Homology in varieties of groups. I
C. R.
Leedham-Green
1-14
Abstract: Well-known techniques allow one to construct a (co-) homology theory relative to a variety. After two paragraphs which discuss the modules to be considered and the construction of the (co-) homology groups, we come to our main homological result, namely that the theory is not always equivalent to a Tor or Ext. In the fourth paragraph we prove our main group-theoretic result; two covering groups of a finite group generate the same variety ``up to exponent". Finally we produce a restricted version of the Künneth formula.
Homology in varieties of groups. II
C. R.
Leedham-Green
15-25
Abstract: The study of (co-) homology groups ${\mathfrak{B}_n}(\Pi ,A)$, $ {\mathfrak{B}^n}(\Pi ,A),\mathfrak{B}$ a variety, II a group in $\mathfrak{B}$, and A a suitable II-module, is pursued. They are compared with a certain Tor and Ext. The definition of the homology of an epimorphism due to Rinehart is shown to agree with that due to Barr and Beck (whenever both are defined). The edge effects of a spectral sequence are calculated.
Homology in varieties of groups. III
C. R.
Leedham-Green
27-33
Abstract: A spectral sequence is used to calculate approximately the homology groups ${\mathfrak{B}_2}(\Pi ,Z)$ as defined in the first paper in this series, for $\Pi$ a finitely generated abelian group and $\mathfrak{B}$ the variety of all nilpotent groups of class at most c.
Essential extensions of partial orders on groups
Jorge
Martinez
35-61
Abstract: Let (G, P) be an l-group and $ \mathcal{C}(P)$ be the lattice of convex l-subgroups of (G, P). We say that the l-cone Q is essential over P if $\mathcal{C}(Q)$ is contained in $\mathcal{C}(P)$. It is shown that for each nonzero x in G and each Q-value D of x, there is a P-value C of x containing D and no other Q-value of x. We specialize to those essential extensions for which the above C always depends uniquely on x and D; these are called very essential extensions. We show that if (G, P) is a representable l-group then P is the meet of totally ordered very essential extensions of P. Further we investigate connections between the existence of total very essential extensions and both representability and normal valuedness. We also study the role played by the various radicals in the theory. The same two classes of extensions are treated in the context of abelian Riesz groups. Similar questions about existence of such total orders are dealt with. The main result in this connection is that such total extensions always exist for finite valued pseudo lattice groups, and that the original cone is the meet of them.
Bergman minimal domains in several complex variables
Shigeo
Ozaki;
Sadao
Katô
63-69
Abstract: K. T. Hahn has obtained the inequality between the Jacobians of a biholomorphic mapping and a holomorphic automorphism of a Bergman minimal domain. This paper extends Hahn's result. Some inequalities concerning Jacobians of the mappings of minimal domains onto another minimal domain are considered, and an example is given.
Quasi-disjointness in ergodic theory
Kenneth
Berg
71-87
Abstract: We define and study a relationship, quasi-disjointness, between ergodic processes. A process is a measure-preserving transformation of a measure space onto itself, and ergodicity means that the space cannot be written as a disjoint union of invariant pieces, unless one of the pieces is of zero measure. We restrict our attention to spaces of total measure one which also satisfy additional regularity properties. In particular, the associated Hilbert space of square-summable functions is separable. A simple class of examples is given by translation by a fixed element on a compact Abelian metrizable group, such processes being known as Kronecker processes. We introduce the notion of a maximal common Kronecker factor (or quotient) process for two processes. Quasi-disjointness is a notion tied to the homomorphisms from two processes into their maximal common Kronecker factor, and reduces to a previous notion, disjointness, when that factor is trivial. We show that a substantial class of processes, the Weyl processes, are quasi-disjoint from every ergodic process. As a corollary, we show that a Weyl process and an ergodic process are disjoint if and only if they have no nontrivial Kronecker factor in common, or, equivalently, if they form an ergodic product. We give an example which suggests an analogous theory could be constructed in topological dynamics.
The structure of substitution minimal sets
Ethan M.
Coven;
Michael S.
Keane
89-102
Abstract: Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given.
