Finite groups containing an intrinsic $2$-component of Chevalley type over a field of odd order
Morton E.
Harris
1-65
Abstract: This paper extends the celebrated theorem of Aschbacher that classifies all finite simple groups $G$ containing a subgroup $L \cong {\text{SL}}(2,q)$, $ q$ odd, such that $ L$ is subnormal in the centralizer in $G$ of its unique involution. Under the same embedding assumptions, the main result of this work allows $L$ to be almost any Chevalley group over a field of odd order and determines the resulting simple groups $ G$. The results of this paper are an essential ingredient in the current classification of all finite simple groups. Major sections are devoted to deriving various properties of Chevalley groups that are required in the proofs of the three theorems of this paper. These sections are of some independent interest.
Two consequences of determinacy consistent with choice
John R.
Steel;
Robert
Van Wesep
67-85
Abstract: We begin with a ground model satisfying ${\text{ZF}} + {\text{AD}} + {\text{A}}{{\text{C}}_{\mathbf{R}}}$, and from it construct a generic extension satisfying ${\text{ZFC}} + {\mathbf{\delta }}_2^1 = {\omega _2} +$ "the nonstationary ideal on ${\omega _1}$ is $ {\omega _2}$-saturated".
Asymptotic estimates of sums involving the Moebius function. II
Krishnaswami
Alladi
87-105
Abstract: Let $n$ be a positive integer and $ \mu (n)$ the Moebius function. If $n > 1$, let $P(n)$ denote its largest prime factor and put $ P(1) = 1$. We study the asymptotic behavior of the sum ${M^ \ast }(x,y) = \sum\nolimits_{1 \leqslant n \leqslant x,P(n) < y} {\mu (n)}$ as $x,y \to \infty$ and discuss a few applications.
A projective description of weighted inductive limits
Klaus-D.
Bierstedt;
Reinhold
Meise;
William H.
Summers
107-160
Abstract: Considering countable locally convex inductive limits of weighted spaces of continuous functions, if $\mathcal{V} = {\{ {V_n}\} _n}$ is a decreasing sequence of systems of weights on a locally compact Hausdorff space $X$, we prove that the topology of ${\mathcal{V}_0}C(X) = {\text{in}}{{\text{d}}_{n \to }}C{({V_n})_0}(X)$ can always be described by an associated system $\overline V = {\overline V _\mathcal{V}}$ of weights on $X$; the continuous seminorms on ${\mathcal{V}_0}C(X)$ are characterized as weighted supremum norms. If $\mathcal{V} = {\{ {\upsilon _n}\} _n}$ is a sequence of continuous weights on $X$, a condition is derived in terms of $\mathcal{V}$ which is both necessary and sufficient for the completeness (respectively, regularity) of the $(LB)$-space $ {\mathcal{V}_0}C(X)$, and which is also equivalent to ${\mathcal{V}_0}C(X)$ agreeing algebraically and topologically with the associated weighted space $C{\overline V _0}(X)$; for sequence spaces, this condition is the same as requiring that the corresponding echelon space be quasi-normable. A number of consequences follow. As our main application, in the case of weighted inductive limits of holomorphic functions, we obtain, using purely functional analytic methods, a considerable extension of a theorem due to B. A. Taylor [37] which is useful in connection with analytically uniform spaces and convolution equations. The projective description of weighted inductive limits also serves to improve upon existing tensor and slice product representations. Most of our work is in the context of spaces of scalar or Banach space valued functions, but, additionally, some results for spaces of functions with range in certain $ (LB)$-spaces are mentioned.
Level sets of derivatives
David
Preiss
161-184
Abstract: The main result of the paper is the characterization of those triples $S$, $G$ and $E$ of subsets of the reals for which there exists an everywhere differentiable real-valued function $f$ of one real variable such that $({\text{Z))}}$ introduced in the paper. The main result leads to a complete description of the structure of the sets
Homotopy in functor categories
Alex
Heller
185-202
Abstract: If ${\mathbf{C}}$ is a small category enriched over topological spaces the category $ {\mathcal{J}^{\mathbf{C}}}$ of continuous functors from ${\mathbf{C}}$ into topological spaces admits a family of homotopy theories associated with closed subcategories of $ {\mathbf{C}}$. The categories $ {\mathcal{J}^{\mathbf{C}}}$, for various $ {\mathbf{C}}$, are connected to one another by a functor calculus analogous to the $\otimes$, Hom calculus for modules over rings. The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories. Among the functors so described are homotopy limits and colimits and, more generally, homotopy Kan extensions. A by-product of the method is a generalization to functor categories of E. H. Brown's representability theorem.
