Group extensions and cohomology for locally compact groups. III
Calvin C.
Moore
1-33
Abstract: We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro's lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups $ {H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.
Group extensions and cohomology for locally compact groups. IV
Calvin C.
Moore
35-58
Abstract: In this paper we shall apply the cohomology groups constructed in [14] 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 defined topologies and we find conditions under which this topology is ${T_2}$. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].
Inclusions and noninclusion of spaces of convolution operators
Michael G.
Cowling;
John J. F.
Fournier
59-95
Abstract: Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type $ (p,q)$ by $L_p^q(G)$. It is shown that, if $\vert 1/q - 1/2\vert < \vert 1/p - 1/2\vert$, then $L_q^q(G)$ is not included in $L_p^p(G)$. This result follows from estimates on the norms, in these spaces, of Rudin-Shapiro measures. The same method leads to a simple example of a convolution operator that is of strong type (q, q) for all q in the interval $(p,p')$ but is not of restricted weak type (p, p) or of restricted weak type $ (p',p')$. Other statements about noninclusion among the spaces $L_p^q(G)$ also follow from various assumptions about G. For instance, if G is unimodular, but not compact, $1 \leqslant p,q,r,s \leqslant \infty$, with $p \leqslant q$, and $\min (s,r') < \min (q,p')$, then $ L_p^q(G)$ is not included in $L_r^s(G)$. Using Zafran's multilinear interpolation theorem for the real method, it is shown that, if $ 1 < p < 2$, then there exists a convolution operator on G that is of weak type (p, p) but not of strong type (p, p); it is not known whether such operators exist when $p > 2$, but it is shown that if $p \ne 1,2,\infty$, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.
The regularity of the locally integrable and continuous solutions of nonlinear functional equations
Halina
Światak
97-118
Abstract: The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class ${C^m}$. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class $ {C^m}$ or ${C^\infty }$. Such a property is characteristic for the solutions $ f:{R^n} \to R$ of the equations $\displaystyle \sum\limits_{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),}$ ($ \ast$) where $x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1$ and where the functions ${\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R$ satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating $(\ast)$ and fixing t is of constant strength, hypoelliptic at a point ${x_0}$. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of $(\ast)$, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.
Dualities for equational classes of Brouwerian algebras and Heyting algebras
Brian A.
Davey
119-146
Abstract: This paper focuses on the equational class $ {{\mathbf{S}}_n}$ of Brouwerian algebras and the equational class ${{\mathbf{L}}_n}$ 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 $ {{\mathbf{S}}_n}$ and ${{\mathbf{L}}_n}$ are characterized. Finally, free products and the finitely generated free algebras in $ {{\mathbf{S}}_n}$ and ${{\mathbf{L}}_n}$ are described.
Homomorphisms of commutative cancellative semigroups into nonnegative real numbers
Mohan S.
Putcha;
Takayuki
Tamura
147-157
Abstract: Let S be a commutative cancellative semigroup and ${T_0}$ be a cofinal subsemigroup of S. Let ${h_0}$ be a homomorphism of ${T_0}$ into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi's condition [2] is necessary and sufficient for $ {h_0}$ to be extended to S. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the ``boundedness condition'' is introduced. For further study, several examples are given.
Two extremal problems
Andrew
Harrington;
Marvin
Ortel
159-167
Abstract: The class H consists of all functions f which are analytic in the disk D and satisfy $\displaystyle \int\limits_D {\int {\vert f(z)\vert dx\;dy} < \infty .}$ Linear extremal problems for functionals of the type $\Lambda(f) = \smallint {\smallint _D}f(z)\phi (z)\;dx\;dy,\phi \in {L^\infty }(D),f \in H$, are studied.
Linking forms and maps of odd prime order
J. P.
Alexander;
G. C.
Hamrick;
J. W.
Vick
169-185
Abstract: A differentiable orientation preserving map of odd prime period on a closed oriented differentiable manifold gives rise to two invariants taking values in a Witt group of bilinear forms. One is globally defined in terms of the rational cohomology of the manifold and the other is locally defined in terms of the fixed point set and its normal bundle. We show that these two invariants are, in fact, equal and apply this result to relate the structure of the manifold to that of the fixed point set and the quotient space.
Multipliers of $L\sp p\sb E$. I
Daniel M.
Oberlin
187-198
Abstract: Let X be an abelian group, the character group of a compact group G. For a subset E of X let $ L_E^p$ be the subspace of E-spectral functions in ${L^p}(G)$. We show that if X is infinite and $1 \leqslant p < 2$, then E can be chosen so that not every multiplier of $\widehat{L_E^p}$ extends to a multiplier of $\widehat{{L^p}}(G)$.
Admissible and singular translates of measures on vector spaces
Alan
Gleit;
Joel
Zinn
199-211
Abstract: We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on $D[0,1]$. Further, we show that in many cases convex, balanced, bounded, and complete subsets of the admissible translates are compact. In addition, we generalize Sudakov's theorem on the characterization of certain quasi-invariant sets to separable reflexive spaces which have the Central Limit Property.
Docility at infinity and compactifications of ANR's
R. B.
Sher
213-224
Abstract: Various conditions of contractibility and extensibility at $ \infty$ for locally compact metric spaces are studied. These are shown to be equivalent if the space under consideration is an absolute neighborhood retract (ANR) and an ANR satisfying them is called docile at $\infty$. Docility at $\infty$ is invariant under proper homotopy domination. The ANR X is docile at $ \infty$ if and only if FX (the Freudenthal compactification of X) is an ANR and $FX - X$ is unstable in FX; the inclusion of X into FX is a homotopy equivalence.
Monotonicity, convexity and symmetric derivates
Clifford E.
Weil
225-237
Abstract: If the first lower symmetric derivate of a continuous function is nonnegative, then it is nondecreasing. If the second lower symmetric derivate of a continuous function is nonnegative, then it is convex. In this paper it is shown that if continuity is replaced by Baire one, Darboux in each of these, then the resulting statements are true.
Every weak proper homotopy equivalence is weakly properly homotopic to a proper homotopy equivalence
David A.
Edwards;
Harold M.
Hastings
239-248
Abstract: We prove that every weak proper homotopy equivalence of $ \sigma$-compact, locally compact Hausdorff spaces is weakly properly homotopic to a proper homotopy equivalence.
Erratum to: ``Isolated invariant sets for flows on vector bundles'' (Trans. Amer. Math. Soc. {\bf 203} (1975), 359--390)
James F.
Selgrade
249
Abstract: Let (F, f) be a flow on a vector bundle (E, M). If f is minimal on M, then the exponential growth rate intervals associated with the factors of an invariant splitting of E, corresponding to a Morse decomposition of PE, are disjoint intervals.