Showing papers in "Annals of Mathematics in 1951"

Non-cooperative games

Abstract: we would call cooperative. This theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our theory, in contradistinction, is based on the absence of coalitions in that it is assumed that each participant acts independently, without collaboration or communication with any of the others. The notion of an equilibrium point is the basic ingredient in our theory. This notion yields a generalization of the concept of the solution of a two-person zerosum game. It turns out that the set of equilibrium points of a two-person zerosum game is simply the set of all pairs of opposing "good strategies." In the immediately following sections we shall define equilibrium points and prove that a finite non-cooperative game always has at least one equilibrium point. We shall also introduce the notions of solvability and strong solvability of a non-cooperative game and prove a theorem on the geometrical structure of the set of equilibrium points of a solvable game. As an example of the application of our theory we include a solution of a

Empty space-times admitting a three parameter group of motions

Abstract: where RV is the Ricci tensor, R is the scalar curvative and K is Einstein's gravitational constant. Equations (1.1) are second order partial differential equations for the g,, and their solutions involve arbitrary functions which are to be determined from physical data expressed as initial values or boundary conditions. On the basis of Mach's principle one would like to conclude that if T, = 0 and there are no singularities anywhere in the g9, then equations (1.1) which take the form

An iterative method of solving a game

Abstract: : In the paper, demonstration is made of the validity of an iterative procedure suggested by George W. Brown for a two-person game. This method corresponds to each player choosing in turn the best pure strategy against the accumulated mixed strategy of his opponent up to then.

Isometries of Operator Algebras

Abstract: Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of all real-valued, continuous functions on the compact Hausdorff space X). Such isometries are the maps induced by homeomorphisms of the spaces involved followed by possible changes of sign in the function values on the various closed and open sets. An internal characterization of these isometries would classify them as an algebra isomorphism of the C(X)'s followed by a real unitary multiplication, i.e., multiplication by a real continuous function whose absolute value is 1. The situation in the case of the ring of complex continuous functions (which we denote by 'C'(X)' throughout) is exactly the same; the real unitary multiplication being replaced, of course, by a complex unitary multiplication. It is the purpose of this paper to present the non-commutative extension of the results stated above. A comment as to why this noncommutative extension takes form in a statement about algebras of operators on a Hilbert space seems to be in order. The work of Gelfand-Neumark [2T has as a very particular consequence the fact that each C'(X) is faithfully representable as a self-adjoint, uniformly closed algebra of operators (C*algebra) on a Hilbert space. The representing algebra of operators is, of course, commutative. A statement about the norm and algebraic structure of C' (X) finds then its natural non-commutative extension in the corresponding statement about not necessarily commutative C*algebras. A cursory examination shows that one cannot hope for a word for word transference of the C'(X) result to the non-commutative situation. An isometry between operator algebras is as likely to be an anti-isomorphism as an isomorphism. The direct sum of two C* algebras, which is again a C* algebra, by [2], with an automorphism in one component and an anti-automorphism in the other shows that isomorphisms and anti-isomorphisms together do not encompass all isometries. It is slightly surprising, in view of these facts, that any orderly classification of the isometries of a C* algebra is at all possible. It turns out, in fact, that all isometric maps are composites of a unitary multiplication and a map preserving the C*or quantum mechanical structure (see Segal [7])of the operator algebra in question. More specifically, such maps are linear isomorphisms which commute with the * operation and are multiplicative on powers, composed with a multiplication by a unitary operator in the algebra.