Holomorphic curves in Lorentzian CR-manifolds
Robert L.
Bryant
203-221
Abstract: A CR-manifold is said to be Lorentzian if its Levi form has one negative eigenvalue and the rest positive. In this case, it is possible that the CR-manifold contains holomorphic curves. In this paper, necessary and sufficient conditions are derived (in terms of the "derivatives" of the CR-structure) in order that holomorphic curves exist. A "flatness" theorem is proven characterizing the real Lorentzian hyperquadric ${Q_5} \subseteq {\mathbf{C}}{P^3}$, and examples are given showing that nonflat Lorentzian hyperquadrics can have a richer family of holomorphic curves than the flat ones.
Nonseparability of quotient spaces of function algebras on topological semigroups
Heneri A. M.
Dzinotyiweyi
223-235
Abstract: Let $S$ be a topological semigroup, $ C(S)$ the space of all bounded real-valued continuous functions on $S$. We define $WUC(S)$ the subspace of $C(S)$ consisting of all weakly uniformly continuous functions and $WAP(S)$ the space of all weakly almost periodic functions in $C(S)$. Among other results, for a large class of topological semigroups $S$, for which noncompact locally compact topological groups are a very special case, we prove that the quotient spaces $ WUC(S)/WAP(S)$ and, for nondiscrete $S$, $ C(S)/WUC(S)$ are nonseparable. (The actual setting of these results is more general.) For locally compact topological groups, parts of our results answer affirmatively certain questions raised earlier by Ching Chou and E. E. Granirer.
Simplexes of extensions of states of $C\sp{\ast} $-algebras
C. J. K.
Batty
237-246
Abstract: Let $B$ be a ${C^\ast}$-subalgebra of a ${C^\ast}$-algebra $A$, $F$ a compact face of the state space $S(B)$ of $B$, and ${S_F}(A)$ the set of all states of $A$ whose restrictions to $ B$ lie in $F$. It is shown that ${S_F}(A)$ is a Choquet simplex if and only if (a) $F$ is a simplex, (b) pure states in $ {S_F}(A)$ restrict to pure states in $F$, and (c) the states of $A$ which restrict to any given pure state in $ F$ form a simplex. The properties (b) and (c) are also considered in isolation. Sets of the form ${S_F}(A)$ have been considered by various authors in special cases including those where $B$ is a maximal abelian subalgebra of $A$, or $A$ is a ${C^\ast}$-crossed product, or the Cuntz algebra $ {\mathcal{O}_n}$.
Generalized Lefschetz numbers
S. Y.
Husseini
247-274
Abstract: Given $ [C;f]$, where $ C$ is a finitely-generated $ \pi$-projective chain complex, and $f:C \to C{\text{a(}}\pi {\text{,}}\varphi {\text{)}}$-chain map, with $\varphi :\pi \to \pi$ being a homomorphism, then the generalized Lefschetz number ${L_{(\pi ,\varphi )}}[C;f]$ of $ [C;f]$ is defined as the alternating sum of the $ (\pi ,\varphi )$-Reidemeister trace of $f$. In analogy with the ordinary Lefschetz number, $ {L_{(\pi ,\varphi )}}[C;f]$ is shown to satisfy the commutative property and to be invariant under $ (\pi ,\varphi )$-chain homotopy. Also, when ${H_\ast}C$ is $\pi$-projective, $\displaystyle {L_{(\pi ,\varphi )}}[C;f] = {L_{(\pi ,\varphi )}}[{H_\ast}C;{H_\ast}f]$ If ${[\alpha ;\pi ]_\varphi }$ is essential. If $(\pi ,\varphi )$-classes of $ f:C \to C$. This is expressed as a decomposition of ${L_{(\pi ,\varphi )}}[C;f]$ in terms of $ {L_{(\pi ',{\varphi _\xi })}}[C';{f_\xi }]$ where $f( \cdot ){\xi ^{ - 1}} = {f_\xi }( \cdot )$ and ${\varphi _\xi }( \cdot ) = \xi \varphi ( \cdot ){\xi ^{ - 1}}$. The algebraic theory is applied to the Nielsen theory of a map $f:X \to X$, where $X$ is a finite CW-complex relative to a regular cover $\tilde X \to X$. One can define a generalized Lefschetz number ${L_{(\pi ,\varphi )}}$ using any cellular approximation to $f$, where $\pi$ is the group of covering transformations of $\tilde X \to X$. The quantity ${L_{(\pi ,\varphi )}}$ can be expressed naturally as a formal sum in the $\pi$-Nielsen classes of $f$ with their indices appearing as coefficients. From this expression, one is able to deduce from the properties of the generalized Lefschetz number the usual results of the relative Nielsen theory.
