Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaces

where H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

The non-associative algebras arising in genetics (1), are rather isolated from other branches of non-associative algebra (6). However, in a paper (5), in which he studied these algebras in terms of their transformation algebras, Schafer proved that the gametic and zygotic algebras for a single diploid locus are Jordan algebras.

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.

It is shown that a ring R is a π-regular ring with no infinite trivial subring if and only if R is a subdirect sum of a strongly regular ring and a finite ring. Some other characterizations of such a ring are given. Similar result is proved for a periodic ring. As a corollary, it is shown that every δ-ring is a subdirect sum of a Unite ring and a commutative ring. This was conjectured by Putcha and Yaqub.

In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equation

or to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.

1. The problem has been proposed by Steiner* of finding the envelope of a system of similar conics circumscribed about a given triangle, and of finding the loci of the centres and foci of the conics of the system. He states that the envelope is a curve of the fourth order having three double points, and gives some of its properties. The problem has been treated by P. H. Schoute in a paper entitled Application de la transformation par droites symétriques à un problème de Steiner.† In this paper the author discusses the problem of the envelope in detail by a geometrical method, and gives the order of the locus of centres, of the locus of foci, and of the locus of vertices, and the class of the envelope of asymptotes, of the envelope of axes, and of the envelope of directrices. But in every case (except that of the locus of centres), he gives the degree or class twice as great as it should be. In a paper published in the Annals of Mathematics‡ I have found explicit equations for the locus of centres and the envelope of asymptotes, showing that each asymptote envelopes a three-eusped hypocycloid; and in a paper in the Transactions of the American Mathematical Society,§ I have obtained an equation for the envelope of the axes, showing that each axis also envelopes a three-cusped hypocycloid. I have also obtained the equation of the envelope of the directrices (a curve of the fourth class) and the equation of the locus of the vertices (a curve of the eighth degree); but the results have not been published. In the solution of a problem in the Educational Times (proposed by himself), Cayley* proved that the locus of the foci of the parabolas which pass through three fixed points is a unicursal quintic passing through the two circular points at infinity, by showing that the co-ordinates of the focus of any such parabola may be explicitly expressed as rational functions of a parameter of the fifth degree. As the locus of the foci of a system of similar conies passing through three fixed points is not in general a unicursal curve, it is hardly likely that Cayley's method could be extended to the general case.

We recall that a JC-algebra (Størmer (3)) is a norm closed Jordan algebra of self-adjoint operators on a Hilbert space. Recently, Alfsen, Shultz, and Størmer (1) have introduced a class of abstract normed Jordan algebras called JB-algebras, and have proved that every special JB-algebra is isometrically isomorphic to a JC-algebra. We show that this result brings to a satisfactory conclusion the discussion in (2) of certain wedges W in Banach algebras and their related Jordan algebras W–W, and leads to two characterisations of the bicontinuously isomorphic images of JC-algebras.

Soient K un anneau commutatif à élémént unité, P un K-module unitaire, SK(P) l'algèbre symétrique de P et le sous-K-module de SK(P) des éléments homogénes de degré m≧0. Rappelons que si P est un K-module libre de rang n + 1, l'algèbre SK(P) est isomorphe à l'algèbre des polynômes K[X0,X1,…,Xn] en les indéterminéesX0,X1,…,Xn à coefficients dans K et est le K-module libre dont une base est formée par les monômes tels que ie., Par les moôomes homogènes de degré m.

In a recent paper (these Proceedings (2), 6 (1939), 75–8), C. G. Lambe established, and gave some applications of, the formula

in which Ds is the symbol for the derivative of fractional order s. Lambe's proof of (1) is not quite rigorous and it does not bring out the conditions which have to be imposed upon f(x) in order to make (1) true. Furthermore this proof does not give any evidence as to the definition of fractional derivative which is to be used in connection with (1).

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.

It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.

The geometry associated with the five invariants of two quadrics is well known. There appears to be an omission, however, in the treatment of the Φ-invariant. Salmon gives the vanishing of Φ merely as a necessary condition for the possibility of the construction of a tetrahedron self-conjugate for one of the quadrics and having its six edges tangential to the other. Sommerville proves its sufficiency, and shows that this condition is poristic, giving rise to two systems of ∝ tetrahedra of the requisite type. No investigation appears to have been made of the locus of the vertices of the ∝ tetrahedra of each system and the dual problem regarding the nature of the developable surface arising from their ∝ faces.

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

The method here described applies to the successive integrations of any function whose graph consists of segments of straight lines.