Algebraic results on representations of semisimple Lie groups
J.
Lepowsky
1-44
Abstract: Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let $ \mathfrak{g}$ and $\mathfrak{k}$ denote the respective complexified Lie algebras. Then every irreducible representation $ \pi$ of $\mathfrak{g}$ which is semisimple under $\mathfrak{k}$ and whose irreducible $\mathfrak{k}$-components integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of $ \mathfrak{g}$ belonging to the infinitesimal nonunitary principal series. It follows that $\pi$ integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of $\mathfrak{k}$ in $\pi$ is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of $\mathfrak{g}$ in the infinitesimal nonunitary principal series, as well as certain more general representations of $ \mathfrak{g}$ on which the center of the universal enveloping algebra of $\mathfrak{g}$ acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.
On the determination of irreducible modules by restriction to a subalgebra
J.
Lepowsky;
G. W.
McCollum
45-57
Abstract: Let $\mathcal{B}$ be an algebra over a field, $\mathcal{A}$ a subalgebra of $\mathcal{B}$, and $\alpha$ an equivalence class of finite dimensional irreducible $ \mathcal{A}$-modules. Under certain restrictions, bijections are established between the set of equivalence classes of irreducible $ \mathcal{B}$-modules containing a nonzero $\alpha$-primary $ \mathcal{A}$-submodule, and the sets of equivalence classes of all irreducible modules of certain canonically constructed algebras. Related results had been obtained by Harish-Chandra and R. Godement in special cases. The general methods and results appear to be useful in the representation theory of semisimple Lie groups.
Hankel transforms and GASP
Stanton
Philipp
59-72
Abstract: The inversion of the classical Hankel transform is considered from three viewpoints. The first approach is direct, and a theorem is given which allows inversion in the (C, 1) sense under fairly weak hypotheses. The second approach is via Abel summability, and it is shown that inversion is possible if it is known that the Hankel transform is Abel summable and if certain critical growth conditions are satisfied. The third approach rests on the observation that Abel means of Hankel transforms satisfy a variant of the GASP equation in two arguments. In this setting the inversion problem becomes a boundary value problem for GASP in a quadrant of the plane with boundary values on one of the axes; a uniqueness theorem for this problem is proved which is best possible in several respects.
Free vector lattices
Roger D.
Bleier
73-87
Abstract: An investigation into the algebraic properties of free objects in the category of vector lattices is carried out. It is shown that each ideal of a free vector lattice is a cardinal (direct) sum of indecomposable ideals, and that there are no nonzero proper characteristic ideals. Questions concerning injective and surjective endomorphisms are answered. Moreover, for finitely generated free vector lattices it is shown that the maximal ideals are precisely those which are both prime and principal. These results are preceded by an efficient review of the known properties of free vector lattices. The applicability of the theory to abelian lattice-ordered groups is discussed in a brief appendix.
Bibasic sequences and norming basic sequences
William J.
Davis;
David W.
Dean;
Bor Luh
Lin
89-102
Abstract: It is shown that every infinite dimensional Banach space X contains a basic sequence $({x_n})$ having biorthogonal functionals $({f_n}) \subset {X^\ast}$ such that $ ({f_n})$ is also basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ ({f_n})$ is necessarily basic. If $[{f_n}]$ norms $[{x_n}]$ then $[{x_n}]$ norms $[{f_n}]$. In order that $[{f_n}]$ norms $[{x_n}]$ it is necessary and sufficient that the operators ${S_n}x = \Sigma _1^n{f_i}(x){x_i}$ be uniformly bounded. If $[{f_n}]$ norms $[{x_n}]$ then ${X^\ast}$ has a complemented subspace isomorphic to $ {[{x_n}]^\ast}$. Examples are given to show that $({f_n})$ need not be basic and, if $({f_n})$ is basic, still $[{f_n}]$ need not norm $[{x_n}]$.
On weighted norm inequalities for the Lusin area integral
Carlos
Segovia;
Richard L.
Wheeden
103-123
Abstract: It is shown that the Lusin area integral for the unit circle is a bounded operator on any weighted ${L^p}$ space, $1 < p < \infty$, on which the conjugate function is a bounded operator. Results are also proved for the case $0 < p \leq 1$.
Codominant dimension of rings and modules
Gary L.
