The components of the automorphism group of a Jordan algebra
S. Robert
Gordon
1-52
Abstract: Let $\mathfrak{F}$ be a semisimple Jordan algebra over an algebraically closed field $\Phi$ of characteristic zero. Let $ G$ be the automorphism group of $ \mathfrak{F}$ and $ \Gamma$ the structure groups of $ \mathfrak{F}$. General results on $G$ and $\Gamma$ are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over $\Phi$. Specifically, the algebraic components of the linear algebraic groups $G$ and $\Gamma$ are determined, and a formula for the number of components in each case is given. In the course of this investigation, certain Lie algebras and root spaces associated with $ \mathfrak{F}$ are studied. For each component ${G_i}$ of $G$, the index of $G$ is defined to be the minimum dimension of the $ 1$-eigenspace of the automorphisms belonging to ${G_i}$. It is shown that the index of ${G_i}$ is also the minimum dimension of the fixed-point spaces of automorphisms in $ {G_i}$. An element of $ G$ is called regular if the dimension of its $1$-eigenspace is equal to the index of the component to which it belongs. It is proven that an automorphism is regular if and only if its $1$-eigenspace is an associative subalgebra of $ \mathfrak{F}$. A formula for the index of each component ${G_i}$ is given. In the Appendix, a new proof is given of the fact that the set of primitive idempotents of a simple Jordan algebra over $ \Phi$ is an irreducible algebraic set.
A generalization of the Steenrod classification theorem to $H$-spaces.
Byron
Drachman
53-88
On the domain of normality of an attractive fixpoint
P.
Bhattacharyya
89-98
Abstract: It is proved that an entire function of order less than $\frac{1}{2}$ has no unbounded immediate domains of attraction for any of its fixpoints. Estimates for the growth of functions with large infinite domains of attraction (e.g. including half planes) are obtained. It is shown that an entire function mapping an infinite domain into itself has polynomial growth in such domains.
Ordered inverse semigroups
Tôru
Saitô
99-138
Abstract: In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse semigroup is orderable. The solution of the first question is carried out in terms of three types of mappings. Two of these consist of mappings of an $\mathcal{R}$-class onto an $\mathcal{R}$-class, while one of these consists of mappings of a principal ideal of the semilattice $ E$ constituted by idempotents onto a principal ideal of $E$. As for the second question, we give a theorem which extends a well-known result about groups that a group $G$ with the identity $e$ is orderable if and only if there exists a subsemigroup $P$ of $G$ such that $P \cup {P^{ - 1}} = G,P \cap {P^{ - 1}} = \{ e\}$ and $ xP{x^{ - 1}} \subseteqq P$ for every $x \in G$.
A generalized dual for a $C\sp*$-algebra
Herbert
Halpern
139-156
Abstract: Let $\mathcal{A}$ be a $ {C^ \ast }$-algebra, let $\mathcal{B}$ be its enveloping von Neumann algebra, and let $ \mathcal{F}$ be the center of $\mathcal{B} $. Let ${\mathcal{B}_ \sim }$ be the set of all $ \sigma$-weakly continuous $\mathcal{F}$-module homomorphisms of the $\mathcal{F}$-module $ \mathcal{B}$ into $\mathcal{F}$ and let ${\mathcal{A}^ \sim }$ be the set of all restrictions to $\mathcal{A}$ of elements of ${\mathcal{B}_ \sim }$. Then $\mathcal{A}$ is classified as CCR, GCR, and NGCR in terms of certain naturally occurring topologies on $ {\mathcal{A}^ \sim }$.
Nonlinear mappings in locally convex spaces
Terrence S.
McDermott
157-165
Abstract: A notion of local linear approximation is defined for a nonlinear mapping, $f$, defined on one locally convex linear topological space with values in another. By use of this notion, theorems on the local solvability of the equation $y = f(x)$ and on the existence of a local inverse for $f$ are obtained. The continuity and linear approximability of the inverse are discussed. In addition, a relationship between the notion of linear approximation used in the paper and the notion of Fréchet differentiability is shown in the case the intervening spaces are Banach spaces.
Irreducible congruences of prime power degree
C. B.
Hanneken
167-179
Abstract: The number of conjugate sets of irreducible congruences of degree $ m$ belonging to $GF(p),p > 2$, relative to the group $G$ of linear fractional transformations with coefficients belonging to the same field has been determined for $m \leqq 8$. In this paper the irreducible congruences of prime power degree ${q^\alpha },q > 2$, are considered and the number of conjugate sets relative to $G$ is determined.
