Tensor products of group algebras
J. E.
Kerlin
1-36
Abstract: Let C be a commutative Banach algebra. A commutative Banach algebra A is a Banach C-algebra if A is a Banach C-module and $c \cdot (aa') = (c \cdot a)a'$ for all ${A_1}, \cdots ,{A_n}$ are commutative Banach C-algebras, then the C-tensor product ${A_1}{ \otimes _C} \cdots { \otimes _C}{A_n} \equiv D$ is defined and is a commutative Banach C-algebra. The maximal ideal space ${\mathfrak{M}_D}$ of D is identified with a closed subset of $ {\mathfrak{M}_{{A_1}}} \times \cdots \times {\mathfrak{M}_{{A_n}}}$ in a natural fashion, yielding a generalization of the Gelbaum-Tomiyama characterization of the maximal ideal space of ${A_1}{ \otimes _\gamma } \cdots { \otimes _\gamma }{A_n}$. If $C = {L^1}(K)$ and ${A_i} = {L^1}({G_i})$, for LCA groups K and $ {G_i},i = 1, \cdots ,n$, then the ${L^1}(K)$-tensor product D of $ {L^1}({G_1}), \cdots ,{L^1}({G_n})$ is uniquely written in the form $D = N \oplus {D_e}$, where N and $ {D_e}$ are closed ideals in D, ${L^1}(K) \cdot N = \{ 0\} $, and ${D_e}$ is the essential part of D, i.e. $ {D_e} = {L^1}(K) \cdot D$. Moreover, if $ {D_e} \ne \{ 0\}$, then $ {D_e}$ is isometrically $ {L^1}(K)$-isomorphic to ${L^1}({G_1}{ \otimes _K} \cdots { \otimes _K}{G_n})$, where $ {G_1}, \cdots ,{G_n}$ is a K-tensor product of ${G_1}, \cdots ,{G_n}$ with respect to naturally induced actions of K on ${G_1}, \cdots ,{G_n}$. The above theorems are a significant generalization of the work of Gelbaum and Natzitz in characterizing tensor products of group algebras, since here the algebra actions are arbitrary. The Cohen theory of homomorphisms of group algebras is required to characterize the algebra actions between group algebras. Finally, the space of multipliers $ {\operatorname{Hom}_{{L^1}(K)}}({L^1}(G),{L^\infty }(H))$ is characterized for all instances of algebra actions of ${L^1}(K)$ on ${L^1}(G)$ and ${L^1}(H)$, generalizing the known result when $ K = G = H$ and the module action is given by convolution.
Distance estimates and pointwise bounded density
A. M.
Davie;
T. W.
Gamelin;
J.
Garnett
37-68
Abstract: Let U be a bounded open subset of the complex plane, and let H be a closed subalgebra of ${H^\infty }(U)$, the bounded analytic functions on U. If E is a subset of $\partial U$, let ${L_E}$ be the algebra of all bounded continuous functions on U which extend continuously to E, and set $ {H_E} = H \cap {L_E}$. This paper relates distance estimates of the form $d(h,H) = d(h,{H_E})$, for all $h \in {L_E}$, to pointwise bounded density of $ {H_E}$ in H. There is also a discussion of the linear space $H + {L_E}$, which turns out often to be a closed algebra.
On arithmetical classifications of inaccessable cardinals and their applications
Géza
Fodor;
Attila
Máté
69-99
Abstract: Lately several authors, among them Fodor, Gaifman, Hanf, Keisler, Lévy and Tarski, dug out an interesting and unduly forgotten operation of Mahlo that, loosely speaking, from a sequence of ordinals discards all those that are easy to locate in this sequence. The purpose of these authors was to invent strengthenings and schemes for repetitions of this and similar operations and to study the properties of cardinals that can be discarded in this way when started with a specific class; for example, the class of all inaccessible cardinals. Our attempt here is to consider such schemes for repetitions of operations that can in a sense be described in an arithmetical way, which might also be called constructive; our investigations are akin to the problem of constructive description of possibly large segments of, say, the set of all countable ordinals. Some applications of our classifications scheme are exhibited, questions ranging from definability of inaccessible cardinals in terms of sets of lower ranks to incompactness theorems in infinitary languages. The paper is concluded with an algebraic-axiomatic type study of our scheme.
