Showing papers in "Transactions of the American Mathematical Society in 1976"

Fixed point theorems for mappings satisfying inwardness conditions

Abstract: Let X be a normed linear space and let K be a convex subset of X. The inward set, IK(x), of x relative to K is defined as follows: IK(x) = {x + c(u x):c ; 1, u E K}. A mapping T:K X is said to be inward if Tx E IK(x) for each x E K, and weakly inward if Tx belongs to the closure of IK(x) for each x E K. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces. 0. Introduction. Let X be a topological vector space, K C X, and T a mapping of K into X. An inwardness condition on T is one which asserts that, in some sense, T maps points x of K "toward" K, or more precisely into the set generated by rays emanating from x and passing through other points of K. Such conditions are always weaker than the assumption that T map the boundary of K, MK, into K. They have been formulated in a variety of ways and imposed by several authors recently in connection with studies both in fixed point theory and in certain differential equations. Our purpose in this paper is to illustrate how different types of inwardness assumptions are related, and to prove several new fixed point theorems in which these concepts play a role. Before stating precise definitions we give a brief review of some of the previous work in this area. The study of inward mappings originated with the investigations of B. Halpern in his 1965 doctoral thesis [7] where he obtained a generalization of the Schauder-Tychonov Theorem, a result he and Bergman further generalized in 1968 [9]. Since then many results have appeared in the literature concerning inward and weakly inward mappings in Halpern's sense, for both single and multivalued mappings (cf. [3], [6], [8], [9], [14], [16] -[19]). Another type of inwardness assumption was used by H. Brezis [11 in Received by the editors May 1, 1974 and, in revised form, October 16, 1974. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 54H25.

Reversible diffeomorphisms and flows

Abstract: We generalize the classical notion of reversibility of a mecfcaisietl system. The generic qualitative properties of symmetric orbits of such sx-ttco* are studied using transversality theory. In particular, we prove analogues alette closed orbit, Liapounov, and homoclinic orbit theorems for R-reversible

Group extensions and cohomology for locally compact groups. IV

Abstract: In this paper we shall apply the cohomology groups constructed in [ 14J to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defilned topologies and we find conditions under which this topology is T2. Finally we resolve several questions left open concerning splitting groups in a previous

Composition series and intertwining operators for the spherical principal series. II

Abstract: In this paper, we consider the connected split rank one 1 1 Lie group of real type F4 which we denote by F4. We first exhibit F4 as a group of operators on the complexification of A. A. Albert's exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space 1 16 15 F4/Spin(9) as the unit ball in R with boundary S . After decomposing the space of spherical harmonics under the action of Spin(9), we obtain the matrix of a transvection operator of F4/Spin(9) acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of F4.

Singularities in the nilpotent scheme of a classical group

Abstract: If (X, x) is a pointed scheme over a ring k, we introduce a (generalized) partition ord(x, X/k). If G is a reductive group scheme over k, the existence of a nilpotent subscheme N(G) of Lie(G) is discussed. We prove that ord(x, N(G)/k) characterizes the orbits in N(G), their codimension and their adjacency structure, provided that G is Gln, or SPn and 1/2 E k. For SOn only partial results are obtained. We give presentations of some singularities of N(G). Tables for its orbit structure are added. Introduction. Let G be a reductive algebraic group over a field of characteristic p. Let g be its Lie-algebra and N(G) the closed subset of the nilpotent elements of g, cf. [19]. The G-orbits in N(G) are characterized by weighted Dynkin diagrams,cf. [20, III]. Consider the following question. Is it possible to classify the orbits in N(G) using only the local structure of the variety N(G)? We prove in (4.3) that the answer is positive if G is Gln or if G is SPn and p * 2. To this end we introduce a local invariant "ord" for any pointed scheme in ? 1. We develop the theory of N(G) over an arbitrary ground ring k in ?2. In ?3 we restrict our attention to the classical group schemes. Using a cross section we obtain information about the orbit structure of N(G). Our main theorem (4.2) relates ord(x, N(G)/k) to the Jordan normal form of the nilpotent endomorphism induced by x in the classical representation. This paper is a condensed version of [13]. The author wishes to express his gratitude to his thesis adviser, Professor T. A. Springer. Conventions and notations. The cardinality of a set V is denoted by # V Any infinite cardinal is represented by oo. If x is a real number then [x] is thc greatest integer in x. All rings are commutative with 1. Let M be a module over a ring A. If M is free the rank of M is denoted by rgAM. An element r E A is called M-regular if a: M M is injective. Let a = (a,, .. . , ar) be a Received by the editors March 18, 1975. AMS (MOS) subject classifications (1970). Primary 14B05, 14L15; Secondary 05A17, 10C30, 13H15, 20G35.

