The analytic continuation of the discrete series. I
Nolan R.
Wallach
1-17
Abstract: In this paper the analytic continuation of the holomorphic discrete series is defined. The most elementary properties of these representations are developed. The study of when these representations are unitary is begun.
The analytic continuation of the discrete series. II
Nolan R.
Wallach
19-37
Abstract: This is the second in a series of papers on the analytic continuation of the holomorphic discrete series. In this paper necessary and sufficient conditions for unitarizability are given in the case of line bundles. The foundations for the vector valued case are begun.
Uniformly continuous functionals on the Fourier algebra of any locally compact group
Anthony To Ming
Lau
39-59
Abstract: Let G be any locally compact group. Let $VN\,(G)$ be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace $UBC\mathop {(G)}\limits^ \wedge$ of $ VN\, (G)$ consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian, $UBC\mathop {(G)}\limits^ \wedge$ is precisely the bounded uniformly continuous functions on the dual group Ĝ. We prove among other things that if G is amenable, then the Banach algebra $ UBC\mathop {(G)}\limits^ \wedge {\ast}$ (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore, $UBC\mathop {(G)}\limits^ \wedge {\ast}$ is commutative if and only if G is discrete. We characterize $ W\mathop {(G)}\limits^ \wedge$, the weakly almost periodic functionals, as the largest subspace X of $VN\, (G)$ for which the Arens product makes sense on ${X^ {\ast} }$ and ${X^ {\ast} }$ is commutative. We also show that if G is amenable, then for certain subspaces Y of $VN\, (G)$ which are invariant under the action of the Fourier algebra $A\, (G)$, the algebra of bounded linear operators on Y commuting with the action of $A\, (G)$ is isometric and algebra isomorphic to ${X^ {\ast} }$ for some $X \subseteq UBC(\mathop {G)}\limits^ \wedge$.
Regular points of Lipschitz functions
Alexander D.
Ioffe
61-69
Abstract: Let f be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for f relative to S, and give a sufficient condition for a point $z \, \in \, S$ to be regular. This condition is then expressed in the particular case when f is ${C^1}$, and is used to obtain a new proof of Hoffman's inequality in linear programming.
Stable measures and central limit theorems in spaces of stable type
Michael B.
Marcus;
Wojbor A.
Woyczyński
71-102
Abstract: Let X be a symmetric random variable with values in a quasinormed linear space E. X satisfies the central limit theorem on E with index p, $0 \, < \, p \, \leqslant \, 2$, if $\mathcal{L}{n^{ - 1/p}}({X_1} + \cdots + {{\text{X}}_n}))$ converges weakly to some probability measure on E. Hoffman-Jorgensen and Pisier have shown that Banach spaces of stable type 2 provide a natural environment for the central limit theorem with index $p = 2$. In this paper we show that, for $0 < p < 2$, quasi-normed linear spaces of stable type p provide a natural environment for the central limit theorem with index p. A similar result holds also for the weak law of large numbers with index p.
Canonical subgroups of formal groups
Jonathan
Lubin
103-127
Abstract: Let R be a complete local domain of mixed characteristic. This paper gives a complete answer to the question: ``If F is a one-dimensional formal group over R of finite height, when is there a canonical morphism
On $3$-manifolds that have finite fundamental group and contain Klein bottles
J. H.
Rubinstein
129-137
Abstract: The closed irreducible 3-manifolds with finite fundamental group and containing an embedded Klein bottle can be identified with certain Seifert fibre spaces. We calculate the isotopy classes of homeomorphisms of such 3-manifolds. Also we prove that a free involution acting on a manifold of this type, gives as quotient either a lens space or a manifold in this class. As a corollary it follows that a free action of ${Z_8}$ or a generalized quaternionic group on $ {S^3}$ is equivalent to an orthogonal action.