Fredholm equations on a Hilbert space of analytic functions
Clasine
van Winter
103-139
Abstract: It is shown that the Hardy class $ {\mathfrak{H}^2}$ for the upper half-plane is equal to the set of functions $f[r\exp \,(i\phi )]$ which are analytic in the open half-plane and square-integrable with respect to r for $0 < \phi < \pi$. A function f is in $ {\mathfrak{H}^2}$ if and only if its Mellin transform with respect to r is a constant times $f(t)\exp \,(\phi t - i\phi /2)$, where f must belong to a certain $ {\mathfrak{L}^2}$-space. This result enables f in ${\mathfrak{H}^2}$ to be constructed from its boundary values on the positive real axis. A study is made of a class $ \mathfrak{N}$ consisting of integral operators K on ${\mathfrak{H}^2}$ having kernels $K(r,r',\phi )$ of operators in $\mathfrak{N}$ form a Hardy class ${\mathfrak{H}^2}(2)$ of functions of two variables, one complex and one real. A generalization leads to Hardy classes $ {\mathfrak{H}^2}(n)$ of functions of n variables. On ${\mathfrak{H}^2}(n)$, there is a class of operators $ \mathfrak{N}(n)$ whose kernels form a class $ {\mathfrak{H}^2}(2n)$. This formalism was developed with a view to the n-body problem in quantum mechanics. It is explained that the results on ${\mathfrak{H}^2}(n - 1)$ are instrumental in evaluating quantities which occur in the theory of n-particle scattering.
Spectral concentration and virtual poles. II
James S.
Howland
141-156
Abstract: Spectral concentration at an isolated eigenvalue of finite multiplicity of the selfadjoint operator ${H_\varepsilon } = {T_\varepsilon } + {A_\varepsilon }{B_\varepsilon }$ is shown to arise from a pole of an analytic continuation of ${A_\varepsilon }{({H_\varepsilon } - z)^{ - 1}}{B_\varepsilon }$. An application to quantum mechanical barrier penetration is given.
Constructive polynomial approximation on spheres and projective spaces.
David L.
Ragozin
157-170
Abstract: This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approximation. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the polynomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, G, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a G-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to k-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space.
Some characterizations of the spaces $L\sp{1}(\mu )$
Kenneth L.
Pothoven
171-183
Abstract: Answers are given to the question of when the so-called hom and tensor functors in categories of Banach spaces preserve certain short exact sequences. The answers characterize the spaces of integrable, real-valued functions ${L^1}(\mu )$.
Dirichlet spaces and strong Markov processes
Masatoshi
Fukushima
185-224
Abstract: We show that there exists a suitable strong Markov process on the underlying space of each regular Dirichlet space. Potential theoretic concepts due to A. Beurling and J. Deny are then described in terms of the associated strong Markov process. The proof is carried out by developing potential theory for Dirichlet spaces and symmetric Ray processes and by using a method of transformation of underlying spaces.
On Cartan subalgebras of alternative algebras
D. M.
Foster
225-238
Abstract: In 1966, Jacobson introduced the notion of a Cartan subalgebra for finite-dimensional Jordan algebras with unity over fields of characteristic not 2. Since finite-dimensional Jordan, alternative, and Lie algebras are known to be related through their structure theories, it would seem logical that such an analogue would also exist for finite-dimensional alternative algebras. In this paper, we show that this is the case. Moreover, the linear transformation we define that plays the role in alternative algebras that ``ad ( )'' plays in Lie algebras is identical with that used in the Jordan theory, and can be used in the Lie case as well. Hence we define Cartan subalgebras relative to this linear transformation for finite-dimensional alternative, Jordan, and Lie algebras, and observe that in the Lie case, they coincide with the classical definition of a Cartan subalgebra.
Concerning $n$-mutual aposyndesis in products of continua
Leland E.
Rogers
239-251
Abstract: This paper is concerned with Cartesian products of regular Hausdorff continua and certain conditions on the factors that make the product n-mutually aposyndetic (given n distinct points, there are n disjoint subcontinua, each containing one of the points in its interior). It is proved that the product of any three regular Hausdorff continua is n-mutually aposyndetic for each $n \geqq 2$. Next, certain conditions on factors of products of two continua are shown to be sufficient for the product to be n-mutually aposyndetic. In connection with this, the concepts of n-semiaposyndesis and aposyndetic-terminal points are introduced. Finally, it is proved that the product of a simple closed curve (or any other ``super n-mutually aposyndetic'' continuum) with every compact Hausdorff continuum is n-mutually aposyndetic for each $n \geqq 2$.
Hermitian functionals on $B$-algebras and duality characterizations of $C\sp{\ast} $-algebras
Robert T.
Moore
253-265
Abstract: The hermitian functionals on a unital complex Banach algebra are defined here to be those in the real span of the normalized states (tangent functionals to the unit ball at the identity). It is shown that every functional f in the dual A' of A can be decomposed as $f = h + ik$, where h and k are hermitian functionals. Moreover, this decomposition is unique for every $f \in A'$ iff A admits an involution making it a ${C^\ast}$-algebra, and then the hermitian functionals reduce to the usual real or symmetric functionals. A second characterization of ${C^\ast}$-algebras is given in terms of the separation properties of the hermitian elements of A (real numerical range) as functionals on A'. The possibility of analogous theorems for vector states and matrix element functionals on operator algebras is discussed, and potential applications to the representation theory of locally compact groups are illustrated.