Signature of links

Abstract: Let L be an oriented tame link in the three sphere S3. We study the Murasugi signature, a(L), and the nullity, r(L). It is shown that a(L) is a locally flat topological concordance invariant and that r(L) is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings. 0. Introduction. Let L be an (oriented) tame link of multiplicity ,u in the three-sphere S3. That is, L consists of,u oriented circles K1, ... , Kg disjointly imbedded in S3. Various authors have investigated a numerical invariant, the signature of L (notation: u(L)). The signature was first defined for knots (, = 1) by H. Trotter [21]. J. Milnor found another definition for this knot signature (see [1 2]) in terms of the cohomology ring structure of the infinite cyclic cover of the knot complement. In [2], D. Erle showed that the definitions of Milnor and Trotter are equivalent. In [15], K. Murasugi formulated a definition of signature for arbitrary links. In this paper we investigate the Murasugi signature in the context of branched covering spaces. To be specific, let D4 denote the four dimensional ball with 3D4 = S3, and let L C S3 be a link and F C D4 a properly imbedded, orientable, locally flat surface with 3F = L C S3. Let M denote the double branched cover of D4 along F. Then we show that u(L) is the signature of the four manifold M (see Lemma 1.1 and Theorem 3.1). Our proof of Theorem 3.1 contains the technicalities necessary to show this in the topological category. Using this viewpoint we are able to prove that u(L) is a topological concordance invariant (Theorem 3.8). We also rederive many of Murasugi's results, generalizing some of them (see Theorems 3.9-3.16). The paper is organized as follows: ? 1 contains the classical definitions of the signature and nullity of a link. It also deals with necessary background concerning branched coverings. Received by the editors November 4, 1974 and, in revised form, December 4, 1974 and April 11, 1975. AMS (MOS) subject classifications (1970). Primary SSA25.

Units and one-sided units in regular rings

Abstract: A ring R is unit regular if for every a E R, there is a unit x E R such that axa = a, and one-sided unit regular if for every a E R, there is a right or left invertible element x E R such that axa = a. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and R = EndA M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation "is isomorphic to a submodule of" (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings. In [1], a ring R with identity was defined to be unit regular if for every a E R there is a unit x E R such that axa = a. The class of all unit regular rings includes [1] all semisimple Artinian rings, all continuous von Neumann rings [7], all strongly regular rings (in particular, all commutative regular rings [3] ). Using the characterization (cf. [6, p. 117] ) of regular group rings as the group rings of locally finite groups, it is easy to show that all regular group rings are unit regular. Unit regular rings are von Neumann finite [4, Proposition 1] and are elementary divisor rings [4, Theorem 3]. Every element of a unit regular ring in which 2 is a unit is equal to the sum of two units [1, Theorem 6]. An example of a regular ring which is not unit regular is the endomorphism ring of an infinite dimensional vector space [1]. In this paper, we define one-sided unit regularity and characterize both unit regularity and one-sided unit regularity within the lattice of principal right ideals of the ring (Theorem 3). For an A-module M with regular endomorphism ring R we prove that R is unit regular if and only if any two isomorphic complemented Presented to the Society, March 19, 1974; received by the editors April 4, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 16A30; Secondary 16A42, 16A48, 16A64, 20K30.

Differential Equations on Closed Subsets of a Banach Space

Abstract: The problem of existence of solutions to the initial value problem x' = f(t, x), x(to) = x0 E F, where f E C[[t0, t0 + al X F, El, F is a locally closed subset of a Banach space E is considered. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunov-like functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general setup.

Coefficient multipliers of bloch functions

Abstract: The class B of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by c/(1 - r) on Iz I < r. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from IP into B and from B into IP. We find the second Kothe dual of B and show its relevance to the multiplier problem. We identify all power series :an,zn such that Ewnanzn is a Bloch function for every choice of the bounded sequence {wnj. Analogous problems for HM spaces are discussed briefly. 1. Introduction. The class of functions f, analytic for Iz I < 1 with f(O) = 0, for which

Isotropic transport process on a Riemannian manifold

Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class C0. The special case of a manifold of negative curvature is considered as an illustration.

