Ergodic equivalence relations, cohomology, and von Neumann algebras. I
Jacob
Feldman;
Calvin C.
Moore
289-324
Abstract: Let $(X,\mathcal{B})$ be a standard Borel space, $R \subset X \times X$ an equivalence relation $\in \mathcal{B} \times \mathcal{B}$. Assume each equivalence class is countable. Theorem 1: $ \exists$ a countable group G of Borel isomorphisms of $(X,\mathcal{B})$ so that $R = \{ (x,gx):g \in G\} $. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of ``module over R'' is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let $\alpha ,\beta $ be rationally independent irrationals on the circle $ \mathbb{T}$, and f Borel: $ \mathbb{T} \to \mathbb{T}$. Then $\exists$ Borel $g,h:\mathbb{T} \to \mathbb{T}$ with $f(x) = (g(ax)/g(x))(h(\beta x)/h(x))$ a.e. The notion of ``skew product action'' is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the ``normalized proper range'' of c, defined in terms of the skew action. See also Schmidt [1].
Ergodic equivalence relations, cohomology, and von Neumann algebras. II
Jacob
Feldman;
Calvin C.
Moore
325-359
Abstract: Let R be a Borel equivalence relation with countable equivalence classes, on the standard Borel space $(X,\mathcal{A},\mu )$. Let $\sigma$ be a 2-cohomology class on R with values in the torus $ \mathbb{T}$. We construct a factor von Neumann algebra ${\mathbf{M}}(R,\sigma )$, generalizing the group-measure space construction of Murray and von Neumann [1] and previous generalizations by W. Krieger [1] and G. Zeller-Meier [1]. Very roughly, ${\mathbf{M}}(R,\sigma )$ is a sort of twisted matrix algebra whose elements are matrices $({a_{x,y}})$, where $ (x,y) \in R$. The main result, Theorem 1, is the axiomatization of such factors; any factor M with a regular MASA subalgebra A, and possessing a conditional expectation from M onto A, and isomorphic to $ {\mathbf{M}}(R,\sigma )$ in such a manner that A becomes the ``diagonal matrices"; $ (R,\sigma )$ is uniquely determined by M and A. A number of results are proved, linking invariants and automorphisms of the system (M, A) with those of $(R,\sigma )$. These generalize results of Singer [1] and of Connes [1]. Finally, several results are given which contain fragmentary information about what happens with a single M but two different subalgebras $ {{\mathbf{A}}_1},{{\mathbf{A}}_2}$.
Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables
L. Š.
Grinblat
361-379
Abstract: Let $ {\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any functional f continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions $x(t)(0 \leqslant t \leqslant 1)$ without discontinuities of the second kind. For a set $T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1]$ the metric ${\rho _T}$ is defined on X as in 2.3. The metric ${\rho _T}$ defines on the X the minimal topology in which all functional continuous in Skorohod's metric and also the functional $ x({t_1} - 0),x({t_1}), \ldots ,x({t_n} - 0),x({t_n}), \ldots$ are continuous. We will give necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics ${\rho _T}$ defined in 2.3.
The porous medium equation in one dimension
Barry F.
Knerr
381-415
Abstract: We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time. In this paper we study the lateral boundary, called the interface, of the support $ P[u]$ of the solution in ${R^1} \times (0,T)$. It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near $t = 0$. We conclude by proving some facts about the behavior of the solution in $ P[u]$.
The broken-circuit complex
Tom
Brylawski
417-433
Abstract: The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic?, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex
Cartan subalgebras of simple Lie algebras
Robert Lee
Wilson
435-446
Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let H be a Cartan subalgebra of L, let $L = H + {\Sigma _{\gamma \in \Gamma }}{L_\gamma }$ be the Cartan decomposition of L with respect to H, and let $ \bar H$ be the restricted subalgebra of Der L generated by ad H. Let T denote the maximal torus of $ \bar H$ and I denote the nil radical of $\bar H$. Then $ \bar H = T + I$. Consequently, each $ \gamma \in \Gamma$ is a linear function on H.