Eerkes
125-139
Abstract: Expanding Nakayama's original concept of dominant dimension, Tachikawa, Müller and Kato have obtained a number of results pertaining to finite dimensional algebras and more generally, rings and their modules. The purpose of this paper is to introduce and examine a categorically dual notion, namely, codominant dimension. Special attention is given to the question of the relation between the codominant and dominant dimensions of a ring. In particular, we show that the two dimensions are equivalent for artinian rings. This follows from our main result that for a left perfect ring R the dominant dimension of each projective left R-module is greater than or equal to n if and only if the codominant dimension of each injective left R-module is greater than or equal to n. Finally, for computations, we consider generalized uniserial rings and show that the codominant dimension, or equivalently, dominant dimension, is a strict function of the ring's Kupisch sequence.
Initial-boundary value problems for hyperbolic systems in regions with corners. I
Stanley
Osher
141-164
Abstract: In recent papers Kreiss and others have shown that initial-boundary value problems for strictly hyperbolic systems in regions with smooth boundaries are well-posed under uniform Lopatinskiĭ conditions. In the present paper the author obtains new conditions which are necessary for existence and sufficient for uniqueness and for certain energy estimates to be valid for such equations in regions with corners. The key tool is the construction of a symmetrizer which satisfies an operator valued differential equation.
On manifolds with the homotopy type of complex projective space
Bruce
Conrad
165-180
Abstract: It is known that in every even dimension greater than four there are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of complex projective space. In this paper we provide an explicit construction of homotopy complex projective spaces. Our initial data will be a manifold X with the homotopy type of $ {\mathbf{C}}{{\mathbf{P}}^3}$ and an embedding ${\gamma _3}:{S^5} \to {S^7}$ . A homotopy 7-sphere ${\Sigma ^7}$ is constructed and an embedding ${\gamma _4}:{\Sigma ^7} \to {S^9}$ may be chosen. The procedure continues inductively until either an obstruction or the desired dimension is reached; in the latter case the final obstruction is the class of ${\Sigma ^{2n - 1}}$ in ${\Theta _{2n - 1}}$. Should this obstruction vanish, the final choice is of a diffeomorphism $ {\gamma _n}:{\Sigma ^{2n - 1}} \to {S^{2n - 1}}$. There results a manifold, denoted $(X,{\gamma _3}, \cdots ,{\gamma _{n - 1}},{\gamma _n})$, with the homotopy type of $ {\mathbf{C}}{{\mathbf{P}}^n}$. We describe the obstructions encountered, but are able to evaluate only the primary ones. It is shown that every homotopy complex projective space may be so constructed, and in terms of this construction, necessary and sufficient conditions for two homotopy complex projective spaces to be diffeomorphic are stated.
Groups whose homomorphic images have a transitive normality relation
Derek J. S.
Robinson
181-213
Abstract: A group G is a T-group if $H \triangleleft K \triangleleft G$ implies that $H \triangleleft G$, i.e. normality is transitive. A just non-T-group (JNT-group) is a group which is not a T-group but all of whose proper homomorphic images are T-groups. In this paper all soluble JNT-groups are classified; it turns out that these fall into nine distinct classes. In addition all soluble $ JN\bar T$-groups and all finite $JN\bar T$-groups are determined; here a group G is a $\bar T$-group if $H \triangleleft K \triangleleft L \leq G$ implies that $H \triangleleft L$. It is also shown that a finitely generated soluble group which is not a T-group has a finite homomorphic image which is not a T-group.
Inverse limits on graphs and monotone mappings
J. W.
Rogers
215-225
Abstract: In 1935, Knaster gave an example of an irreducible continuum (i.e. compact connected metric space) K which can be mapped onto an arc so that each point-preimage is an arc. The continuum K is chainable (or arc-like). In this paper it is shown that every one-dimensional continuum M is a continuous image, with arcs as point-preimages, of some one-dimensional continuum $ M'$. Moreover, if M is G-like, for some collection G of graphs, then $M'$ can be chosen to be G-like. A corollary is that every chainable continuum is a continuous image, with arcs as point-inverses, of a chainable (and hence, by a theorem of Bing, planar) continuum. These investigations give rise to the study of certain special types of inverse limit sequences on graphs.