A quasi-Anosov diffeomorphism that is not Anosov

Abstract: In this note, we give an example of a diffeomorphism / on a three dimensional manifold M such that /has a property called quasiAnosov but such that / does not have a hyperbolic structure (is not Anosov). Mané has given a method of extending / to a diffeomorphism g on a larger dimensional manifold V such that g has a hyperbolic structure on M as a subset of V. This gives a counterexample to a question of M. Hirsch. M. Hirsch asks in [2], if a diffeomorphism g: V —► V has a compact invariant submanifold M C V with a hyperbolic structure as a subset of V, does it folfollow that g restricted to M is Anosov (has a hyperbolic structure). He proves this is true in certain cases if g has a dense orbit in V. Ricardo MarTé notes that g restricted to M has a property he calls quasi-Anosov [5]. He asks if a quasiAnosov diffeomorphism is always Anosov. C. Robinson [6] gives an example of a quasi-Anosov flow (not a diffeomorphism) that is not Anosov on an eleven dimensional manifold. In this note, we give an example of a quasi-Anosov diffeomorphism / on a three dimensional manifold. (This is the minimal dimension.) Mane gives a method in [5] of embedding our result in a diffeomorphism g of a manifold V such that g has a hyperbolic structure on M. This gives a counterexample to the question of Hirsch as stated above. However, we do not know if g can be constructed so M is contained in the nonwandering set of g. Also, the results of Hirsch [2] show that if our f:M3 —*■ M3 is embedded in g: V—► V so that g has a hyperbolic structure on M and the dimension of Kis four or five, then g cannot have a point with a dense orbit in all of V. 1. Definitions and Theorem. A diffeomorphism is quasi-Anosov if the fact that | Tf\"v\\ is bounded for all n £ Z implies that v = 0. Here Tf is the induced map on tangent vectors of M, v E TM. If A C M is invariant by a diffeomorphism /, we say that /has a hyperbolic structure on A if there are 0 < X < 1, C> 0, and a splitting TM IA = F\" 0 Es such that for n > 0, Received by the editors June 30, 1975. AMS (MOS) subject classifications (1970). Primary 58F15. Í1) Research was partially supported by the National Science Foundation, GP42329X. Copyright © 1976. American Mathematical Society 267 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 268 JOHN FRANKS AND CLARK ROBINSON \\Tf\"us\\ 0 such that f\"(U)C\\ U¥:0. We define the stable and unstable manifolds at all points jc G M by rv\"s(jc) = {y EM: d(fnx, fny) —> 0 as n —> ~} and W(x) ={yEM: d(f\"x, f\"y) -> 0 as n —»• <*>}. If /has a hyperbolic structure on the nonwandering set then these are actually manifolds [3]. If we also assume the periodic points are dense in Í2, then by [4] or [10] each point v of M is on the stable and unstable manifolds of some point in SI. Also {u G TXM: \\ Tfnv\\ is bounded for n < 0} = Tx [W\"(x)] and {v E TXM: | Tf\"v\\ is bounded for n > 0} = Tx [Ws(x)]. See [3]. Therefore under these conditions if Tx [Wu(x)] n Tx [W*(x)] = {0X } for all x in M, then / is quasi-Anosov. See [4] or [10] for more definitions and basic facts of the theory. Theorem. Let M be the connected sum of two copies of the three torus. There is a diffeomorphism fon M that is quasi-Anosov but not Anosov. Remarks. 1. The diffeomorphism/has two hyperbolic invariant subsets and the dimension of the stable bundle on the source is two and on the sink is one. Since the dimension of Es is not constant / cannot be Anosov. However Tx [Wu(x)] n Tx [Ws(x)] = {0X } for all x in M, so /is quasi-Anosov. 2. A quasi-Anosov diffeomorphism on a two dimensional manifold is Anosov, so our example is in the lowest possible dimension. To prove this statement, note that iff is quasi-Anosov on a two manifold then/has a hyperbolic structure on Í2 as approved in [5], [7], or [9]. Since / is quasi-Anosov, all the splitting must be one dimensional (exercise). Since the splittings have constant dimensions on SI [2], [5], and [7] all prove that /has a hyperbolic splitting on all of M. Therefore / is Anosov. 