On the Structure of Semigroups

Abstract: The purpose of this paper is to give the basis, and a few fundamental theorems, of a suggested systematic theory of semigroups. By a semigroup is meant a set S closed to a single associative, binary multiplication. Two elements of S are said to be left equivalent (I) if they generate the same left ideal in S. Similarly a right equivalence (r) can be defined. These equivalences commute (or are associable, Dubreil, 1), and their product equivalence is denoted by b. The equivalence b is to be compared with the two-sided analogue f of I and r; x 3 y(f) means that x and y generate the same two-sided ideal. In the case where S is finite, these equivalences b and f coincide, but this is not true in general. A b-class has the property that the 1-classes which it contains generate isomorphic left ideals in S; a theorem on the structure of b-classes is given (Theorem 1). Schwarz (1) and Clifford (1, 3) have both made use of minimal conditions on right and left ideals; and Clifford has shown (1) that the minimum ideal of a semigroup, if it contains both minimal right, and minimal left ideals, is a simple subsemigroup of the type called by Rees completely simple (Rees, 1, 2). In his paper "On Semigroups" (Rees, 1), Rees determined the structure of such simple semigroups. The minimal conditions we use are more stringent, they are the minimal conditions on the partially ordered sets of the principal right, left and two-sided ideals. If the right and left conditions are satisfied, then so is the twosided condition (a fact not obvious in our case, since a principal two-sided ideal is not in general a principal right or left ideal). Further, in this case we do have b = f; and the b-classes take on a simpler aspect. The f-classes of any semigroup S correspond to semigroups called principal factors (cf. Rees, 1, p. 391) of S; our definition does not, however, depend on the existence of a "principal series" of ideals of S. We say that S is semisimple if all these factors are non-nilpotent semigroups. Another, and probably more fruitful idea, is that of regularity. The element a e S is regular if aza = a for some z e S; this is the condition introduced, for rings, by J. v. Neumann (1).1 A semigroup is regular, if all its elements are regular. Regular semigroups are semisimple, but not conversely; however, any semisimple semigroup which satisfies the right and left minimal conditions, is regular. It seems that these last-mentioned semigroups may form the class which will most repay study; the simple ones are completely simple (and therefore of known structure), and already the extension theory of Clifford (2) suggests the possibility of building up more

Projections in banach algebras

Abstract: There have been two main contributions to the theory of operator algebras on Hilbert space. In a series of five memoirs, Murray and von Neumann have made important strides toward the structure theory of the weakly closed case. In work begun by the Russian school, a study has been made of the more general case where merely uniform closure is assumed. In terminology suggested by Segal, we call these W*-algebras and C*-algebras respectively. A notable advantage of the C*-case is the existence of an elegant system of intrinsic postulates due to Gelfand and Neumark [2]; so one can, and does, study C*-algebras in an abstract fashion that pays no attention to any particular representation. A corresponding characterization of TV*-algebras is not known, but nevertheless several substitutes have been suggested. In [6] von Neumann postulated from the start a second topology behaving like the weak topology. Steen [8] assumed completeness relative to a topology induced by positive functionals. In the present paper the entire burden will be thrown upon a more algebraic, and in some sense more elementary assumption; briefly put, our postulate is the assumption of least upper bounds in the partially ordered set of projections-the precise axioms are given in ?2. We call the algebras in question ATV*algebras (the "A" suggesting "abstract"). This work is in essence a continuation of the study that was begun by Rickart

On a Class of Analytic Functions from the Quantum Theory of Collisions

Abstract: The present investigation has its origin in the quantum theory of collisions. It appears appropriate, therefore, to give here a very short review of the subject even though some of the material to be covered in this section may be found in standard publications, such as the Theory of Atomic Collisions by N. F. Mott and H. S. W. Massey (Oxford University Press 1933).

Some Fixed Point Theorems

Abstract: where f, p are continuous, g satisfies a Lipschitz condition, p(t) has period 1, and g(t)/I ? 1 for large t at any rate. Our choice of hypotheses and the main lines of our investigations have been dominated by what is significant in the theory of differential equations, but our results are concerned solely with sets of points and transformations of sets of points. In the first place if a solution of (1) for which o = 0o, = v = no1, when t= 0 has t = 0i, vq = ni, at t = 1, then the transformation 3 of the point x = (6 77o) on to the point xi = (4i, nj) is (1, 1) and continuous, and also orientation preserving in the open I, -q Cartesian plane. That is to say as the point x describes a closed Jordan curve J counter-clockwise, the point xi = 3(x) describes J1 = 3(J) counter clock-wise. Even when we consider closed invariant subsets of the plane, we continue to assume that 3 is (1, 1) continuous and orientation-preserving in the whole plane and not merely in the subset considered; for the tranformations set up by the solutions of the differential equation in which we are especially interested always satisfy these hypotheses. We use methods depending essentially on these hypotheses, and although there may be generalizations of some of our results of one kind or another, we do not attempt to discuss them. We shall also give sotne special consideration to transformations which decrease areas or leave them the same size, as this type of result is easily verified for transformations defined by certain classes of such differential equations. A complete and accurate statement of our aims and results would need various lengthy definitions and the introduction of new notation, but we shall first describe the main lines we have followed, and our reasons for doing so, in terms which to a large extent explain themselves. Precise definitions of the terms used will be given later in the appropriate places. We are indebted to Prof. L. C. Young and to Mr. H. D. Ursell for many helpful criticisms and suggestions.