Principal homogeneous spaces and group scheme extensions
William C.
Waterhouse
181-189
Abstract: Suppose $ G$ is a finite commutative group scheme over a ring $R$. Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for $G$ is isomorphic to $\operatorname{Ext} (G',{G_m})$ vanishes, and from this derives a more general form of Chase's theorem. Our Ext will be in the usual (fpqc) topology, and we show why this gives the same group. We also give an explicit isomorphism and indicate how it is related to the existence of a normal basis.
Classification of generalized Witt algebras over algebraically closed fields
Robert Lee
Wilson
191-210
Abstract: Let $\Phi$ be a field of characteristic $p > 0$ and $m,{n_1}, \ldots ,{n_m}$ be integers $\geqq 1$. A Lie algebra $W(m:{n_1}, \ldots ,{n_m})$ over $ \Phi$ is defined. It is shown that if $\Phi$ is algebraically closed then $W(m:{n_1}, \ldots ,{n_m})$ is isomorphic to a generalized Witt algebra, that every finite-dimensional generalized Witt algebra over $\Phi$ is isomorphic to some $W(m:{n_1}, \ldots ,{n_m})$, and that $W(m:{n_1}, \ldots ,{n_m})$ is isomorphic to $ W(s:{r_1}, \ldots ,{r_s})$ if and only if $m = s$ and ${r_i} = {n_{\sigma (i)}}$ for $1 \leqq i \leqq m$ where $\sigma$ is a permutation of $\{ 1, \ldots ,m\}$. This gives a complete classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The automorphism group of $W(m:{n_1}, \ldots ,{n_m})$ is determined for $ p > 3$.
Approximations and representations for Fourier transforms
Raouf
Doss
211-221
Abstract: $G$ is a locally compact abelian group with dual $\Gamma$. If $p(\gamma ) = \sum\nolimits_1^N {{a_n}({x_n},\gamma )}$ is a trigonometric polynomial, its capacity, by definition is $\Sigma \vert{a_n}\vert$. The main theorem is: Let $ \varphi$ be a measurable function defined on the measurable subset $ \Lambda$ of $ \Gamma$. If $ \varphi$ can be approximated on finite sets in $\Lambda$ by trigonometric polynomials of capacity at most $C$ (constant), then $\varphi = \hat \mu$, locally almost everywhere on $ \Lambda$, where $ \mu$ is a regular bounded measure on $G$ and $ \vert\vert\mu \vert\vert \leqq C$.
Existence and uniqueness of solutions of boundary value problems for two dimensional systems of nonlinear differential equations
Paul
Waltman
223-234
Abstract: The paper considers the nonlinear system $x' = f(t,x,y),y' = g(t,x,y)$ with linear and nonlinear two point boundary conditions. With a Lipschitz condition, an interval of uniqueness for linear boundary conditions is determined using a comparison theorem. A corresponding existence theorem is established. Under the assumption of uniqueness, a general existence theorem is established for quite general nonlinearities in the functions and in the boundary conditions. Examples are provided. The results extend previous work on second order scalar differential equations.
Linear ordinary differential equations with boundary conditions on arbitrary point sets
Michael
Golomb;
Joseph
Jerome
235-264
Abstract: Boundary-value problems for differential operators $\Lambda$ of order $2m$ which are the Euler derivatives of quadratic functionals are considered. The boundary conditions require the solution $F$ to coincide with a given function $f \in {\mathcal{H}_L}(R)$ at the points of an arbitrary closed set $B$, to satisfy at the isolated points of $ B$ the knot conditions of $ 2m$-spline interpolations, and to lie in $ {\mathcal{H}_L}(R)$. Existence of solutions (called ``$\Lambda$-splines knotted on $B$") is proved by consideration of the associated variational problem. The question of uniqueness is treated by decomposing the problem into an equivalent set of problems on the disjoint intervals of the complement of $B'$, where $B'$ denotes the set of limit points of $ B$. It is also shown that $ \Lambda$, considered as an operator from $ {\mathcal{L}_2}(R)$ to $ {\mathcal{L}_2}(R)$), with appropriately restricted domain, has a unique selfadjoint extention $ {\Lambda _B}$ if one postulates that the domain of $ {\Lambda _B}$ contains only functions of $ {\mathcal{H}_L}(R)$ which vanish on $ B.I + {\Lambda _B}$ has a bounded inverse which serves to solve the inhomogeneous equation $\Lambda F = G$ with homogeneous boundary conditions. Approximations to the $ \Lambda$-splines knotted on $B$ are constructed, consisting of $\Lambda $-splines knotted on finite subsets ${B_n}$ of $B$, with $\cup {B_n}$ dense in $B$. These approximations $ {F_n}$ converge to $ F$ in the sense of ${\mathcal{H}_L}(R)$.