2. Proof of Theorem. We first construct a \"734\" diffeomorphism on Mx = T3 with certain linearity properties near the source. Lemma 1. Let Mx = T3; there exists a diffeomorphism /, : Af, —»• Mx which leaves invariant a one dimensional foliation F and has the following properties: (1) TTiere is a point pEMx which is a source for fx and a neighborhood Uofp with local coordinates xx, x2, x3 which are C except at p and such that License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A QUASI-ANOSOV DIFFEOMORPHISM THAT IS NOT ANOSOV 269 (2) If q, fx(q) E U then x¡(fx(q)) = 2xf(í7); Le., in the x¡ coordinates fx is multiplication by 2. (3) The leaves of the foliation F restricted to U are given by x2(q) = constant, x3(q) = constant. (4) If Aj = n„>o f\"ffli ~ ^)>tnen ^i 's a comPact invariant hyperbolic set whose stable manifolds are the leaves of the foliation F restricted to Mx {p}. Proof. The diffeomorphism fx is a \"DA\" on Mx = T3. This is a wellknown construction in dimension 2 [10] or [11] ; we include an exposition of the dimension 3 DA construction as an appendix for completeness. However for the moment we need only that fx is a perturbation of a hyperbolic toral automorphism which had a two dimensional unstable foliation and one dimensional stable foliation. The perturbation changes a hyperbolic fixed point to a source, but preserves the stable foliation so it remains invariant under fx. If p is the source and Kis any sufficiently small neighborhood of p, then Ax =Ç\\ B>0 fx(Mx V) has a hyperbolic structure and its stable manifolds are the leaves of F (the stable foliation of the hyperbolic toral automorphism) restricted to Mx — {p} (see Appendix for proof). We assume the original hyperbolic toral automorphism g was based on a matrix which has distinct real eigenvalues, one between 0 and 1 and the other two greater than 1. For example, ( 0 0 1 1. \\l -6 5/ We choose coordinates uv u2, u3 on a neighborhood of p in directions parallel to the eigenspaces of g and such that p = (0, 0, 0) in these coordinates. We assume (see Appendix) that fx is constructed to be linear on a neighborhood of p in these coordinates, so that the ux direction, the contracting direction ofg, is an eigendirection for fx with eigenvalue 2, and so that fx =gox\\ the unstable manifold W = {q\\ux(q) = 0}. Let wx, w2 be standard coordinates on R2 and define L: R2 —+ R2 by L(wx, w2) = (2wx, 2w2). Since any two expanding linear maps of R2 are locally conjugate by a homeomorphism which is a C°° diffeomorphism except at the fixed points, we know that if D = {w ER2\\\\w\\<9} there exists yp: D —► W which is C except at 0 and has C°° inverse except at p and satisfies M2 be another copy of the same thing, with one dimensional foliation F'. Let F be an open set on M2 and yx, y2, y3 local coordinates analogous to U and jc,, x2, x3 on Mx, but such that F' is given by yx(q) = const,y2(q) = const. Define | \\x on U and I |a on Fby \\q\\\\ = 23=1 XAjtf and \\q'\\\\ = 23=1 y¡(q')2. UtD¡ = {z G U\\ \\z\\. < 1/8}. We will attach Mx Dx and M2 D2 along a collar neighborhood of these boundaries to form a new manifold M diffeomorphic to the connected sum of two copies of T3. Let Ax = {z E U\\ 1/8 < \\z\\x < 8}and A2 = {z E V\\ 1/8 < \\z\\2 < 8}. We define an attaching diffeomorphismg: A2 —*■ Ax by g( Vp y2, y3) = ÇZy?)~1(yl, y2, y3) in jc,coordinates. Thus g sends the circle of radius r in A2 to the circle of radius l/r in Ax, so the outer boundary of A2 is taken to the inner boundary of Ax and vice versa. Note also that g ° f2~1(z)=flog(z). We will say that zx ~ z2 if zx = g(z2) and define M to be (Mx Dx) U (M2 D2)/~. Then M is a C°° manifold and we define a diffeomorphism /„: M-* M by f0(z) = fx(z) if zEMx-Dx and/0(z) =f2\\z) if zEM2 D2. Notice if z G (Mx Dx)C\\ (M2 D2) = Ax U AJ ~ and if z is the equivalence class of q E A2 and q E A x, then g » ffl (q) = /, (g(q)) = fx (q) so f2 ' (q) ~ fx(q'), and hence /0 is well defined. We will consider the annulus A =AXU A2/~ and use the coordinates xx, x2, x3 which come from Ax. Then if \\z\\2 = \\z\\\\ = Sx,(z)2, we have A = {z|l/8<|z|<8}. There are two one dimensional foliations on A, the restrictions of F on Mx and F' on M2. We will denote these also by F and F'. The foliation F consists of straight lines in the JCjcoordinates but the foliation F' is a more complicated \"dipole\" foliation in these coordinates which will be discussed later. Since there are tangencies of F and F', we want to modify /0 and F' to eliminate these tangencies. Lemma 2. 77zere exists a C°° isotopy htof A such that: (1) h0 = id: A —► A (2) ht(z) = z for all t and all z in a neighborhood of the boundary of A. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A QUASI-ANOSOV DIFFEOMORPHISM THAT IS NOT ANOSOV 271 (3) 7/5 = {z £ A11/4 < |z|<4} then the foliations F and hx(F') are nowhere tangent o