Modular permutation representations
L. L.
Scott
101-121
Abstract: A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauer's theory of blocks to Green's work on indecomposable modules.
The existence of ${\rm Irr}(X)$
M. W.
Mislove
123-140
Abstract: If X is a compact totally ordered space, we obtain the existence of an irreducible semigroup with idempotents X, ${\text{Irr}}(X)$, with the property that any irreducible semigroup with idempotents X is the idempotent separating surmorphic image of ${\text{Irr}}(X)$. Furthermore, it is shown that the Clifford-Miller endomorphism on ${\text{Irr}}(X)$ is an injection when restricted to each $ \mathcal{H}$-class of ${\text{Irr}}(X)$. A construction technique for noncompact semigroups is given, and some results about the structure of such semigroups are obtained.
Infinite matroids
Samuel S.
Wagstaff
141-153
Abstract: Matroids axiomatize the related notions of dimension and independence. We prove that if S is a set with k matroid structures, then S is the union of k subsets, the ith of which is independent in the ith matroid structure, iff for every (finite) subset A of S, $\vert A\vert$ is not larger than the sum of the dimensions of A in the k matroids. A matroid is representable if there is a dimension-preserving imbedding of it in a vector space. A matroid is constructed which is not the union of finitely many representable matroids. It is shown that a matroid is representable iff every finite subset of it is, and that if a matroid is representable over fields of characteristic p for infinitely many primes p, then it is representable over a field of characteristic 0. Similar results for other kinds of representation are obtained.
The growth of subuniform ultrafilters
S.
Negrepontis
155-165
Abstract: Some of the results on the topology of spaces of uniform ultrafilters are applied to the space $\Omega ({\alpha ^ + })$ of subuniform ultrafilters (i.e., the set of ultrafilters which are $ \alpha$-uniform but not ${\alpha ^ + }$-uniform) on ${\alpha ^ + }$ when $\alpha$ is a regular cardinal. The main object is to find for infinite cardinals $\alpha$, such that $\alpha = {\alpha ^{\underbar{a}}}$, a topological property that separates the space $\beta (\Omega ({\alpha ^ + }))\backslash \Omega ({\alpha ^ + })$ (the growth of $\Omega ({\alpha ^ + })$) from the space $U({\alpha ^ + })$ of uniform ultrafilters on ${\alpha ^ + }$. Property ${\Phi _\alpha }$ fulfils this rôle defined for a zero-dimensional space X by the following condition: every nonempty closed subset of X of type at most $\alpha$ is not contained in the uniform closure of a family of $\alpha$ pairwise disjoint nonempty open-and-closed subsets of X. The ``infinitary'' properties of $ \Omega ({\alpha ^ + })$, as they are measured by $ {\Phi _\alpha }$, are more closely related to those of $ U(\alpha )$ than to those of $U({\alpha ^ + })$. A consequence of this topological separation is that the growth of $\Omega ({\alpha ^ + })$ is not homeomorphic to $U({\alpha ^ + })$ and, in particular, that $ \Omega ({\alpha ^ + })$ is not ${C^ \ast }$-embedded in the space $\Sigma ({\alpha ^ + })$ of $\alpha $-uniform ultrafilters on ${\alpha ^ + }$. These results are related to, and imply easily, the Aronszajn-Specker theorem: if $\alpha = \alpha^{\underbar{a}}$ then ${\alpha ^ + }$ is not a ramifiable cardinal. It seems possible that similar questions on the ${C^ \ast }$-embedding of certain spaces of ultrafilters depend on (and imply) results in partition calculus.