Quadratic Jordan algebras and cubing operations
Kevin
McCrimmon
265-278
Abstract: In this paper we show how the Jordan structure can be derived from the squaring and cubing operations in a quadratic Jordan algebra, and give an alternate axiomatization of unital quadratic Jordan algebras in terms of operator identities involving only a single variable. Using this we define nonunital quadratic Jordan algebras and show they can be imbedded in unital algebras. We show that a noncommutative Jordan algebra $ \mathfrak{A}$ (over an arbitrary ring of scalars) determines a quadratic Jordan algebra $ {\mathfrak{A}^ + }$.
Representations of quadratic Jordan algebras
Kevin
McCrimmon
279-305
Abstract: Although representations do not play as much of a role in the theory of Jordan algebras as they do in the associative or Lie theories, they are important in considering Wedderburn splitting theorems and other applications. In this paper we develop a representation theory for quadratic Jordan algebras over an arbitrary ring of scalars, generalizing the usual theory for linear Jordan algebras over a field of characteristic $\ne 2$. We define multiplication algebras and representations, characterize these abstractly as quadratic specializations, and relate them to bimodules. We obtain first and second cohomology groups with the usual properties. We define a universal object for quadratic specializations and show it is finite dimensional for a finite-dimensional algebra. The most important examples of quadratic representations, those obtained from commuting linear representations, are discussed and examples are given of new ``pathological'' representations which arise only in characteristic 2.
Representations of free metabelian $\mathcal{D}_\pi$-groups
John F.
Ledlie
307-346
Abstract: For $\pi$ a set of primes, a ${\mathcal{D}_\pi }$-group is a group $G$ with the property that, for every element $g$ in $G$ and every prime $p$ in $\pi ,g$ has a unique $p$th root in $G$. Two faithful representations of free metabelian $ {\mathcal{D}_\pi }$-groups are established: the first representation is inside a suitable power series algebra and shows that free metabelian $ {\mathcal{D}_\pi }$-groups are residually torsion-free nilpotent; the second is in terms of two-by-two matrices and is analogous to W. Magnus' representation of free metabelian groups using two-by-two matrices. In a subsequent paper [12], these representations will be used to derive several properties of free metabelian ${\mathcal{D}_\pi }$-groups.
$\sigma $-finite invariant measures on infinite product spaces
David G. B.
Hill
347-370
Abstract: A necessary and sufficient condition in terms of Hellinger integrals is established for the existence of a $\sigma $-finite invariant measure on an infinite product space. Using this it is possible to construct a wide class of transformations on the unit interval which have no $\sigma$-finite invariant measure equivalent to Lebesgue measure. This class includes most of the previously known examples of such transformations.
Generalizations of ${\rm QF}-3$ algebras
R. R.
Colby;
E. A.
Rutter
371-386
Abstract: This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation-- QF-$3$ algebras--in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: $ R$ contains a faithful $ \Sigma$-injective left ideal, $R$ contains a faithful $II$-projective injective left ideal; the injective hulls of projective left $R$-modules are projective, and the projective covers of injective left $R$-modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.
On branch loci in Teichm\"uller space
W. J.
Harvey
387-399
Abstract: The branch locus of the ramified covering of the space of moduli of Fuchsian groups with fixed presentation by the corresponding Teichmüller space is decomposed into a union of Teichmüller spaces, each characterised by a description of the action of the conformal self-mappings admitted by the underlying Riemann surfaces. Equivalence classes of subloci under the action of the modular group are studied, and counted in certain simple cases. One may compute as a result the number of conjugacy classes of elements of prime order in the mapping class group of closed surfaces.
