Showing papers in "Transactions of the American Mathematical Society in 1973"

Orbits of families of vector fields and integrability of distributions

Abstract: Let D be an arbitrary set of Cc vector fields on the Cc manifold M. It is shown that the orbits of D are C' submanifolds of M, and that, moreover, they are the maximal integral submanifolds of a certain C9? distribution PD. (In general, the dimension of PD(m) will not be the same for all m EM.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution A such that every vector field X in the Lie algebra generated by D belongs to A (i.e. X(m) e A(m) for every m EM). Their work therefore requires the additional assumption that A be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of A is not assumed in proving the first main result. It turns out that A is integrable if and only if A = PD' and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.

On the existence of invariant measures for piecewise monotonic transformations

Abstract: A class of piecewise continuous, piecewise C transforma­tions on the interval [0,1] is shown to have absolutely continuous invariant measures.

Inner Product Modules Over B ∗ -Algebras

Abstract: This paper is an investigation of right modules over a B*algebra B which posses a B-valued "inner product" respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a W*-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and W*-algebras. The extension of an inner product module over B by a B*-algebra A containing B as a *-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.

Properties of fixed-point sets of nonexpansive mappings in Banach spaces

Abstract: Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either F = 0 or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if T : C -» C is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings. Introduction. In this paper, A' always denotes a Banach space (either real or complex) and C a nonempty, closed and convex subset of X. Most published results on nonexpansive mappings have centered on existence theorems for fixed points of nonexpansive F : C —> X. In this paper we initiate the study of the structure of the fixed-point set F(T) = {x : Tx = x}. In this connection it is useful to assume that a conditional fixed point property is satisfied: Either F has no fixed points, or F has a fixed (CFP) point in every nonempty bounded closed convex set that F leaves invariant. It is obvious that existence theorems serve to define classes of nonexpansive mappings which satisfy (CFP). However, (CFP) holds even in contexts where no existence theorem can be hoped for. For example (Theorem 4), if C is locally weakly compact and X is strictly convex, then every nonexpansive F : C -» X satisfies (CFP). Our principal structure result is Theorem 2: if C is locally weakly compact, T : C -» C is nonexpansive, and F satisfies (CFP), then F(T) is a nonexpansive retract of C. Therefore our approach is to study the structure of F(T) by studying nonexpansive retracts of C. Although we are not primarily interested in existence results, our main structure theorem does permit us to prove an existence result (Theorem 7) under more general hypotheses than before, by a more transparent argument. Received by the editors November 13, 1970 and, in revised form, May 19, 1972. A MS (MOS) subject classifications (1970). Primary 47H10.

Joint measures and cross-covariance operators

Abstract: Let H1 (resp., H2) be a real and separable Hilbert space with Borel o-field r1 (resp., r2), and let (H1 x H2, r, x r2) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (H1 x H2, rJ x r2), i.e., joint measures, and the projections of such measures on (H1, rl) and (H2, r2). In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are ob- tained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized. (*) IH ~~~~~~~~~llx 11 2 djLi(x) < oo

Prime ideals and sheaf representation of a pseudo symmetric ring

Abstract: Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn. Introduction. Recently Koh [91 has obtained a sheaf representation of a ring without nilpotent elements. While Lambek [121 has unified this and the commutative case by introducing symmetric rings, Hofmann [7, Theorems 1.17 and 1.241 has extended the representation to semiprime rings. Using the maximal modular ideal space, Koh [11] has also obtained the representation for strongly harmonic rings. In this paper, the result of Lambek [121 is extended to a larger class of ringspseudo symmetric rings (Theorem 3.5). Example 5.1(e) is an example of a pseudo symmetric ring whose representation does not fall under any other types mentioned above. (See [7, p. 3111.) Almost symmetric rings are also introduced. A symmetric ring is almost symmetric and an almost symmetric ring is pseudo symmetric, but not conversely in either case. Some properties of these rings are discussed in the first two sections. In pseudo symmetric rings, the minimal prime ideals have the same characterization as for the commutative case. Mewborn's characterization of a commutative ring with compact minimal prime ideal space is generalized to pseudo symmetric rings with certain right quotient ring. For a pseudo symmetric ring, its prime ideal space is a T1-space iff it is a completely regular T2-space iff its usual basic open sets are closed as well. Presented to the Society, November 24, 1972 and January 28, 1973 under the title Prime ideal space and sheaf representation; received by the editors October 16, 1972. AMS (MOS) subject classifications (1970). Primary 16A64, 16A66, 18F20, 16A34; Secondary 16A48, 54D10, 54D20, 54H10.