Weighted norm inequalities for the conjugate function and Hilbert transform
Richard
Hunt;
Benjamin
Muckenhoupt;
Richard
Wheeden
227-251
Abstract: The principal problem considered is the determination of all non-negative functions $W(x)$ with period $2\pi$ such that $\displaystyle \int_{ - \pi }^\pi {\vert\tilde f(\theta ){\vert^p}W(\theta )\;d\... ...a \leq C} \;\int_{ - \pi }^\pi {\vert f(\theta ){\vert^p}W(\theta )\;d\theta }$ where $1 < p < \infty$, f has period $2\pi$, C is a constant independent of f, and $\tilde f$ is the conjugate function defined by $\displaystyle \tilde f(\theta ) = \mathop {\lim }\limits_{\varepsilon \to {0^ +... ...rt\phi \vert \leq \pi } {\frac{{f(\theta - \phi )\;d\phi }}{{2\tan \phi /2}}.}$ The main result is that $W(x)$ is such a function if and only if $\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {W(\theta )\;d\theta } } \... ...ert}}\int_I {{{[W(\theta )]}^{ - 1/(p - 1)}}d\theta } } \right]^{p - 1}} \leq K$ where I is any interval, $ \vert I\vert$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the nonperiodic case, the discrete case, an application to weighted mean convergence of Fourier series, and an estimate for one of the functions in the Fefferman and Stein decomposition of functions of bounded mean oscillation.
The modulus of the boundary values of bounded analytic functions of several variables
Chester Alan
Jacewicz
253-261
Abstract: One necessary condition and one sufficient condition are given in order that a nonnegative function be the modulus of the boundary values of a bounded analytic function on the polydisc. As a consequence, a weak version of a theorem of F. Riesz is generalized to several variables. For special classes of functions several conditions are given which are equivalent to a function's being the modulus of the boundary values of a bounded analytic function. Finally, an algebraic structure is provided for these special classes of functions.
On the asymptotic representation of analytic solutions of first-order algebraic differential equations in sectors
Steven
Bank
263-283
Abstract: In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field. (Our results include as a special case, the case when the coefficients are rational functions.) In our main result, we obtain precise asymptotic representations for a broad class of solutions of such equations.
A necessary and sufficient condition for a ``sphere'' to separate points in euclidean, hyperbolic, or spherical space
J. E.
Valentine;
S. G.
Wayment
285-295
Abstract: The purpose of this paper is to give conditions wholly and explicitly in terms of the mutual distances of $n + 3$ points in n-space which are necessary and sufficient for two of the points to lie in the same or different components of the space determined by the sphere which is determined by $n + 1$ of the points. Thus in euclidean space we prove that if the cofactor $[{p_i}{p_j}^2]$ of the element ${p_i}{p_j}^2\;(i \ne j)$ in the determinant $ \vert{p_i}{p_j}^2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${E_n} - \Omega $ (where $\Omega$ denotes the sphere or hyperplane containing the remaining $n + 1$ points) if and only if $\operatorname{sgn} [{p_i}{p_j}^2] = {( - 1)^n}$ or ${( - 1)^{n + 1}}$, respectively. In hyperbolic space the result is: if the cofactor $[{\sinh ^2}\;{p_i}{p_j}/2]$ of the element ${\sinh ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $\vert{\sinh ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 1)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${H_n} - \Omega $ (where $\Omega$ denotes the hyperplane, sphere, horosphere, or one branch of an equidistant surface containing the remaining $n + 1$ points) if and only if $\operatorname{sgn} [{\sinh ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or $ {( - 1)^{n + 1}}$, respectively. For spherical space we obtain: if the cofactor $ [{\sin ^2}\;{p_i}{p_j}/2]$ of the element ${\sin ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $\vert{\sin ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${S_n} - \Omega $ (where $\Omega$ denotes the sphere containing the remaining $n + 1$ points which may be an $(n - 1)$ dimensional subspace) if and only if $ \operatorname{sgn} [{\sin ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or ${( - 1)^{n + 1}}$ respectively.
Locally $B\sp{\ast} $-equivalent algebras. II
Bruce A.
Barnes
297-303
Abstract: Let A be a locally ${B^\ast}$-equivalent Banach $ ^\ast$-algebra. Then A possesses a unique norm $\vert \cdot \vert$ with the property that $\vert{a^\ast}a\vert = \vert a{\vert^2}$ for all $a \in A$. Let B be the ${B^\ast}$-algebra which is the completion of A in the norm $\vert \cdot \vert$. In this paper it is shown that there exists a closed ${B^\ast}$-equivalent $^\ast$-ideal of A which contains the maximal GCR ideal of B. In particular, when B is a GCR algebra, then $A = B$.