Improbability of collisions in Newtonian gravitational systems
Donald Gene
Saari
267-271
Abstract: It is shown that the set of initial conditions leading to a collision in finite time has measure zero.
The Baer sum functor and algebraic $K$-theory
Irwin S.
Pressman
273-286
Abstract: The Baer sum operation can be described in such a way that it becomes a functorial product on categories of exact sequences of a fixed length. This product is proven to be coherently associative and commutative. The Grothendieck groups and Whitehead groups of some of these categories are computed.
Bonded projections, duality, and multipliers in spaces of analytic functions
A. L.
Shields;
D. L.
Williams
287-302
Abstract: Let $\varphi$ and $\psi$ be positive continuous functions on $ [0,1)$ with $\varphi (r) \to 0$ as $r \to 1$ and $\smallint _0^1\psi (r)\;dr < \infty$. Denote by ${A_0}(\varphi )$ and ${A_\infty }(\varphi )$ the Banach spaces of functions f analytic in the open unit disc D with $ \vert f(z)\vert\varphi (\vert z\vert) = o(1)$ and $\vert f(z)\vert\varphi (\vert z\vert) = O(1),\vert z\vert \to 1$, respectively. In both spaces $\left\Vert f\right\Vert _\varphi = {\sup _D}\vert f(z)\vert\varphi (\vert z\vert)$. Let ${A^1}(\psi )$ denote the space of functions analytic in D with $\left\Vert f\right\Vert _\psi = \smallint {\smallint _D}\vert f(z)\vert\psi (\vert z\vert)\;dx\;dy < \infty$. The spaces ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are identified in the obvious way with closed subspaces of ${C_0}(D),{L^1}(D)$, and ${L^\infty }(D)$, respectively. For a large class of weight functions $ \varphi ,\psi$ which go to zero at least as fast as some power of $(1 - r)$ but no faster than some other power of $(1 - r)$, we exhibit bounded projections from $ {C_0}(D)$ onto ${A_0}(\varphi )$, from ${L^1}(D)$ onto $ {A^1}(\psi )$, and from ${L^\infty }(D)$ onto ${A_\infty }(\varphi )$. Using these projections, we show that the dual of $ {A_0}(\varphi )$ is topologically isomorphic to $ {A^1}(\psi )$ for an appropriate, but not unique choice of $\psi$. In addition, ${A_\infty }(\varphi )$ is topologically isomorphic to the dual of $ {A^1}(\psi )$. As an application of the above, the coefficient multipliers of $ {A_0}(\varphi ),{A^1}(\psi )$, and $ {A_\infty }(\varphi )$ are characterized. Finally, we give an example of a weight function pair $ \varphi ,\psi$ for which some of the above results fail.
Some splitting theorems for algebras over commutative rings
W. C.
Brown
303-315
Abstract: Let R denote a commutative ring with identity and Jacobson radical p. Let $ {\pi _0}:R \to R/p$ denote the natural projection of R onto $R/p$ and $ j:R/p \to R$ a ring homomorphism such that ${\Pi _0}j$ is the identity on $R/p$. We say the pair (R, j) has the splitting property if given any R-algebra A which is faithful, connected and finitely generated as an R-module and has $A/N$ separable over R, then there exists an $(R/p)$-algebra homomorphism $(R[[x]],1)$ with the splitting property. Two examples are given at the end of the paper which show that $R/p$ being integrally closed is necessary but not sufficient to guarantee (R, j) has the splitting property.
Equivariant bordism and Smith theory. II
R. E.
Stong
317-326
Abstract: This paper analyzes the homomorphism from equivariant bordism to Smith homology for spaces with an action of a finite group G.
Subordination principle and distortion theorems on holomorphic mappings in the space $C\sp{n}$
Kyong T.
Hahn
327-336
Abstract: Generalizing the notion of subordination principle in the complex plane to the space of several complex variables, we obtain various distortion theorems on holomorphic mappings of one bounded domain into another in terms of geometrical quantities of the domains and the Bergman metric furnished, thus obtaining a generalization of the Koebe-Faber distortion theorem among others.
Normal operations on quaternionic Hilbert spaces
K.
Viswanath
337-350
Abstract: Simple modifications of standard complex methods are used to obtain a spectral theorem, a functional calculus and a multiplicity theory for normal operators on quaternionic Hilbert spaces. It is shown that the algebra of all operators on a quaternionic Hilbert space is a real $ {C^\ast}$-algebra in which (a) every normal operator is unitarily equivalent to its adjoint and (b) every operator in the double commutant of a hermitian operator is hermitian. Unitary representations of locally compact abelian groups in quaternionic Hilbert spaces are studied and, finally, the complete structure theory of commutative von Neumann algebras on quaternionic Hilbert spaces is worked out.