Statistical mechanics on a compact set with ^{} action satisfying expansiveness and specification

Abstract: We consider a compact set Q with a homeomorphism (or more generally a Z' action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on ¿?(ß)—the real continuous functions on ft. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated. 0. Introduction. Invariant measures for an Anosov diffeomorphism have been studied by Sinai [16], [17]. More generally, Bowen [2], [3] has considered invariant measures on basic sets for an Axiom A diffeomorphism. The problems encountered are strongly reminiscent of those of statistical mechanics (for a classical lattice system—see [14, Chapter 7]). In fact Sinai [18] has explicitly used techniques of statistical mechanics to show that an Anosov diffeomorphism does not in general have a smooth invariant measure. In this paper, we rewrite a part of the general theory of statistical mechanics for the case of a compact set ß satisfying expansiveness and the specification property of Bowen [2]. Instead of a Z action we consider a Z' action as is usual in lattice statistical mechanics, where Q = Fz' (F: a finite set). This rewriting gives a more general and intrinsic formulation of (part of) statistical mechanics; it presents a number of technical problems, but the basic ideas are contained in the papers of Gallavotti, Lanford, Miracle, Robinson, and Ruelle [7], [11], [12], [13], etc. The ideas of Bowen [2] and Goodwyn [8] on the relation between topological and measure-theoretical entropy are also used. We describe now some of our results in the case of a homeomorphism F of a metrizable compact set ñ satisfying expansiveness and specification (see §1). Let na = {x G ñ: T\"x = {x}}, and let <3(ß) be the Banach space of real continuous functions on ñ. The pressure F is a continuous convex function on ú(fi) defined by 1 \" P(w) = Hm -logZ(

The measure algebra of a locally compact hypergroup

Abstract: A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups.-§1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. Such a structure can arise in several ways in harmonic analysis. Two major examples are furnished by the space of conjugacy classes of a compact nonabelian group, and by the two-sided cosets of certain nonnormal closed subgroups of a compact group. Another example is given by series of Jacobi polynomials. The class of hypergroups includes the class of locally compact topological semigroups. In this paper we will show that many well-known group theorems extend to the commutative hypergroup case. In §1 we discuss some basic structure and determine the idempotent probability measures. In §2 we present some elementary theory of characters of a hypergroup. In §3 we look at a restricted class of hypergroups, namely those on which there is a generalized translation in the sense of Levitan ([11], or see [12, p. 427]). (The notation of the present paper would seem to have two advantages over Levitan's: ours is compatible with current notation for compact groups, and in Levitan's notation, it is almost impossible to express correctly commutativity and associativity.) The theory for these hypergroups looks much like locally compact abelian group theory, yet covers a much wider range of examples. Finally in §4 we discuss some examples and further questions. 1. Basic properties. We recall some standard notation (in the following, X is a locally compact Hausdorff space): Received by the editors January 13, 1972. AMS (MOS) subject classifications (1970) Primary 22A20, 22A99, 43A10; Secondary 33A65, 42A60.

Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques

Abstract: Nous etablissons une inegalite' entre les sommes de Cesaro et la fonction maximale associees a une fonction definie sur la sphere, et nous en de'duisons divers re'sultats de convergence en norme L , convergence presque partout, localisation des developpements en harmoniques spheriques, ainsi qu'un theoreme de multiplicateurs qui ge'ne'ralise le theoreme classique de Marcinkiewicz sur les series trigonometriques. Lameme etude est faite pour les developpements suivant les polynomes ultraspheriques. Nous montrons de plus que les sommes partielles du developpement en harmoniques spheriques d'une fonction de L (2. ), p 4- 2, ne convergent pas forcement en norme.

Square integrable representations of nilpotent groups

Abstract: We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if r is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in L2(Nr/J), and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.

The polynomial identities of the Grassmann algebra

Abstract: By using the theory of codimensions the ¿\"-ideal of polynomial identities of the Grassmann (exterior) algebra is computed.

Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics

Abstract: Let A j, A-, • • • be dissipative sets that generate semigroups of nonlinear contractions T At), T St) • • « . Conditions are given on \\A } Which imply the existence of a limiting semigroup Tit). The results include types of convergence besides strong convergence. As an application, it is shown that solutions of the pair of equations „2, 2 2* u = — aux + a. [v — u ) and 2 2 2 vf = avx + a. (u u ), a a constant, approximate the solutions of ut = yÁ(d2/dx2) log u as o. goes to infinity.

On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities

Abstract: We consider a holomorphic family IvtitED of projective algebraic varieties Vt parametrized by the unit disc D {t e C: |ti < 1f and where Vt is smooth for t 0 but V0 may have arbitrary singularities. Displacement of cycles around a path t = t0e'0 (0 <0 < 277) leads to the Picard-Lefschetz transformation T: H*(Vto, Z) H*(V t, Z) on the homology of a smooth Vto. We prove that the eigenvalues of T are roots of unity and obtain an estimate on the elementary divisors of T. Moreover, we give a global inductive procedure for calculating T in specific examples, several of which are worked out to illustrate the method.