On monotone matrix functions of two variables
Harkrishan
Vasudeva
305-318
Abstract: The theory of monotone matrix functions has been developed by K. Loewner; he first gives some necessary and sufficient conditions for a function to be a monotone matrix function of order n, and then, as a result of further deep investigations including questions of interpolation he arrives at the following criterion: A real-valued function $f(x)$ defined in (a, b) is monotone of arbitrary high order n if and only if it is analytic in (a, b), can be analytically continued onto the entire upper half-plane, and has there a nonnegative imaginary part. The problem of monotone operator functions of two real variables has recently been considered by A. Koranyi. He has generalized Loewner's theorem on monotone matrix functions of arbitrary high order n to two variables. We seek a theory of monotone matrix functions of two variables analogous to that developed by Loewner and show that a complete analogue to Loewner's theory exists in two dimensions.
The support of Mikusi\'nski operators
Thomas K.
Boehme
319-334
Abstract: A class of Mikusiński operators, called regular operators, is studied. The class of regular operators is strictly smaller than the class of all operators, and strictly larger than the class of all distributions with left bounded support. Regular operators have local properties. Lions' theorem of supports holds for regular operators with compact support. The fundamental solution to the Cauchy-Riemann equations is not regular, but the fundamental solution to the heat equation in two dimensions is regular and has support on a half-ray.
Purely inseparable, modular extensions of unbounded exponent
Linda Almgren
Kime
335-349
Abstract: Let K be a purely inseparable extension of a field k of characteristic $p \ne 0$. Sweedler has shown in [2, p. 403] that if K over k is of finite exponent, then K is modular over k if and only if K can be written as the tensor product of simple extensions of k. This paper grew out of an attempt to find an analogue to this theorem if K is of unbounded exponent over k. The definition of a simple extension is extended to include extensions of the form $ k[x,{x^{1/p}},{x^{1/{p^2}}}, \cdots ][{x^{1/{p^\infty }}}]$. If K is the tensor product of simple extensions, then K is modular. The converse, however, is not true, as several counterexamples in §4 illustrate. Even if we restrict $ [k:{k^p}] < \infty$, the converse is still shown to be false. Given K over k modular, we construct a field $\cap _{i = 1}^\infty k{K^{{p^i}}} \otimes M( = Q)$ that always imbeds in K where M is the tensor product of simple extensions in the old sense. In general $K \ne Q$. For K to be the tensor product of simple extensions, we need $K = Q$, and $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k( \cap _{i = 1}^\infty {K^{{p^i}}})$. If for some finite N, $k{K^{{p^N}}} = k{K^{{p^{N + 1}}}}$, then we have (by Theorem 11) that $K = Q$. This finiteness condition guarantees that M is of finite exponent. Should $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k$, then we would have the condition of Sweedler's original theorem. The counterexamples in §4 will hopefully be useful to others interested in unbounded exponent extensions. Of more general interest are two side theorems on modularity. These state that any purely inseparable field extension has a unique minimal modular closure, and that the intersection of modular extensions is again modular.
Monads of infinite points and finite product spaces
Frank
Wattenberg
351-368
Abstract: The notion of ``monad'' is generalized to infinite (i.e. non-near-standard) points in arbitrary nonstandard models of completely regular topological spaces. The behaviour of several such monad systems in finite product spaces is investigated and we prove that for paracompact spaces X such that $X \times X$ is normal, the covering monad $ \mu$ satisfies $\mu (x,y) = \mu (x) \times \mu (y)$ whenever x and y have the same ``order of magnitude.'' Finally, monad systems, in particular non-standard models of the real line, R, are studied and we show that in a minimal nonstandard model of R exactly one monad system exists and, in fact, $\mu (x) = \{ x\} $ if x is infinite.