A contribution to differential geometry in the large

Abstract: The contribution can be simply formulated in the following theorem. I first formulate HYPOTHESIS A: Let Mn n > 2, be a manifold of class C'(3 ? a? w) with a Riemannian metric given by a positive-definite tensor gij(i, j = 1, ***, n) of class Ca such that M' is complete with respect to the metric. THEOREM 1. Assume A. If the Riemannian curvature K(P, -y) (P e yf, y a 2-dimensional plane through P) formed from gij satisfies

The Homotopy Groups of a Triad. III

Abstract: The principal purpose of this paper is to prove a rather general theorem about the homotopy groups of a triad in what may be called the "critical dimension," i.e., the lowest dimension for which the homotopy groups of a triad are non-zero. This theorem may be stated roughly as follows. Let (X j A, B) be a triad such that X = A u B. If 1I"1'(A, A n B) = 0 for p ~ m and 1I"q(B, A n B) = 0 for q ~ n, then the authors have shown previously [2] that ·lIAXj A, B) = 0 for r ~ m + n under very general conditions. We now show that 1I"m+>l.H(X; A, B) is isomorphic to the tensor product, 1I"mH(A, A n B) ® 1I">l.+,(B, A n B), under rather general conditions. Moreover, this isomorphism is defined in a very natural manner by means of a generalized Whitehead product. This theorem includes as special cases some results we have announced previously without prooe The proof which we give below depends heavily on a recent paper of J. C. Moore, [81. This proof is much simpler than the authors' original, unpublished proofs for the previously announced results. In sections 2 and 3 we give some applications of our main theorem to some problems of current interest in algebraic topology. This paper is essentially a continuation of our earlier papers, [1], [2], and [31. For the explanation of any terminology or notation that is not contained in the present paper, the reader is referred to these previous papers. In general, it is assumed that the reader is familiar with the basic properties of triad homotopy groups and generalized Whitehead products.

Sur la Theorie des Representations Unitaires

Abstract: Le but de ce travail est de pr6senter quelques r6sultats nouveaux-ainsi que d'autres-relatifs A la th6orie des representations unitaires des algebres involutives. Le Chapitre I concerne la th6orie des decompositions spectrales, que nous exposons sur des bases apparemment nouvelles (quoique l'auteur ne se fasse aucune illusion sur l'originalit6 r6elle de ses r6sultats), et dont nous faisons par la suite un usage constant. Que notre m6thode soit plus naturelle que celles qu'on connait d6jh, c'est ce que l'avenir seul d6cidera. Le Chapitre II traite des propri6t6s 6l6mentaires des representations unitaires, d'une fagon du reste sommaire. II contient une g6n6ralisation du Th6oreme de Plancherel, ainsi qu'un r6sultat partiel sur la structure topologique de l'ensemble des formes positives "'6l6mentaires." Le Chapitre III est consacr6 A la th6orie des sommes continues d'espaces de Hilbert, notion qui joue dans les travaux r6cents de I. Gelfand et M. Neumark un r6le fondamental et naturel, et qui vient de faire l'objet d'un m6moire important de J. von Neumann. Contrairement A ce qui se passe chez ce dernier, nous nous sommes efforc6 de prendre en consideration l'aspect topologique des problemes, en sorte que notre th6orie repose sur la notion de champ de vecteurs continu d'une part, et que nous ne nous restreignons pas, d'autre part, aux fonctions d6finies sur la droite. Nous pensons aussi avoir montr6 que cette th6orie est le domaine naturel de la theorie de l'intdgration, et que, loin de se construire A l'aide de celle-ci, la th6orie des sommes continues doit redonner comme simples cas particuliers les r6sultats connus relatifs aux fonctions num6riques ou vectorielles. Le Chapitre IV applique les notions pr6c6dentes A la demonstration d'un r6sultat remarquable annonc6 r6cemment par F. I. Mautner; A vrai dire, ce n'est pas exactement ce r6sultat que nous d6montrons, puisque les sommes continues utilis6es ici sont relatives A des espaces g6n6raux, et non pas seulement A la droite; mais il est parfaitement clair qu'une th6orie limit6e A la droite est inutilisable pratiquement: la meilleure preuve en est administr6e par F. I. Mautner lui-meme qui, appliquant son th6oreme aux fonctions de type positif d6finies sur un groupe non ab6lien, obtient un r6sultat qui ne se reduit pas, dans le cas ab6lien, au classique th6oreme de S. Bochner: et ce parce que, comme on 1'a d6jA dit, la th6orie de J. von Neumann n6glige completement les circonstances topologiques du probleme. En ce qui concerne l'originalit6 des r6sultats expos6s dans ces deux derniers Chapitres, elle est 6videmment douteuse-et du reste c'est la lecture de la Note