Balanced subgroups of abelian groups

Abstract: The balanced subgroups of Fuchs are generalised to arbitrary abelian groups. Projectives and injectives with respect to general balanced exact sequences are classified; a new class of groups is introduced in order to classify these projectives.

Universally torsionless and trace modules

Abstract: We study, over an arbitrary ring R, a class of right modules intermediate between the projective and the flat content modules. Over the ring of rational integers these modules are the locally free abelian groups. Over any commutative ring they are the modules which remain torsionless under all scalar extensions. They each possess a certain separability property exactly when R is left semihereditary. We define M to be universally torsionless if the natural map M 0 A Hom(M*, A) is monic for all left modules A. We give various equivalent conditions for M to be universally torsionless, one of which is that M is a trace module, i.e. that x E M.M*(x) for all x E M. We show the countably generated such modules are projective. Chase showed that rings over which products of projective or flat modules are also, respectively, projective or flat have other interesting properties and that they are characterized by certain left ideal theoretical conditions. We show similar results hold when the trace or content properties are preserved by products.

Mean convergence of generalized Walsh-Fourier series

Abstract: Paley proved that Walsh-Fourier series converges in I? (1 < p < °°). We generalize Paley's result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.

Topologically defined classes of going-down domains

Abstract: Let R be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on R and its overrings relate to the topological space Spec(R). Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the "D + M construction" is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.

Analysis with weak trace ideals and the number of bound states of Schrödinger operators

Abstract: We discuss interpolation theory for the operator ideals Ip p defined on a separable Hilbert space as those operators A whose singular values

Lie group representations and harmonic polynomials of a matrix variable

Abstract: The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices In this context, a generalization of the socalled "classical separation of variables theorem" for the symmetric algebra is obtained The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups The results of the first part lead to generalizations of several well-known theorems in the theory of group representations The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained Introduction Let Eo = Rn xk and E = Cn xk with k > n Let G SO(k) and G = SO(k,C) The group GL(n, C) operates on E to the left; the group GL(k, C) and its subgroups U(k), G, and Go act linearly on E to the right When n = 1 and k > 2 the following theorems are well known: THEOREM 01 The representations of U(k) and its complexification Received by the editors August 16, 1974 AMS (MOS) subject classifications (1970) Primary 13F20, 22E30, 22E45; Secondary 31C05, 33A45, 43A85

Representations of the ¹-algebra of an inverse semigroup

Abstract: In this paper the star representations on Hubert space of the /'-algebra of an inverse semigroup are studied. It is shown that the set of all irreducible star representations form a separating family for the /'-algebra. Then specific examples of star representations are constructed, and some theory of star representations is developed for the /'-algebra of a number of the most important examples of inverse semigroups. Introduction. Let 5 be a semigroup (as defined in [2, p.l]). If a, b E S, we write ab for the semigroup product of a with b. Let ll(S) be the set of all complex-valued functions fon S such that 11/11, = L l/(a)l<~. aGS lif.gE /'(S), then the convolution product / * g is given by the definition (f*gXc)= Z f(fl)s(b), cES. a,b with ab=c With convolution multiplication and norm II • II1, I1 (S) is a Banach algebra. If a E S, we identify a with the function which takes the value 1 at a and is 0 everywhere else. In this way S is embedded in /'(S). Having made this identification, when /G P(S) we have /= £ f(a)a. A map a —*■ a* of S into S is called an involution on S if (ab)* = b*a* all a, b E S, and (a*)* a all a G S. If S has an involution *, then P(S) has an involution * defined by the rule /* = E /(a)*«*, feHs), a£5 Received by the editors September 12, 1974. AMS (MOS) subject classifications (1970). Primary 43A65; Secondary 43A20.