On a functional calculus for decomposable operators and applications to normal, operator-valued functions
Frank
Gilfeather
369-383
Abstract: Whenever $A = {\smallint _\Lambda } \oplus A(\lambda )\mu (d\lambda )$ is a decomposable operator on a direct integral $H = {\smallint _\Lambda } \oplus H(\lambda )\mu (d\lambda )$ of Hilbert spaces and f is a function analytic on a neighborhood of $ \sigma (A)$, then we obtain that $ f(A(\lambda ))$ is defined almost everywhere and $f(A)(\lambda ) = f(A(\lambda ))$ almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic function A is a normal operator. Two main results are obtained. Let f be an analytic function on a neighborhood of the spectrum of an operator A. If $ f''(z) \ne 0$ for all z in the spectrum of A and if $ f(A)$ is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal operators. As above if $ f(A)$ is normal and in addition, $ f(z) - {\zeta _0}$ has at most two roots counted to their multiplicity for each ${\zeta _0}$ in the spectrum of N, then A is a binormal operator.
Local finite cohesion
W. C.
Chewning
385-400
Abstract: Local finite cohesion is a new condition which provides a general topological setting for some useful theorems. Moreover, many spaces, such as the product of any two nondegenerate generalized Peano continua, have the local finite cohesion property. If X is a locally finitely cohesive, locally compact metric space, then the complement in X of a totally disconnected set has connected quasicomponents; connectivity maps from X into a regular ${T_1}$ space are peripherally continuous; and each connectivity retract of X is locally connected. Local finite cohesion is weaker than finite coherence [4], although these conditions are equivalent among planar Peano continua. Local finite cohesion is also implied by local cohesiveness [l2] in locally compact ${T_2}$ spaces, and a converse holds if and only if the space is also rim connected. Our study answers a question of Whyburn about local cohesiveness.
Cells and cellularity in infinite-dimensional normed linear spaces
R. A.
McCoy
401-410
Abstract: Certain concepts such as cells, cellular sets, point-like sets, and decomposition spaces are studied and related in normed linear spaces. The relationships between these concepts in general resemble somewhat the corresponding relationships in Euclidean space.
Embedding rings with a maximal cone and rings with an involution in quaternion algebras
Carl W.
Kohls;
William H.
Reynolds
411-419
Abstract: Sufficient conditions are given for an algebra over a totally ordered field F to be isomorphic to a subring of the algebra of quaternions over the real closure of F. These conditions include either the requirement that the nonnegative scalars form a maximal cone in the algebra, or that the algebra have an involution such that the scalars are the only symmetric elements. For many matrix algebras, the cone requirement alone is imposed.
Prehomogeneous vector spaces and varieties
Frank J.
Servedio
421-444
Abstract: An affine algebraic group G over an algebraically closed field k of characteristic 0 is said to act prehomogeneously on an affine variety W over k if G has a (unique) open orbit $o(G)$ in W. When W is the variety of points of a vector space V, $G \subseteq GL(V)$ and G acts prehomogeneously and irreducibly on V (We say an irreducibly prehomogeneous pair (G, V).), the following conditions are shown to be equivalent: 1. the existence of a nonconstant semi-invariant P in $ k[V] \cong S({V^\ast})$, 2. $(G',V)$ is not a prehomogeneous pair ($ G'$ is the commutator subgroup of G, a semisimple closed subgroup of G.), 3. if $ X \in o(G)$, then $ B \subseteq G$ acting prehomogeneously on W'' is shown to be sufficient for $G\backslash W$, the set of G-orbits in the affine variety W to be finite. These criteria are then applied to a class of irreducible prehomogeneous pairs (G, V) for which $ G'$ is simple and three further conjectures, one due to Mikio Sato, are stated.
Boundary values of solutions of elliptic equations satisfying $H\sp{p}$ conditions
Robert S.
Strichartz
445-462
Abstract: Let A be an elliptic linear partial differential operator with ${C^\infty }$ coefficients on a manifold ${\mathbf{\Omega }}$ with boundary ${\mathbf{\Gamma }}$. We study solutions of $Au = \sigma$ which satisfy the ${H^p}$ condition that ${\sup _{0 < t < 1}}{\left\Vert {u( \cdot ,t)} \right\Vert _p} < \infty$, where we have chosen coordinates in a neighborhood of ${\mathbf{\Gamma }}$ of the form ${\mathbf{\Gamma }} \times [0,1]$ with ${\mathbf{\Gamma }}$ identified with $ t = 0$. If A has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data $ u( \cdot ,0) = {f_0},{(\partial /\partial t)^j}u( \cdot ,0) = {f_j},j = 1, \cdots ,m - 1$ as follows: $ {f_0} \in {L^p}$ (or $\mathfrak{M}$ if $p = 1$) and $ {f_j} \in {\mathbf{\Lambda }}( - j;p,\infty ),j = 1, \cdots ,m$ . Here ${\mathbf{\Lambda }}$ denotes the Besov spaces in Taibleson's notation. If $m = 1$ then u has nontangential limits almost everywhere.