Metrically complete regular rings
K. R.
Goodearl
275-310
Abstract: This paper is concerned with the structure of those (von Neumann) regular rings $R$ which are complete with respect to the weakest metric derived from the pseudo-rank functions on $ R$, known as the ${N^ \ast }$-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all $ {\aleph _0}$-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group $ {K_0}(R)$, which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of $R$. For instance, it is proved that the simple homomorphic images of $R$ are right and left self-injective rings, and $ R$ is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective $R$-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of $ R$. As another example of the results derived, it is proved that if all simple artinian homomorphic images of $R$ are $n \times n$ matrix rings (for some fixed positive integer $n$), then $R$ is an $n \times n$ matrix ring.
A fake topological Hilbert space
R. D.
Anderson;
D. W.
Curtis;
J.
van Mill
311-321
Abstract: We give an example of a topologically complete separable metric AR space $ X$ which is not homeomorphic to the Hilbert space ${l^2}$, but which has the following properties: (i) $X$ imbeds as a convex subset of ${l^2}$ (ii) every compact subset of $ X$ is a $Z$-set; (iii) $X \times X \approx {l^2};$ (iv) $ X$ is homogeneous; (v) $ X \approx X\backslash G$ for every countable subset $G$.
Analysis of spectral variation and some inequalities
Rajendra
Bhatia
323-331
Abstract: A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if $ U$, $V$ are unitary matrices and $K$ is a skew-Hermitian matrix such that $U{V^{ - 1}} = \exp K$, then for every unitary-invariant norm the distance between the eigenvalues of $U$ and those of $V$ is bounded by $\vert\vert K\vert\vert$. This generalises two earlier results which used particular unitary-invariant norms.
The restriction of admissible modules to parabolic subalgebras
J. T.
Stafford;
N. R.
Wallach
333-350
Abstract: This paper studies algebraic versions of Casselman's subrepresentation theorem. Let $ \mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field $ F$ of characteristic zero and $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ be an Iwasawa decomposition for $\mathfrak{g}$. Then $ (\mathfrak{g},\mathfrak{k})$ is said to satisfy property $(\mathfrak{n})$ if $M \ne M$ for every admissible $ (\mathfrak{g},\mathfrak{k})$-module $M$. We prove that, if $ (\mathfrak{g},\mathfrak{k})$ satisfies property $ (\mathfrak{n})$, then $\mathfrak{n}N \ne N$ whenever $ N$ is a $ (\mathfrak{g},\mathfrak{k})$-module with $\dim N < \operatorname{card} F$. This is then used to show (purely algebraically) that $ (\mathfrak{s}l(n,F),\mathfrak{s}o(n,F))$ satisfies property $(\mathfrak{n})$. The subrepresentation theorem for $ \mathfrak{s}l(n)$ is an easy consequence of this.
Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of ${\bf C}\sp{n}$
Al
Boggess
351-374
Abstract: The relationship between the Levi geometry of a submanifold of ${{\mathbf{C}}^n}$ and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of $ {{\mathbf{C}}^n}$, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in ${{\mathbf{C}}^n}$. This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of $ {{\mathbf{C}}^n}$. In fact, we show that if $S$ is a real analytic, generic, submanifold of ${{\mathbf{C}}^n}$ (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then $S$ is not extendible to any open set in ${{\mathbf{C}}^n}$.
A new solution to the word problem in the fundamental groups of alternating knots and links
Mark J.
Dugopolski
375-382
Abstract: A new solution to the word problem for alternating knots and links is given. The solution is based on Waldhausen's algorithm, but is greatly simplified.