Automorphisms of Siegel domains
O. S.
Rothaus
351-382
Abstract: This paper studies nonaffine biholomorphisms from one tube domain to a second. A sequel will carry out the same study for arbitrary Siegel domains. With the help of the Bergman kernel function, we can give an explicit form for such biholomorphisms; and with the use of structure theory for Jordan algebras, we can give an algebraic and geometric description of the nature of such tube domains.
Bounds on the ratio $n(r,\,a)/S(r)$ for meromorphic functions
Joseph
Miles
383-393
Abstract: Let f be a meromorphic function in the plane. We prove the existence of an absolute constant K such that if ${a_1},{a_2}, \ldots ,{a_q}$ are distinct elements of the Riemann sphere then $\lim {\inf _{r \to \infty }}\;(\Sigma _{j = 1}^q\vert n(r,{a_j})/S(r) - 1\vert) < K$. We show by example that in general no such bound exists for the corresponding upper limit. These results involving the unintegrated functionals of Nevanlinna theory are related to previous work of Ahlfors, Hayman and Stewart, and the author.
Locally univalent functions with locally univalent derivatives
Douglas Michael
Campbell
395-409
Abstract: S. M. Shah and S. Y. Trimble have discovered that the behavior of an analytic function $f(z)$ is strongly influenced by the radii of univalence of its derivatives ${f^{(n)}}(z)\;(n = 0,1,2, \ldots )$. In this paper many of Shah and Trimble's results are extended to large classes of locally univalent functions with locally univalent derivatives. The work depends on the concept of the $ {\mathcal{U}_\beta }$-radius of a locally univalent function that is introduced and developed in this paper. Ch. Pommerenke's definition of a linear invariant family of locally univalent functions and the techniques of that theory are employed in this paper. It is proved that the universal linear invariant families $ {\mathcal{U}_\alpha }$ are rotationally invariant. For fixed $f(z)$ in ${\mathcal{U}_\alpha }$, it is shown that the function $r \to {\text{order}}\;[f(rz)/r]\;(0 < r \leqq 1)\;$ is a continuous increasing function of r.
Inseparable splitting theory
Richard
Rasala
411-448
Abstract: If L is a purely inseparable field extension of K, we show that, for large enough extensions E of K, the E algebra $L{ \otimes _K}E$ splits to become a truncated polynomial algebra. In fact, there is a unique smallest extension E of K which splits $L/K$ and we call this the splitting field $S(L/K)$ of $L/K$. Now $L \subseteq S(L/K)$ and the extension $S(L/K)$ of K is also purely inseparable. This allows us to repeat the splitting field construction and obtain inductively a tower of fields. We show that the tower stabilizes in a finite number of steps and we study questions such as how soon must the tower stabilize. We also characterize in many ways the case when L is its own splitting field. Finally, we classify all K algebras A which split in a similar way to purely inseparable field extensions.
Martingale convergence to infinitely divisible laws with finite variances
B. M.
Brown;
G. K.
Eagleson
449-453
Abstract: Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely divisible laws with bounded variances, of the row sums of elementary systems of independent random variables.
Symmetrization of distributions and its application
Kuang-ho
Chen
455-471
Abstract: Let P be a polynomial such that k of the $n - 1$ principal curvatures are different from zero at each point of $N(P) = \{ s \in {R^n}:P(s) = 0\} ;N(P)$ is assumed to be nonempty, bounded, and $n - 1$ dimensional. If $ {\text{Supp}}\;\varphi \subset {U^\delta } = \{ s \in {R^n}:\vert P(s)\vert < \delta \}$ with $\delta$ small and $\varphi \in C_c^\infty ({R^n})$, let ${\varphi ^\rho }$ be the integral of $\varphi$ over $N(P - q)$ if $q \in [ - \delta ,\delta ]$ and $ {\varphi ^\sigma }(s) = {\varphi ^\rho }(P(s))$ on $ {U^\delta }$ and $ = 0$ outside ${U^\delta }$. Then ${\varphi ^\sigma } \in C_c^\infty ({R^n})$. We define the symmetrization $ {v^\sigma }$ of a distribution v, with ${\text{Supp}}\;v \subset {U^\delta }$, in a natural way. Setting $u = {\mathcal{F}^{ - 1}}\{ v\}$ and $ {u_0} = {\mathcal{F}^{ - 1}}\{ {v^\sigma }\}$, we prove that ${u_0}$ is the integral of the product of u with some function $w(,)$ which depends only on P. This result is used to prove a Liouville type theorem for entire solutions of $ P( - i{D_x})u(x) = f(x)$, with $ f \in C_c^\infty ({R^n})$.
Errata to ``Mappings onto the plane''
Dix H.
Pettey
473
Erratum to ``Under the degree of some finite linear groups''
Harvey I.
Blau
475