Banach spaces whose duals contain $l\sb{1}(\Gamma )$ with applications to the study of dual $L\sb{1}(\mu )$ spaces
C.
Stegall
463-477
Abstract: THEOREM I. If E is a separable Banach space such that $E'$ has a complemented subspace isomorphic to ${l_1}({\mathbf{\Gamma }})$ with ${\mathbf{\Gamma }}$ uncountable then $ E'$ contains a complemented, $ M({\mathbf{\Delta }})$, the Radon measures on the Cantor set. THEOREM II. If E is a separable Banach space such that $ E'$ has a subspace isomorphic to ${l_1}({\mathbf{\Gamma }})$ with ${\mathbf{\Gamma }}$ uncountable, then E contains a subspace isomorphic to ${l_1}$, THEOREM III. Let E be a Banach space. The following are equivalent: (i) $ E'$ is isomorphic to ${l_1}({\mathbf{\Gamma }})$; (ii) every absolutely summing operator on E is nuclear; (iii) every compact, absolutely summing operator on E is nuclear; (iv) if X is a separable subspace of E, then there exists a subspace Y such that $ X \subseteq Y \subseteq E$ and $Y'$ is isomorphic to ${l_1}$. THEOREM IV. If E is a $ {\mathcal{L}_\infty }$ space then (i) $E'$ is isomorphic to ${l_1}({\mathbf{\Gamma }})$ for some set ${\mathbf{\Gamma }}$ or (ii) $ E'$ contains a complemented subspace isomorphic to $M({\mathbf{\Delta }})$. COROLLARY. If E is a separable $ {\mathcal{L}_\infty }$ space, then $E'$ is (i) finite dimensional, or (ii) isomorphic to ${l_1}$, or (iii) isomorphic to $ M({\mathbf{\Delta }})$. COROLLARY. If $ {L_1}(\mu )$ is isomorphic to the conjugate of a separable Banach space, then ${L_1}(\mu )$ is isomorphic to $ {l_1}$ or $M({\mathbf{\Delta }})$.
Infinite compositions of M\"obius transformations
John
Gill
479-487
Abstract: A sequence of Möbius transformations $\{ {t_n}\} _{n = 1}^\infty $, which converges to a parabolic or elliptic transformation t, may be employed to generate a second sequence $\{ {T_n}\} _{n = 1}^\infty$ by setting $ {T_n} = {t_1} \circ \cdots \circ {t_n}$. The convergence behavior of $\{ {T_n}\}$ is investigated and the ensuing results are shown to apply to continued fractions which are periodic in the limit.
Uniqueness of Haar series which are $(C,\,1)$ summable to Denjoy integrable functions
William R.
Wade
489-498
Abstract: A Haar series $\Sigma \;{a_k}{\chi _k}$ satisfies Condition H if ${a_k}{\chi _k}/k \to 0$ uniformly as $k \to \infty$. We show that if such a series is (C, 1) summable to a Denjoy integrable function f, except perhaps on a countable subset of [0, l], then that series must be the Denjoy-Haar Fourier series of f.
On the multiplicative completion of certain basic sequences in $L\sp{p},$ $1<p<\infty $
Ben-Ami
Braun
499-508
Abstract: Boas and Pollard proved that given any basis $\{ {f_n}\} _{n = 1}^\infty $ for ${L^2}(E)$ one can delete the first k basis elements and then find a bounded measurable function M such that $\{ M{f_n}\} _{n = k + 1}^\infty$ is total in $ {L^2}(E)$, that is, the closure of the linear span of the set $\{ M{f_n}:n \geq k + 1\} $ is ${L^2}(E)$. We improve this result by weakening the hypothesis to accept bases of ${L^p}(E),1 < p < \infty$, and strengthening the conclusion to read serially total, that is, given any $f \in {L^2}(E)$ one can find a sequence of reals $\{ {a_n}\} _{n = k + 1}^\infty$ such that $ \Sigma _{n = k + 1}^\infty {a_n}M{f_n}$ converges to f in the norm. We also show that certain infinite deletions are possible.