An isomorphism (as groups) is established between an arbitrary connected module over a structural matrix near-ring and a direct sum of appropriate modules over the base near-ring. This isomorphism leads to a characterization of the 2-primitive ideals of a structural matrix near-ring.

Given a Banach space E, let us denote by Max(E) the largest operator space structure on E. Recently Paulsen-Pisier and, independently, Junge proved that for any Banach spaces E, F, isomorphically where and respectively denote the Haagerup tensor product and the spatial tensor product of operator spaces. In this paper we show that, in general, this equality does not hold completely isomorphically.

In Bonnycastle's Arithmetic a rule is given for finding any root of a number by approximation which in substance reduces to this statement:—If a and b are nearly equal, then the nth root of a/b is nearly equal to

Let A be a complex Banach algebra. By an ideal in A we mean a two-sided idealunless otherwise specified. As in (7, p. 59) by the strong radical of A we mean theintersection of the modular maximal ideals of A (if there are no such ideals we set =A). Our aim is to discuss the nature of and the relation of to A for a specialclass of Banach algebras. Henceforth A will denote a semi-simple modular annihilatorBanach algebra (one for which the left (right) annihilator of each modular maximalright (left) ideal is not (0)). For the theory of such algebras see (2) and (9).

If we consider the circle circumscribing any triangle ABC (see figures 11, 12), and diminish its radius still causing it to pass through A and B: then if ACB be an acute angle, C passes without the circle, but if ACB be an obtuse angle, C remains within the circle. If C be a right angle, the radius of the circle, being ½AB, cannot be farther diminished.

Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M is

here z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

In this paper, we consider interpolants on h·ℤn from the closure of the space spanned by translates of the function (‖·‖2 + 1)β/2 (β>−n and not an even nonnegative integer) along h·ℤn. We show that these interpolants approximate a function, whose Fourier transform satisfies certain asymptotic conditions, up to an error of order hp, on any compact domain in ℝn, where p is only restricted by the smoothness of the function.

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

Let k ≧ 2 be an integer and each of ν1, ν2, …, νk and δ1, δ2, …, δk be 0 or 1. Then given any positive integer M and non-negative reals a1, a2, …, aM we put

where

and

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.

§ 1. In space of three dimensions the properties of self-conjugate tetrads, pentads and hexads with regard to a quadric are well known (see Baker's Principles of Geometry, vol. iii). The general theorem in space of n dimensions Sn is to establish the existence of a set of n + p + 1 points A0, A1, …, An+p (0 ≤ p ≤ n−1) such that the pole, with respect to a given quadric, of the (n−1)-flat determined by any set of n of the points lies in the p-flat determined by the remaining p + 1 points.

Consider the Fredholm equation of the second kind

where

and Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.