The semilattice tensor product of distributive lattices

Abstract: We define the tensor product A 3 B for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see 4J, [71, 181) and different from one studied by A. Waterman [121, D. Mowat [9J, and Z. Shmuely 1101. We show that the semilattice A 0 B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of A 0 B and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices. 1. Preliminaries. For terminology and basic results of lattice theory and universal algebra, consult Birkhoff [3] and Gratzer [5], [6]. The join and meet of elements al, . . ., an of a lattice are denoted by I.n 1 ai and H1' l ai respectively. All semilattices considered are join-semilattices. The smallest and largest elements of a lattice, if they exist, are denoted by 0 and 1 respectively. We denote by 2 the two element lattice consisting of 0 and 1. The category of distributive lattices is denoted by V. 2. Existence of the semilattice tensor product. DEFINITION 2.1. Let A, B and C be semilattices. A function f: A x B C is a bihomomorphism if the functions ga: B C defined by g,(b) = f(a, b) and hb :A -+ C defined by hb(a) = f(a, b) are homomorphisms for all a EA and b EB. DEFINITION 2.2. Let A and B be semilattices. A semilattice C is a tensor product of A and B if there is a bihomomorphism f: A x B C such that C is generated by f(A x B) and for any semilattice D and any bihomomorphism g: A x B -+D there is a homomorphism h: C D satisfying g = hf. Note that since f(A x B) generates C, the homomorphism h is necessarily unique. THEOREM 2.3. Let A and B be semilattices. Then a tensor product of A and B exists and is unique up to isomorphism. PROOF. Let K be the free semilattice on A x B and let w be the canonical inclusion map of A x B into K. Let p be the set of all ordered pairs of the Received by the editors November 4, 1974. AMS (MOS) subject classifications (1970). Primary 06A20, 06A35. 183 Copyright

Sobolev Inequalities for Weight Spaces and Supercontractivity

Abstract: For e C 2(R n) with q(x) = aIxII +s for Ixl > xo, a, s > 0, define the measure d,u = exp (-24)dx on Rn. We show that for any k GE Z+ f I f 121 Ig (I f 1)j2sk/(s+ 1) du. 4 c { llD f ll2(dL ) + ll f jjLfjL(d))lS() As a consequence we prove e t\ V: Lq(R , d,u) Lp(R , d,), p, q # 1, 00, is bounded for all t > 0.

Dualities for equational classes of Brouwerian algebras and Heyting algebras

Abstract: This paper focuses on the equational class Sn of Brouwerian algebras and the equational class Ln of Heyting algebras generated by an n-element chain. Firstly, duality theories are developed for these classes. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in Sn and Ln are characterized. Finally, free products and the finitely generated free algebras in Sn and Ln are described. Recently there has been considerable interest in distributive pseudocomplemented lattices, Brouwerian algebras and Heyting algebras. In particular, activity has centered around the equational subclasses ([8], [11], [24], [35], [36]), and steps have been made towards the determination of the injectives, absolute subretracts, free products and free algebras in these classes ([1], [2], [3], [12], [19], [20], [21], [27], [31], [32], [33], [34], [46], [47]). In this work attention is focused upon the equational class Sn of Brouwerian algebras and the equational class Ln of Heyting algebras generated by an n-element chain. Firstly, a duality theory is developed for each of these classes, the dual of an algebra being a Boolean space endowed with a continuous action of the endomorphism monoid of the n-element chain. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in Sn and Ln are characterized. Finally, free products and the finitely generated free algebras in Sn and Ln are described. 1. The categories. Our standard references on category theory, universal algebra, and lattice theory are S. Mac Lane [37], G. Gr'atzer [17], and G. Griatzer [18] respectively; for our general topological requirements we refer to J. Dugundji [13] and for a discussion of Boolean a spaces we call on P. R. Halmos [23]. Received by the editors August 13, 1974. AMS (MOS) subject classifications (1970). Primary 06A35, 18A40, 54H10; Secondary 02C05, 08A10, 08A25, 18C05, 54F05, 54G05.