The support of Mikusiński operators

Abstract: A class of Mikusiriski operators, called regular operators, is studied. The class of regular operators is strictly smaller than the class of all operators, and strictly larger than the class of all distributions with left bounded support. Regular operators have local properties. Lions' theorem of supports holds for regular operators with compact support. The fundamental solution to the Cauchy- Riemann equations is not regular, but the fundamental solution to the heat equation in two dimensions is regular and has support on a half-ray.

The strong law of large numbers when the mean is undefined

Abstract: Let S,, = XI + ** +X. where (X") are i.i.d. random variables with EX,t = oo. An integral test is given for each of the three possible alternatives lim(S./n) = +oo a.s.; lim(S./n) = -oo a.s.; lim sup(S,,/n) = +oo and lim inf (S,,/n) 8 -00 a.s. Some applications are noted.

Krull dimension in power series rings

Abstract: Let R denote a commutative ring with identity. If there exists a chain PqCP jC...cP„ of zz + 1 prime ideals of R, where P X R, but no such chain of zz + 2 prime ideals, then we say that R has dimension n. The power series ring r?[LX]J may have infinite dimension even though R has finite dimension.

On the character of Weil’s representation

Abstract: The importance of certain representations of symplectic groups, usually called Weil representations, for the general problem of finding representations of certain group extensions is made explicit. Some properties of the character of Weil's representation for a finite symplectic group are given and discussed, again in the context of finding representations of group extensions. As a by-product, the structure of anisotropic tori in symplectic groups is given. I. In the title above a pun is intended, for this paper is concerned with two aspects of the celebrated Weil representation-first with its character in the sense of character theory in group representations, then with its character in the more everyday sense of its nature. I think also that the character (in the technical sense) of the Weil representation says something about the character in the general sense. In any case, both facts presented here seem to me rather striking. We proceed to describe them. Let F be a field of characteristic not 2, and let J{(F) be the group of (n + 2) x (n + 2) matrices of the form /1 X1 . .x X Z\ XI xn 1 1 Q Yi

Algebraic cohomology of topological groups

Abstract: A general cohomology theory for topological groups is described, and shown to coincide with the theories of C. C. Moore [12] and other authors. We also recover some invariants from algebraic topology. This article contains proofs of results announced in [151. We consider algebraic cohomology groups of topological groups, which are shown to include the invariants considered by Van Est [6], Hochschild and Mostow [7], C. C. Moore [12], and Tate (see [5]). We identify some of these groups as invariants familiar from algebraic topology. Let G be a topological group. A topological G-module is an abelian topological group A together with a continuous map G x A -4 A satisfying the usual relations g(a + a) = ga + ga', (gg ')a = g(g 'a), la = a. The category of topological G-modules and equivariant continuous homomorphisms is a quasiabelian category in the sense of Yoneda [16], and hence we get Ext functors just as in an abelian category. A proper short exact sequence will be a sequence O ) A -4 B -A C 0 of topological G-modules which is exact as a sequence of abstract groups and such that A has the subspace topology induced by its embedding in B, and such that u be an open map. For any G-module A we define the algebraic cohomology groups HZ(G, A) to be the ith Ext group Ext2 (Z, A), where Z denotes the group of integers with the discrete topology and trivial G-action. There is another set of short exact sequences we might have chosen which also give the category of topological G-modules the structure of a quasi-abelian S-category in the sense of Yoneda. We might have demanded that in addition to being exact in the previous sense, there be a continuous map s: C -* B such that the composition u o s be the identity on C. If G is locally compact we recover the "continuous cochains" theory, which is discussed in [5], [6], and [7]. If G is not locally compact it must be shown that continuous cochains are effaceable, i.e. that for any continuous cocycle c: GC -n A there is a short exact sequence 0 -O A B -4 C -0 such that r 0 c is the coboundary of a Received by the editors September 24, 1971. AMS (MOS) subject classifications (1970). Primary 22A05. 83 Copyrdit 0) 1973, American Mathematical Society This content downloaded from 157.55.39.253 on Sat, 11 Jun 2016 06:04:56 UTC All use subject to http://about.jstor.org/terms

Algebraic results on representations of semisimple Lie groups

Abstract: Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let g and f denote the respective complexified Lie algebras. Then every irreducible representation n of g which is semisimple under f and whose irreducible tcomponents integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of g belonging to the infinitesimal nonunitary principal series. It follows that 77 integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of t in 77 is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of g in the infinitesimal nonunitary principal series, as well as certain more general representations of g on which the center of the universal enveloping algebra of g acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.