Cylindric algebras and algebras of substitutions
Charles
Pinter
167-179
Abstract: Several new formulations of the notion of cylindric algebra are presented. The class $ C{A_\alpha }$ of all cylindric algebras of degree $\alpha$ is shown to be definitionally equivalent to a class of algebras in which only substitutions (together with the Boolean $+ , \cdot$, and $-$) are taken to be primitive operations. Then $C{A_\alpha }$ is shown to be definitionally equivalent to an equational class of algebras in which only substitutions and their conjugates (together with $+ , \cdot$, and $- $) are taken to be primitive operations.
A characterization of $U\sb{3}(2\sp{n})$ by its Sylow $2$-subgroup
Robert L.
Griess
181-186
Abstract: We determine all the finite groups having a Sylow 2-subgroup isomorphic to that of $ {U_3}({2^n}),n \geq 3$. In particular, the only such simple groups are the ${U_3}({2^n})$.
Boundedly complete $M$-bases and complemented subspaces in Banach spaces
William J.
Davis;
Ivan
Singer
187-194
Abstract: Subsequences of boundedly complete M-bases need not be boundedly complete. An example of a somewhat reflexive space is given whose dual and one of whose factors fail to be somewhat reflexive. A geometric description of boundedly complete M-bases is given which is equivalent to the definitions of V. D. Milman and W. B. Johnson. Finally, certain M-bases for separable spaces give rise to proper complemented subspaces.
Averaging operators in $C(S)$ and lower semicontinuous sections of continuous maps
Seymour Z.
Ditor
195-208
Abstract: For certain kinds of compact Hausdorff spaces S, necessary and sufficient topological conditions are provided for determining if there exists a norm 1 projection of $ C(S)$ onto any given separable selfadjoint subalgebra A, the conditions being in terms of the decomposition that A induces on S. In addition, for arbitrary S and selfadjoint closed subalgebra A of $ C(S)$, some results on lower bounds for the norms of projections of $ C(S)$ onto A are obtained. An example is given which shows that the greatest lower bound of the projection norms need not be attained.
Class numbers of totally imaginary quadratic extensions of totally real fields
Judith S.
Sunley
209-232
Abstract: Let K be a totally real algebraic number field. This paper provides an effective constant $C(K,h)$ such that every totally imaginary quadratic extension L of K with ${h_L} = h$ satisfies $\vert{d_L}\vert < C(K,h)$ with at most one possible exception. This bound is obtained through the determination of a lower bound for $L(1,\chi )$ where $\chi$ is the ideal character of K associated to L. Results of Rademacher concerning estimation of L-functions near $s = 1$ are used to determine this lower bound. The techniques of Tatuzawa are used in the proof of the main theorem.
Projective groups of degree less than $4p/3$ where centralizers have normal Sylow $p$-subgroups
J. H.
Lindsey
233-247
Abstract: This paper proves the following theorem: Theorem 1. Let $ \bar G$ be a finite primitive complex projective group of degree n with a Sylow p-subgroup $\bar P$ of order greater than p for p prime greater than five. Let $n \ne p,n < 4p/3$, and if $p = 7,n \leqslant 8$. Then $p \equiv 1 \pmod 4,\bar P$ is a trivial intersection set, and for some nonidentity element $ \bar x\;in\;\bar G,C(\bar x)$ does not have a normal Sylow p-subgroup.
The structure of $n$-uniform translation Hjelmslev planes
David A.