Generalization of a construction of antoine

Abstract: The principal purpose of this paper is to construct in Euclidean space R n, n > 3, a compact zero-dimensional set, A, whose complement is not simply connected. The construction constitutes an affirmative answer to a problem proposed by Borsuk [1], and is a generalization of Antoine's set [2] for the case n = 3. The paper is divided into three parts. In Part 1 the set A is constructed and proved to be zero-dimensional. In Part 2, 7rl(S nA) is computed explicitly and represented non-trivially2 in the symmetric group S6 . In Section 3 it is shown that the construction cannot be accomplished in Hilbert space. More precisely, it is shown that the complement of any compact subset of Hilbert space is contractible. I have been unable to determine whether a set of finite dimension can leave the Hilbert cube multiply connected, but have shown, however, that if there be such a set, then for some n its projection in the subspace xi = x2 = * = Xn = 0 is of infinite dimension. An analogous result is shown to hold for subsets of Rn; namely, the projection of such a set in any (n 1)-dimensional hyperplane has positive dimension. An interesting corollary to this, pointed out to me by E. E. Floyd, is the fact that the set A C Rn C Rn+l leaves Rn+l simply connected. Finally in Section 3, a remark by J. WV. Alexander [4] on Antoine's construction is generalized to show that for each 1 < q < n there is a q-cell in Rn whose complement is not simply connected.

Group extensions of Lie groups II

Abstract: morphism group of Q, then every extension of Q by H which is associated with 4, is a composite of any one of them, singled out arbitrarily, with an extension of C by H which is associated with the induced homomorphism of H into the group of automorphisms of C. The proof given by Eilenberg and MacLane makes essential use of factor sets and is therefore not applicable to topological group extensions. In ?1, we shall give a different proof which is applicable to general topological groups. However, in the general case, the theorem requires an additional topological restriction (see Defn. 1.1). Because of the necessity of verifying that our various constructions do not compel us to leave the category of Lie groups, we shall word our proof for Lie groups. In the rest of this paper, we shall confine our attention to extensions of abelian Lie groups. In particular, if C is an abelian Lie group, and if C1 denotes the connected component of the identity in C, we shall study the relationships between the extensions, by a fixed Lie group H, of C1, C, and C/Cl . For the main results, we assume that H is connected. In that case, if 4,t is a homomorphism of II into the group A(C) of the continuous open automorphisms of C which is determined by some extension of C by H the automorphisms belonging to Av(H) map every component of C onto itself, and the induced homomorphism,

On Direct Product Decomposition of Partially Ordered Sets

Abstract: Now we want to show that the proposition holds also when m, n are infinite. A partially ordered set is called connected if it cannot be decomposed into the sum of any two sets, and irreducible if it cannot be decomposed into the product of any two sets which have two or more elements. For instance directed sets are connected. LEMMA 1. If a partially ordered set S is connected, there exist for any two elements xo , x e S some finite number of xi, xi e S such that

Arithmetic on Algebraic Varieties

Abstract: D. G. Northcott has recently contributed some interesting new theorems ([4a], [4b]) to a subject which I introduced in my thesis [1] under the above-given title, and which had been further developed by Siegel [2] and myself [3]. It is my purpose here, by making explicit some concepts which had remained implicit in these papers, to supply what seems to be the proper algebraic foundations for that theory, and to give a comprehensive account of its results, including some new ones, up to date.