The generalized Fredholm operators

Abstract: Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if T(X) is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces ( finite-dimensional spaces, weakly compact operators ( > compact operators, generalized Fredholm operators Fredholm operators, Tauberian operators with closed range ( > semi-Fredholm operators. 1. Preliminaries. Let k denote either the real field or the complex field. Let B be the category whose objects are Banach spaces over k and whose morphisms are continuous linear operators T: X -+ Y. As usual, B(X, Y) denotes the set of all continuous linear operators from X to Y. With the norm ITI = supl,1<1IT(x)I, B(X, Y) becomes a Banach space over k. We let B(X, k) = X*, and B(T, k) = T*. Ix denotes the identity operator on X. The sequence of continuous linear operators X S y T z is said to be exact at Y if S(X) = Ker T. The sequence of continuous linear operators TT T Xi1 0 X2 2X X3* yXn nXn+ 1 is exact if it is exact at each Xi (i = 2, 3, . . , n). An exact sequence of the type 0 -+ X -+ Y -+ Z 0 is called a short exact sequence in B; by the open mapping theorem, there exists a closed linear subspace Y1 of Y such that X-Y1 and Z _ Y/Y1, where denotes (and will denote) a topological isomorphism. Exact sequences and diagram lemmas are the main tools employed in this Received by the editors January 10, 1975 and, in revised form, February 28, 1975. AMS (MOS) subject classifications (1970). Primary 47B30, 18A99.

Mean convergence of Fourier series on compact Lie groups

Abstract: The main result is an LP mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semisimple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the LP mean convergence theorem, the theory of the Hilbert transform with weight function is needed. Introduction. A theorem of M. Riesz says that if f is in LP of the circle, 1 < p < oo, and if SNf(x)__Nakeikx is the Nth partial sum of the Fourier series of f, then SN f converges to f in the LP norm as N -* . Pollard [15] proved a similar result for Jacobi polynomials on the interval [1, 1]. If fis inLP([-1, 1]; (1 -x)'(l -x)dx) and if N SN f (X) =E do ', Oa Rk ' (X) k=O is the Nth partial sum of the Jacobi series of f, then SN f converges to f in the LP norm provided 4 ma+ ? 1< <4min o+l 3 +1 Xe+ '2,B + 3, 2 F1 + 1 Here R'' 0 is a normalized Jacobi polynomial and dk' p is an appropriate constant. It is well known that for suitable choices of a, 3 the {R 'kn} are the elementary spherical functions for the rank 1 symmetric spaces of compact type. Cast in this setting Pollard's theorem is an LP mean convergence result for bi-K invariant functions on the rank 1 compact symmetric space U/K. In this paper we investigate extending this result to higher rank symmetric spaces. Even in the abelian case of the n-torus Tn the result depends drastically on how the multiple series is summed. Consider for example f in LP(Tn) with Received by the editors October 3, 1974 and, in revised form, March 12, 1975. AMS (MOS) subject classifications (1970). Primary 43A90, 43A75; Secondary 33A45, 33A75. 61 Copyright @ 1976, American Mathematical Society This content downloaded from 207.46.13.58 on Sat, 17 Sep 2016 05:41:43 UTC All use subject to http://about.jstor.org/terms

Geodesics in piecewise linear manifolds

Abstract: A simplicial complex M is metrized by assigning to each simplex a £ M a linear simplex a* in some Euclidean space Rfc so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric pmThis paper studies geodesies in metric complexes. Let P e M; then the tangent space 7p(M) is canonically isometric to an orthogonal product of cones from P, Rk x i>p(M); once k is as large as possible. vpQA) is called the normal geometry at P in M. Let PX be a tangent direction at P in vp(M). I define numbers k+(PX) and kJ¡PX), called the maximum and minimum curvatures at P in the direction PX. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l n-manifold without boundary. Assume k+(PX) < 0 for all P e M and all PX Ç vp(M). Then M is p.l. isomorphic to R". This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.L n-manifold without boundary. Assume (1) there is a number k ^ 0 such that k_(PX) > k whenever P is in the (n — 2)-skeleton of M and whenever PX Ç pp(M); (2) the Simplexes of M are bounded in size and shape. Then M Is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.