Drake
249-282
Abstract: Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If $n > 1$, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane $\mathfrak{A}$ with the following property: for each point P of $ \mathfrak{A}$, the substructure $^{n - 1}P$ of all neighbor points of P is an $(n - 1)$-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods $ ^1P \subset {\;^2}P \subset \cdots \subset {\;^n}P = \mathfrak{A}$. For $i < n,{\;^i}P$ is an i-uniform affine Hjelmslev plane under the induced incidence relation (for some parallel relation). Hjelmslev planes are called strongly n-uniform if they are n-uniform and possess one additional property; the additional property is designed to assure that the planes have epimorphic images which are strongly $(n - 1)$-uniform. Henceforth, assume that $ \mathfrak{A}$ is a strongly n-uniform translation (affine) Hjelmslev plane. Let $ {{(^i}P)^ \ast }$ denote the incidence structure $^iP$ together with the parallel relation induced therein by the parallel relation holding in $\mathfrak{A}$. Then for all positive integers $ i \leq n$ and all points P and Q of $ \mathfrak{A}$, ${{(^i}P)^ \ast }$ and ${{(^i}Q)^\ast}$ are isomorphic strongly i-uniform translation Hjelmslev planes. Let $^i\mathfrak{A}$ denote this common i-uniform plane; $ {{(^i}\mathfrak{A})_j}$, denote the ``quotient'' of $^i\mathfrak{A}$ modulo $^j\mathfrak{A}$. The invariant $ r = {p^x}$ of $\mathfrak{A}$ is the order of the ordinary translation plane ${{(^n}\mathfrak{A})_{n - 1}}$. Then the translation group of $ \mathfrak{A}$ is an abelian group with 2xk cyclic summands, k an integer $\leq n$; one calls k the width of $ \mathfrak{A}$. If $0 \leq j < i \leq n$, then ${{(^i}\mathfrak{A})_j}$ is a strongly $(i - j)$-uniform translation Hjelmslev plane; if also $j \geq k,{{(^i}\mathfrak{A})_j}$ and ${{(^{i - k}}\mathfrak{A})_{j - k}}$ are isomorphic. Then if $\mathfrak{A}(i)$ denotes ${{(^i}\mathfrak{A})_{i - 1}},\mathfrak{A}(1), \cdots ,\mathfrak{A}(n)$ is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if ${T_1}, \cdots ,{T_k}$ is an arbitrary sequence of translation planes with common order and if $ n \geq k$, then there exists a strongly n-uniform translation Hjelmslev plane $ \mathfrak{A}$ of width k such that $\mathfrak{A}(i) \cong {T_i}$ for $i \leq k$. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the $ (n - 1)$-uniform planes by means of group congruences.
Embedding theorems and generalized discrete ordered abelian groups
Paul
Hill;
Joe L.
Mott
283-297
Abstract: Let G be a totally ordered commutative group. For each nonzero element $g \in G$, let $L(g)$ denote the largest convex subgroup of G not containing g. Denote by $U(g)$ the smallest convex subgroup of G that contains g. The group G is said to be generalized discrete if $U(g)/L(g)$ is order isomorphic to the additive group of integers for all $g \ne 0$ in G. This paper is principally concerned with the structure of generalized discrete groups. In particular, the problem of embedding a generalized discrete group in the lexicographic product of its components, $U(g)/L(g)$, is studied. We prove that such an embedding is not always possible (contrary to statements in the literature). However, we do establish the validity of this embedding when G is countable. In case F is o-separable as well as countable, the structure of G is completely determined.
The Brauer group of graded Azumaya algebras
L. N.
Childs;
G.
Garfinkel;
M.
Orzech
299-326
Abstract: We study G-graded Azumaya R-algebras for R a commutative ring and G a finite abelian group, and a Brauer group formed by such algebras. A short exact sequence is obtained which relates this Brauer group with the usual Brauer group of R and with a group of graded Galois extensions of R. In case G is cyclic a second short exact sequence describes this group of graded Galois extensions in terms of the usual group of Galois extensions of R with group G and a certain group of functions on $ {\text{Spec}}(R)$.
Steenrod squares in spectral sequences. I
William M.
Singer
327-336
Abstract: We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. We show these operations compatible with the differentials of the spectral sequence, and with the Steenrod squares defined on the cohomology of the total complex. We give a general rule for computing the operations on ${E_2}$.
Steenrod squares in spectral sequences. II
William M.
Singer
337-353
Abstract: We apply the results of the previous paper to three special cases. We obtain Steenrod operations on the change-of-rings spectral sequence, on the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces, and on the Serre spectral sequence.