Continuous functions defined on spheres

Abstract: This theorem was conjectured some years ago by Kakutani [2]. He proved the case n = 2, generally known as Kakutani's theorem. The object of the present paper is to prove the following analog of Kakutani's theorem. THEOREM. Let S be the surface of a sphere center Z in Euclidean 3-space R3, and let f(x) be a continuous real-valued function defined on S. Then there exist four points X1 , X2 , X3 , X4 on S forming the vertices of a square with center Z, such that

On the Theory of Obstructions

Abstract: N. E. Steenrod has solved1 the homotopy classification problem for maps K -+ S', where n > 2 and K is an (n + 1)-dimensional polyhedron. His solution is stated in terms of separation cochains and depends on a certain theorem concerning obstructions (cf. (6.1) below). The latter, likewise the homotopy classification theorem, has been extended by M. M\. Postnikov and also, for n = 2, by Hassler Whitney to the case of maps in an (n 1)-connected space X, which is arbitrary except that 7r1(X) is finitely generated.2 The main purpose of this note is to show how the theory of composite chain systems ([11], ?4) and the secondary boundary operator can be used in proving the theorems of Postnikov and Whitney. Our conditions are more restrictive than theirs, since we eventually confine ourselves to finite complexes. This is because our definitions of the squaring operations (?5 below) do not apply to infinite complexes. However the same theorems can be proved, by an elaboration of our methods, for maps of a finite complex in an arbitrary (n 1)-connected space. This was done in an earlier draft. But the simplification due to the relation (5.9) below, for which the image space is also required to be a finite complex, seems to justify the loss of generality. In revising the first draft I have been greatly helped by a series of discussions with Steenrod, who suggested the use of the difference homomorphism and the relation (5.9).

The Complex of a Group Relative to a Set of Generators: Part II

Abstract: In order to describe the contents of the present paper we begin with some definitions. An extension of a local group Y is a pair (E, 4) where E is a local group and-? is a strong homomorphism of E onto X. Let (E, 4) be an extension of Y. It follows from the definition of strong homomorphism (?31) that the set N = 0_1 is a subgroup of E; N will naturally be called the kernel of (E, 4). A function u on Y to E such that qu(y) = y is a selector of (E, 4). A selector u of (E, 0) is symmetric if u(y'1) = u(y)V'. A selector u(y) is multiplicative if u(yiy2) = u(yI)u(y2) whenever YiY2 is defined. (Since the homomorphism 4 is strong, u(y1)u(y2) is defined whenever YiY2 is defined.) The extension (E, 0) of Y is inessential if it admits a multiplicative selector. Let Y be a local subgroup of a group Q generated by Y. We call an extension (E, 0) of Y extendible over Q if E can be imbedded in a group which is mapped onto Q by a homomorphism whose contraction to E is 0. Assume now (and during the remainder of the introduction) that Q is simply connected relative to Y (?34) and that Y contains no elements of order 2. Consider the abelian group P2(Y) defined in ??13, 34. Although it is determined by Y alone, P2(Y) is isomorphic to the second homotopy group of the complex K(Q, Y) (?35). The main theorem in the present paper is the following: in order that every extension of Y be extendible over Q it is necessary and sufficient that P2(Y) = { 1}. We leave open the question of the relationship between extendibility and the structure of P2(Y) when the latter is non-trivial. Consider the natural homomorphism aQY:G(Y) -4 Q defined in ?34. (G(Y) is a free group since Y has no elements of order 2.) Let C = C(Q, Y) be the preimage of X under aQY ; C is a local subgroup of G(Y). The problem of determining whether an extension of Y is extendible over Q is closely related to the problem of determining whether a given extension of C is inessential. We shall show that every extension of C is inessential if p2(Y) = {1}. Since the proof gives a somewhat misleading appearance of complexity, it will be well to outline the steps. It will be shown that the local subgroup C of G(Y) contains the canonical generators of G(Y) (?32) as well as the group R of defining relations of Q (?34).

Forced Oscillations for Several Nonlinear Circuits

Abstract: 1. Let us consider an electric circuit made up of a self-inductance L connected in series with a capacity C [L and C positive constants] and a device [for example, an electronic tube] whose resistance depends on the current x. It will be convenient to denote the resistance by g(x)/x. We also insert a periodic e.m.f. of period T. The time derivative of this e.m.f. [which is also a function of period T] we shall denote by e(t). Then x will satisfy the well known differential equation