liouville numbers and the Besicovitch-Eggleston set of non-normal numbers satisfy these conditions.) We prove that if $h$ is a dimension function that is strongly concave at $0$, then the $h$-dimensional Hausdorff measure $\cal H^{h}(M)$ of $M$ equals $0$ or infinity.

A Banach space is called (almost) transitive if the isometry group acts (almost) transitively on the unit sphere. The main problems around transitivity are the Banach-Mazur conjecture that the only separable and transitive Banach spaces are the Hilbert ones (1930) and the Wood conjecture that C_0(L) cannot be almost transitive in its natural supremum norm unless L is a singleton (1982). In this note we give necessary and sufficient conditions on the locally compact space L for the (almost) transitivity of C_0(L). This will clarify the topological content of Wood's problem.

All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples obtained in [1].

In [4] Dipper and James investigated the representation theory of Hecke algebras of type Bn, H(Bn). Using the results in [4] and exploiting the fact that the Hecke algebra of type F4, denoted by H(W), contains two copies of H(B3) certain right ideals of H(W) will be constructed in this paper. These right ideals will be proved to be irreducible on the assumption that H(W) is semisimple.

In previous papers [1, 2, 3] the sums of a number of series of products of E-functions have been found. For the definitions and properties of the E-functions the reader is referred to [4, pp. 348–358]. In § 3 a further series of this type is given. The proof is based on an integral of an E-function with respect to its parameters, to be established in § 2. Similar integrals were given in [5] and [6].

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

Let A be a Banach *-algebra. We assume neither that A has an identity element, nor that the involution on A is continuous.

1. Equation (4) should read

2. On p. 83 below equation (7) replace

3. On p. 83 two lines above equation (9) replace

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.

1. In this note we consider the formal solution of the dual integral equations

where f(x) and g(x) are given and χ(x) is to be found. The direct solution of these equations has been given by Noble [1] but we shall show that they may be solved more easily if they are first reduced to a form in which g(x) ≡ 0.

The equation of the title in positive integers x, y, z where D is a given integer has been considered for some 300 years [4, pp 634–639]. As observed by V. A. Lebesgue, and probably known to Euler, if x, y, z is one non-trivial solution i.e., one with xy(x2 – y2) ≠0, another is given by . It then follows that there are infinitely many such with (x, y) = 1. The question that remains is to determine for which values of D such solutions exist.

The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).

(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.

(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.

It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.

The paper is related to the area which was recently called topological homology [3, 6, 12, 16, 4]. We consider questions associated with the central Hochschild cohomology of C*-algebras. The study of the latter was begun by J. Phillips and I. Raeburn in [9, 10], when they were investigating some problems of the theory of perturbations of C*-algebras. In [8] we obtained a description of the structure of C*-algebras with central bidimension zero: it was proved that these C*-algebras are unital and have continuous trace. In the special case of separable and a priori unital C*-algebras this statement was proved by J. Phillips and I. Raeburn in [11] with the help of a different approach. The question was raised. Which values can the central bidimension of C*-algebras take? In the present paper it is shown that, for any CCR-algebra A having at least one infinite-dimensional irreducible representation, the central bidimension and the global central homological dimension of A are greater than one. At the same time it is proved that there exist CCR-algebras which are centrally biprojective, but which have both dimensions equal to one. This situation contrasts with the state of affairs in the “traditional” theory of the Banach Hochschild cohomology. Recall [3, Ch. 5] that the bidimension and the global homological dimension of any infinite-dimensional biprojective C*-algebra are equal to two. Besides, there is no CCR-algebra of bidimension one (respectively, global homological dimension one). See [7].

In [1, p. 130] the following partition theorem was deduced from a general theorem concerning the limit of a recurrent sequence.

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the type

If we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.

In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M24, together with its mod-3 cohomology ring. The main results are

Theorem 1. The ring H*(M24, Z)(3)is the commutative graded Z(3)-algebra with generators

and relations v2 = 0 and βθ = 0. The Chern classes of the Todd representation in GL11F2

Let T denote the class of functions

f(z) = z+a2z2+…

that are analytic in U = {|z| <1}, and satisfy the condition

Imf(z). Imz≧ 0 (zεU).

Thus T denotes the class of typically real functions introduced by W. Rogosinski [5].

One of the most striking results in the theory of functions

g(z) = z + b2z2…

that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <¼} ⊂ g(U). In this note we point out that the same result holds for functions in the class T, a fact which seems to have been overlooked previously. We also determine the largest subdomain of U in which every f(z) in T is univalent, extending previous results in [1] and [2].

In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number λ, every nonzero element of G can be written as a difference of two elements of H fin exactly λ ways.

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.