Convex hulls and extreme points of families of starlike and convex mappings

Abstract: The closed convex hull and extreme points are obtained for the starlike functions of order a and for the convex functions of order a. More generally, this is determined for functions which are also Mold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order a. Also, the lower bound on Re(/(z)/z} is found for each z (\\z\\ < 1) where/varies over the convex functions of order a and a > 0. Introduction. In this paper we determine the closed convex hulls and extreme points of families of functions which are generalizations of the starlike and convex mappings. These results allow us to solve a number of extremal problems over related families of analytic functions. Let A denote the unit disk {z G C: \\z\\ < 1) and let A denote the set of functions analytic in A. Then A is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of A. Let S be the subset of A consisting of the functions/ that are univalent in A and satisfy /(0) = 0 and f'(0) = 1. Let K and Si denote the subfamilies of S of convex and starlike mappings; that is,/ e K if /(A) is convex, and/ e St if/(A) is starlike with respect to 0. The problem of studying the convex hulls and the extreme points of various families of univalent functions was initiated by three of the authors in [2]. We shall take advantage of some of the basic results obtained there and generally use the same notation with the exception that !$F shall now denote the closed convex hull of a family of functions F. @[$F] shall denote the set of extreme points of §F. Theorems 2 and 3 in [2] completely determined the sets §K, <£[§#], §S/ and ©[§5/]. The present paper contains generalizations of these results. We consider the family, denoted St(a), of starlike functions of order a introduced in [11] by M. S. Robertson. A function/analytic in A belongs to St(a) Received by the editors May 31, 1972 and, in revised form, November 6, 1972. AMS (MOS) subject classifications (1970). Primary 30A32; Secondary 30A34, 30A40.

The Picard group of noncommutative rings, in particular of orders

Abstract: The strucrure of the Picard group of not necessarily commutative rings, and specifically of orders, and its relation to the automorphism group are studied, mainly with arithmetic applications in mind. This paper is concerned with the Picard group, Pic (A), of a noncommutative (i.e. not necessarily commutative) ring A with identity, defined via the tensor product of invertible bimodules (see e.g. [Bl] or [B2]). Although some general results on Pic (A) are known, relatively little systematic work has been done so far. The real interest of the noncommutative theory has until now been the special case of Azumaya (central separable) algebras, when in fact Pic (A) coincides with Pic of the commutative ring cent (A) (the centre of A) (see [RZ]). On the other hand our principal interest lies in orders. But although our strongest results come in this case, some of these extend without too much weakening to a wider class of rings. In fact in the early part of the paper the treatment is completely general. Although most of the present paper is algebraic in spirit, the whole work was done with applications to the arithmetic of noncommutative orders over the ring Z of integers in mind. Our theory then becomes a tool both for certain local as well as for noncommutative \"local-global\" problems, and it leads up to the consideration of arithmetic subgroups in certain algebraic groups. Moreover the explicit computations for integral group rings given at the end of this paper are really number theoretic, i.e. depend crucially on Z being the base ring. For the general theory one can, without loss of generality, view the ring A as an algebra over some commutative ring R and look at Pic„ (A), the group given by bimodules with R acting the same on both sides. One can always take R = Z, noting that Pic (A) = Pic7(A). For theoretical reasons and for applications it is however the normal subgroup Pic , . .(A) = Picent(A) which is the ° r cent(A ) really interesting object, even when one considers R-orders, and R /= cent (A). Received by the editors July 19, 1972. AMS (MOS) subject classifications (1970). Primary 16A54, 16A18, 16A26, 16A72. (')Most of this work was done while the author was Visiting Professor in the University of Arizona at Tucson and was partly supported by the National Science Foundation. 1 Copyright © 1973, Z\\merican Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Some results on the length of proofs

Abstract: . Given a theory T, let \-^A mean "A has a proof in T of at most k lines". We consider a formulation PA* of Peano arithmetic withfull induction but addition and multiplication being ternary relations. We showthat \-k A is decidable for PA* and hence PA* is closed under a weak enrule. Ananalogue of Godel's theorem on the length of proofs is an easy corollary. 1. Introduction. In this paper we shall consider questions regarding thelengthOjof proofs. Now the length of (the shortest) proof of a given formulain a formal system depends strongly on the way in which the system is pre-sented. E.g. adjoining one of the theorems as an axiom reduces the lengthof some proofs. Thus in order to get significant results, we have either toconfine ourselves to particular formalisations of particular theories or elseto tormulate a criterion which distinguishes "nice" and "not so nice"formalisations of the same theory. We shall take here the second approach.In particular we shall consider theories formalised in some language of thelower predicate calculus by means of a finite number of axioms, axiomschemata and schematic rules of inference. Formalisations of this kind willinclude Hilbert type and Gentzen type formalisations of classical and in-tuitionistic arithmetic.2. Schematic systems. Since axiom schemata and rules of inference aregenerally explained in the literature with the help of formula variables, inorder to define the notion of a schematic system we do the obvious, namely,we expand the notation of the predicate calculus to include metamathematicalsymbols and emphasize substitution as the central idea. Precise details