General position of equivariant maps
Edward
Bierstone
447-466
Abstract: A natural generic notion of general position for smooth maps which are equivariant with respect to the action of a compact Lie group is introduced. If G is a compact Lie group, and M, N are smooth G-manifolds, then the set of smooth equivariant maps $ F:M \to N$ which are in general position with respect to a closed invariant submanifold P of N, is open and dense in the Whitney topology. The inverse image of P, by an equivariant map in general position, is Whitney stratified. The inverse images, by nearby equivariant maps in general position, are topologically ambient isotopic. In the local context, let V, W be linear G-spaces, and $F:V \to W$ a smooth equivariant map. Let ${F_1}, \ldots ,{F_k}$ be a finite set of homogeneous polynomial generators for the module of smooth equivariant maps, over the ring of smooth invariant functions on V. There are invariant functions ${h_1}, \ldots ,{h_k}$ such that $F = U \circ$ graph h, where graph h is the graph of $h(x) = ({h_1}(x), \ldots ,{h_k}(x))$, and $ U(x,h) = \Sigma _{i = 1}^k{h_i}{F_i}(x)$. The isomorphism class of the real affine algebraic subvariety $(U = 0)$ of $V \times {{\mathbf{R}}^k}$ is uniquely determined (up to product with an affine space) by V, W. F is said to be in general position with respect to $0 \in W$ at $0 \in V$ if graph $h:V \to V \times {{\mathbf{R}}^k}$ is transverse to the minimum Whitney stratification of $ (U = 0)$, at $ x \in V$.
The oscillatory behavior of certain derivatives
Richard J.
O’Malley;
Clifford E.
Weil
467-481
Abstract: The derivatives considered are the approximate derivative and the kth Peano derivative. The main results strengthen the Darboux property, which both of these derivatives possess. Theorem. If the approximate derivative ${f'_{{\text{ap}}}} = f'$ and on which $ f'$ attains both M and -- M. The other main theorem is obtained from this one by replacing the approximate derivative by the kth Peano derivative.
The two-generator subgroups of one-relator groups with torsion
Stephen J.
Pride
483-496
Abstract: The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups, and some additional applications of the techniques are included in the paper. In particular, examples are given to show that the following result of K. Honda is no longer true for infinite groups: if g is a commutator in a finite group G then every generator of ${\text{sgp}}\{ g\}$ is a commutator in G. This confirms a conjecture of B. H. Neumann.
Derivatives of entire functions and a question of P\'olya. II
Simon
Hellerstein;
Jack
Williamson
497-503
Abstract: It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with $ f'$, only real zeros, then $ f''$ has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which $f,f'$, and $f''$ have only real zeros are those in the Laguerre-Pólya class, i.e. $\displaystyle f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}},$ $a \geqslant 0,b,c$ and the ${z_n}$ real, and $\Sigma z_n^{ - 2} < \infty$. This gives a strong affirmative version of an old conjecture of Pólya.
The asymptotic behavior of the first eigenvalue of differential operators degenerating on the boundary
Allen
Devinatz;
Avner
Friedman
505-529
Abstract: When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form $\varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i}$ then results are known regarding the behavior of the principal eigenvalue $\lambda = {\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$. These results are very sharp in case the vector $({b_i})$ has a unique asymptotically stable point in the domain $\omega$ where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of $\omega$ is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of ${\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$ are established.
On the degree of convergence of piecewise polynomial approximation on optimal meshes
H. G.
Burchard
531-559
Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces $ {\mathbf{N}}_0^{p,n}$ (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in ${\mathbf{N}}_0^{p,n}$ we define a derivative ${D^{n,\sigma }}f$ in ${L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}$, which agrees with ${D^n}f$ for smooth f, and prove several properties of $ {D^{n,\sigma }}$.
Compact perturbations of certain von Neumann algebras
Joan K.
Plastiras
561-577
Abstract: Let $\mathcal{E}$ be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space $ \mathcal{H}$. If we denote the commutant of $ \mathcal{E}$ by $ \mathcal{D}(\mathcal{E})$ and the ideal of compact operators on $\mathcal{H}$ by $ \mathcal{C}(\mathcal{H})$, then it is easily verified that $\mathcal{D}(\mathcal{E}) + \mathcal{C}(\mathcal{H}) = \{ T + K:T \in \mathcal{D}(\mathcal{E}),K \in \mathcal{C}(\mathcal{H})\}$ is a ${C^\ast}$-algebra. In this paper we classify all such algebras up to $^\ast$-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.