The transfer and compact Lie groups
Mark
Feshbach
139-169
Abstract: Let G be a compact Lie group with H and K arbitrary closed subgroups. Let BG, BH, BK be l-universal classifying spaces, with $\rho (H,G):BH \to BG$ the natural projection. Then transfer homomorphisms $T(H,G):h(BH) \to h(BG)$ are defined for h an arbitrary cohomology theory. One of the basic properties of the transfer for finite coverings is a double coset formula. This paper proves a double coset theorem in the above more general context, expressing ${\rho ^{\ast}}(K,G) \circ T(H,G)$ as a sum of other compositions. The main theorems were announced in the Bulletin of the American Mathematical Society in May 1977.
On parabolic measures and subparabolic functions
Jang Mei G.
Wu
171-185
Abstract: Let D be a domain in $ R_x^n \, \times \, R_t^1$ and ${\partial _p}D$ be the parabolic boundary of D. Suppose ${\partial _p}D$ is composed of two parts B and S: B is given locally by $t = \tau$ and S is given locally by the graph of ${x_n} = f({x_1},{x_2}, \cdots ,{x_{n - 1}},t)$ where f is Lip 1 with respect to the local space variables and Lip $ \tfrac{1} {2}$ with respect to the universal time variable. Let $\sigma$ be the n-dimensional Hausdorff measure in $ {R^{n + 1}}$ and $ \sigma '$ be the $ (n - 1)$-dimensional Hausdorff measure in $ {\textbf{R}^n}$. And let $E \subseteq {\partial _p}D$. We study (i) the relation between the parabolic measure on ${\partial _p}D$ and the measure dm on ${\partial _p}D$ and (ii) the boundary behavior of subparabolic functions on D.
Complete characterization of functions which act, via superposition, on Sobolev spaces
Moshe
Marcus;
Victor J.
Mizel
187-218
Abstract: Given a domain $\Omega \subset {R_N}$ and a Borel function $h:\,{R_m} \to R$, conditions on h are sought ensuring that for every m-tuple of functions ${u_i}$ belonging to the first order Sobolev space ${W^{1,p}}(\Omega )$, the function $ h({u_1}( \cdot ), \ldots ,{u_m}( \cdot ))$ will belong to a first order Sobolev space $ {W^{1,r}}(\Omega )$, $1 \leqslant r \leqslant p < \infty$.In this paper conditions are found which are both necessary and sufficient in order that h have the above property. This result is based on a characterization obtained here for those Borel functions $g:\,{R_m} \times {({R_N})_m} \to R$ satisfying the requirement that for every m-tuple of functions $ {u_i} \in {W^{1,p}}(\Omega )$ the function $g({u_1}( \cdot ), \ldots ,{u_m}( \cdot ),\nabla {u_1}( \cdot ), \ldots ,\nabla {u_m}( \cdot ))$ belongs to ${L^r}(\Omega )$. A needed result on the measurability of the set of ${R_k}$-Lebesgue points of a function on $ {R_N}$ is presented in an appendix.
The Littlewood-Paley theory for Jacobi expansions
William C.
Connett;
Alan L.
Schwartz
219-234
Abstract: The machinery for harmonic analysis utilizing Jacobi polynomial expansions is developed using the explicit form of the convolution kernel discovered by Gasper. Various maximal functions, and the standard Littlewood-Paley functionals are studied and an application is given to multiplier theorems.
Adjacent connected sums and torus actions
Dennis
McGavran
235-254
Abstract: Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension $ n < m$ with embeddings of $ X\, \times \,{D^{m - n}}$ into M and N. Suppose the interior of $X\, \times \,{D^{m - n}}$ is removed from M and N and the resulting manifolds are attached via a homeomorphism $f:\,X \times \,{S^{m - n - 1}}\, \to \,X\, \times \,{S^{m - n - 1}}$. Let this homeomorphism be of the form $ f(x,\,t)\, = \,(x,\,F(x)(t))$ where $ F:\,X \to \,SO(m - n)$. The resulting manifold, written as $M\,{\char93 _X}\,N$, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus ${T^n}$ on closed, compact, connected, simply connected $(n\, + \,2)$-manifolds, $n \geqslant \,4$.