Entropy for group endomorphisms and homogeneous spaces
Rufus
Bowen
401-414
Abstract: Topological entropy $ {h_d}(T)$ is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare ${h_d}(T)$ with measure theoretic entropy $ h(T)$; in particular $h(T) = {h_d}(T)$ for Haar measure and affine maps $ T$ on compact metrizable groups. A particular case of this yields the well-known formula for $h(T)$ when $T$ is a toral automorphism.
On higher-dimensional fibered knots
J. J.
Andrews;
D. W.
Sumners
415-426
Abstract: The geometrical properties of a fibration of a knot complement over $ {S^1}$ are used to develop presentations for the homotopy groups as modules over the fundamental group. Some homotopy groups of spun and twist-spun knots are calculated.
Central separable algebras with purely inseparable splitting rings of exponent one
Shuen
Yuan
427-450
Abstract: Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic.
Integer-valued entire functions
Raphael M.
Robinson
451-468
Abstract: The theory of integer-valued entire functions is organized in an improved fashion. Detailed results are proved when the indicator diagram is a line segment. For the first time, a method is developed for treating completely integer-valued functions with an unsymmetrical growth pattern.
Embedding of abelian subgroups in $p$-groups
Marc W.
Konvisser
469-481
Abstract: Research concerning the embedding of abelian subgroups in $p$-groups generally has proceeded in two directions; either considering abelian subgroups of small index (cf. J. L. Alperin, Large abelian subrgoups of $p$-groups, Trans. Amer. Math. Soc. 117 (1965), 10-20) or considering elementary abelian subgroups of small order (cf. B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin, 1967, p. 303). The following new theorems extend these results: Theorem A. Let $G$ be a $p$-group and $M$ a normal subgroup of $G$. (a) If $M$ contains an abelian subgroup of index $ p$, then $M$ contains an abelian subgroup of index $ p$ which is normal in $ G$. (b) If $p \ne 2$ and $M$ contains an abelian subgroup of index $ {p^2}$, then $ M$ contains an abelian subgroup of index ${p^2}$ which is normal in $G$. Theorem B. Let $G$ be a $p$-group, $p \ne 2, M$ a normal subgroup of $ G$, and let $k$ be 2, 3, 4, or 5. If $M$ contains an elementary abelian subgroup of order ${p^k}$, then $M$ contains an elementary abelian subgroup of order ${p^k}$ which is normal in $G$.
The convergence of rational functions of best approximation to the exponential function
E. B.
Saff
483-493
Abstract: The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function $a{e^{yz}}$ on the disk $\vert z\vert \leqq \rho $. It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for $a{e^{yz}}$ on $ \vert z\vert \leqq \rho$ converge uniformly to $a{e^{yz}}$ on each bounded subset of the plane.
Gelfand theory of pseudo differential operators and hypoelliptic operators
Michael E.
Taylor
495-510
Abstract: This paper investigates an algebra $ \mathfrak{A}$ of pseudo differential operators generated by functions $a(x) \in {C^\infty }({R^n}) \cap {L^\infty }({R^n})$ such that $ {D^\alpha }a(x) \to 0$ as $\vert x\vert \to \infty$, if $\vert\alpha \vert \geqq 1$, and by operators $q(D){Q^{ - 1/2}}$ where $q(D) < P(D),Q = I + P{(D)^ \ast }P(D)$, and $ P(D)$ is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative ${C^ \ast }$ algebra $ \mathfrak{A}/J$ is investigated, where $J$ consists of the elements of $\mathfrak{A}$ which are compact. This gives a necessary and sufficient condition for a differential operator $ q(x,D):{\mathfrak{B}_2}_{,\tilde P} \to {L^2}$ to be Fredholm. (Here and in the rest of this paragraph we assume that the coefficients of all operators under consideration satisfy the conditions given on $a(x)$ in the first sentence.) It is also proved that if $p(x,D)$ is a formally selfadjoint operator on $ {R^n}$ which has the same strength as $P(D)$ uniformly on ${R^n}$, then $p(x,D)$ is selfadjoint, with domain $ {\mathfrak{B}_{2,\tilde P}}({R^n})$, and semibounded, if $n \geqq 2$. From this a Gårding type inequality for uniformly strongly formally hypoelliptic operators and a global regularity theorem for uniformly formally hypoelliptic operators are derived. The familiar local regularity theorem is also rederived. It is also proved that a hypoelliptic operator $ p(x,D)$ of constant strength is formally hypoelliptic, in the sense that for any $ {x_0}$, the constant coefficients operator $ p({x_0},D)$ is hypoelliptic.