Sweedler's two-cocycles and generalizations of theorems on Amitsur cohomology
Dave
Riffelmacher
255-265
Abstract: For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by $ {\mathcal{H}^2}(C/k)$, which generalizes Amitsur's second cohomology group ${H^2}(C/k)$. In this paper, if I is a nilpotent ideal of C and $\bar C\, \equiv \,C/I$ is K-projective, a natural bijection $ {\mathcal{H}^2}(C/k)\tilde \to {\mathcal{H}^2}(\bar C{\text{/}}k)$ is established. Also, when $k \subset B$ are fields and C is a commutative B-algebra, the sequence $\{ 1\} \to {H^2}(B{\text{/}}k)\xrightarrow{{{l^{\ast}}}}{H^2}(C/k)\xrightarrow{r}{H^2}(C/B)$ is shown to be exact if the natural map $C{ \otimes _k}C \to C{ \otimes _B}C$ induces a surjection on units, $ {l^ {\ast} }$ is induced by the inclusion, and r is the ``restriction'' map.
The Albanese mapping for a punctual Hilbert scheme. I. Irreducibility of the fibers
Mark E.
Huibregtse
267-285
Abstract: Let $f:\,X \to A$ be the canonical mapping from an algebraic surface X to its Albanese variety A, $X(n)$ the n-fold symmetric product of X, and $H_X^n$ the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length n on X. The latter is a nonsingular and irreducible variety of dimension $2n$, and the ``Hilbert-Chow'' morphism $ {\sigma _n}:\,H_X^n \to X(n)$ is a birational map which desingularizes $ X(n)$. This paper studies the composite morphism $\displaystyle {\varphi _n}:\,H_X^n\xrightarrow{{{\sigma _n}}}X(n)\xrightarrow{{{f_n}}}A ,$ where ${f_n}$ is obtained from f by addition on A. The main result (Part 1 of the paper) is that for $n \gg 0$, all the fibers of ${\varphi _n}$ are irreducible and of dimension $ 2n - q$, where $q = \dim A$. An interesting special case (Part 2 of the paper) arises when $X = A$ is an abelian surface; in this case we show (for example) that the fibers of ${\varphi _n}$ are nonsingular, provided n is prime to the characteristic.
Wall manifolds
R. E.
Stong
287-298
Abstract: In the calculation of the oriented cobordism ring, it is standard to consider so-called Wall manifolds, for which the first Stiefel-Whitney class is the reduction of an integral class. This paper studies the Wall-type structures in the equivariant case.
A pointwise ergodic theorem for the group of rational rotations
Lester E.
Dubins;
Jim
Pitman
299-308
Abstract: Let f be a bounded, measurable function defined on the multiplicative group $\Omega$ of complex numbers of absolute value 1, and define $\displaystyle {{f_n}(\omega ) = \frac{1} {n}\sum\limits_{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,$ ($(1)$) where ${z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $n(k)$ is an increasing sequence of positive integers with $n(k)$ dividing $n(k')$ whenever $k < k'$, then $ {f_{n(k)}}$ converges almost surely as $ k \to \infty$.
An axiom for nonseparable Borel theory
William G.
Fleissner
309-328
Abstract: Kuratowski asked whether the Lebesgue-Hausdorff theorem held for metrizable spaces. A. Stone asked whether a Borel isomorphism between metrizable spaces must be a generalized homeomorphism. The existence of a Q set refutes the generalized Lebesgue-Hausdorff theorem. In this paper we discuss the consequences of the axiom of the title, among which are ``yes'' answers to both Kuratowski's and Stone's questions. The axiom states that a point finite analytic additive family is $\sigma$ discretely decomposable. We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to ${\omega _2}$ using Lévy forcing. Our proof displays relationships between $\sigma$ discretely decomposable families, analytic additive families and d families.
Projective geometries as projective modular lattices
Ralph
Freese
329-342
Abstract: It is shown that the lattice of subspaces of a finite dimensional vector space over a finite prime field is projective in the class of modular lattices provided the dimension is at least 4.
On a sufficient condition for proximity
Ka Sing
Lau
343-356
Abstract: A closed subspace M in a Banach space X is called U-proximinal if it satisfies: $(1 + \rho )S \cap (S + M) \subseteq S + \varepsilon (\rho )(S \cap M)$, for some positive valued function $\varepsilon (\rho )$, $\rho > 0$, and $\varepsilon (\rho ) \to 0$ as $\rho\, \to\, 0$, where S is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are U-proximinal, for example, the subspaces with the 2-ball property (semi M-ideals) and certain subspaces of compact operators in the spaces of bounded linear operators.
On Castelnuovo's inequality for algebraic curves. I
Robert D. M.
Accola
357-373
Abstract: Let ${W_p}$ be a Riemann surface of genus p admitting a simple linear series $g_n^r$ where $ n\, =\, m(r - 1)\, +\, q,\,\, q\, =\, 2,\,3,...,\,r - 1$, or r. Castelnuovo's inequality states that (1) $2p\, \leqslant\, 2f(r,n,1)\, =\, m(m - 1)(r - 1)\, +\, 2m(q - 1)$. By further work of Castelnuovo, equality in (1) and $q\, <\, r$ implies that ${W_p}$ admits a plane model of degree $n\, -\, r\, +\, 2$ with $r\, -\, 2$ m-fold singularities and one $ (n\, -\, r\, +\, 1\, -\, m)$-fold singularity. Formula (1) generalizes as follows. Suppose ${W_p}$ admits s simple linear series $ g_n^r$ where $n\, =\, m(rs - 1)\, +\, q$ and $ q = - (s - 1)r + 2,\, - (s - 1)r + 3,\ldots,r - 1$, or r. For q consider the cases $ v = 0,1,\ldots,s - 1$ as follows: case $v\, =\, 0:2 \leqslant q \leqslant r$, case $ v > 0:2 \leqslant q + vr \leqslant r + 1$. Then (2) $2p \,\leqslant\, 2f(r,\,n,\,s)\, =\, {m^2}(r{s^2}\, -\, s) \,+\, ms(2q \,-\, 1\, -\, r)\, -\, v (v \, -\, 1)r\, -\, 2v (q \,-\, 1)$. Examples show that (2) is sharp. Finally, if $n\, = \,m'r\, + \,q'$,
Jordan rings with nonzero socle
J. Marshall
Osborn;
M. L.
Racine
375-387
Abstract: Let $\mathcal{J}$ be a nondegenerate Jordan algebra over a commutative associative ring $ \Phi$ containing $\tfrac{1}{2}$. Defining the socle $\mathcal{G}$ of $ \mathcal{J}$ to be the sum of all minimal inner ideals of $\mathcal{J}$, we prove that $\mathcal{G}$ is the direct sum of simple ideals of $\mathcal{J}$. Our main result is that if $\mathcal{J}$ is prime with nonzero socle, then either (i) $ \mathcal{J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $\mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $ \mathcal{J}''$ contains $H\, \cap \,S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $\mathcal{J}$ is simple then $\mathcal{J}$ contains a completely primitive idempotent if and only if either $ \mathcal{J}$ is unital and satisfies DCC on principal inner ideals or $\mathcal{J}$ is isomorphic to the Jordan algebra of symmetric elements of a $*$-simple associative algebra A with involution $*$ containing a minimal one-sided ideal.
On invariant operator ranges
E.
Nordgren;
M.
Radjabalipour;
H.
Radjavi;
P.
Rosenthal
389-398
Abstract: A matricial representation is given for the algebra of operators leaving a given dense operator range invariant. It is shown that every operator on an infinite-dimensional Hilbert space has an uncountable family of invariant operator ranges, any two of which intersect only in 0.