Integral representation of functions and distributions positive definite relative to the orthogonal group
A. E.
Nussbaum
355-387
Abstract: A continuous function f on an open ball B in ${R^N}$ is called positive definite relative to the orthogonal group $O(N)$ if f is radial and $\smallint \smallint f(x - y)\phi (x)\overline {\phi (y)} \;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. It is shown that f is positive definite in B relative to $O(N)$ if and only if f has an integral representation $f(x) = \smallint {e^{ix \cdot t}}d{\mu _1}(t) + \smallint {e^{x \cdot t}}d{\mu _2}(t)$, where $ {\mu _1}$ and $ {\mu _2}$ are bounded, positive, rotation invariant Radon measures on $ {R^N}$ and ${\mu _2}$ may be taken to be zero if, in addition to f being positive definite relative to $O(N),\smallint \smallint f(x - y)( - \Delta \phi )(x)\phi (y)\;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. Both conditions are satisfied if f is a radial positive definite function in B. Thus the theorem yields as a special case Rudin's theorem on the extension of radial positive definite functions. The result is extended further to distributions.
On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms
A. Edward
Nussbaum
389-408
Abstract: To every continuous function f on an interval $0 \leq x < a(0 < a \leq \infty )$ and every positive number $\nu$ associate the kernel $\displaystyle f(x,y) = \int_0^\pi {f({{({x^2} + {y^2} - 2xy\;\cos \;\theta )}^{1/2}}){{(\sin \;\theta )}^{2\nu - 1}}d\theta ,\quad 0 < x,y < a/2.}$ Let $ \Omega (z) = \Gamma (\nu + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\ke... ...iptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$, where $ {J_{\nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$ is the Bessel function of index $\nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. It is shown that f has an integral representation $f(x) = \smallint_{ - \infty }^\infty {\Omega (x\sqrt \lambda )d\gamma (\lambda )}$, where $\gamma$ is a finite, positive Radon measure on R, if and only if the kernel $f(x,y)$ is positive definite. If $\nu = (N - 1)/2$, where N is an integer $ \geq 2$, this condition is equivalent to ${f_N}(x) = f(\vert x\vert),\;x \in {R^N},\;\vert x\vert < \alpha$, is positive definite relative to the orthogonal group $O(N)$. The results of this investigation extend the preceding one of the author on functions positive definite relative to the orthogonal group. In particular they yield the result of Rudin on the extensions of radial positive definite functions.
Space-time processes, parabolic functions and one-dimensional diffusions
Tze Leung
Lai
409-438
Abstract: In this paper, we study the properties of the space-time process and of parabolic functions associated with a Markov process. Making use of these properties and the asymptotic behavior of the first passage probabilities near the boundary points, we prove certain theorems concerning when $ u(X(t),t)$ is a martingale, where $X(t)$ is a conservative regular one-dimensional diffusion with inaccessible boundaries. A characterization of the class of parabolic functions associated with classical diffusions is also obtained.
A generalized operational calculus developed from Fredholm operator theory
Jack
Shapiro;
Martin
Schechter
439-467
Abstract: Let A be a closed operator on the Banach space X. We construct an operator, \begin{displaymath}\begin{array}{*{20}{c}} {(\lambda - A){{R'}_\lambda }(A) = I ... ... {{{R'}_\lambda }(A)(\lambda - A) = I + {F_2}} \end{array} \end{displaymath} where ${F_1}$ and ${F_2}$ are bounded finite rank operators. $\lambda \in {\Phi _A}$ except for at most a countable set containing no accumulation point in ${\Phi _A}$. Let ${\sigma _\Phi }(A)$ be the complement of $ {\Phi _A}$, and let ${\sigma _\Phi }(A)$ and at $(\infty )$. We then use the operator, $ N(A - \lambda )$.
Projections and approximate identities for ideals in group algebras
Teng-Sun
Liu;
Arnoud
van Rooij;
Ju Kwei
Wang
469-482
Abstract: For a locally compact group G with property $({{\text{P}}_1})$, if there is a continuous projection of ${L^1}(G)$ onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2-sided approximate identity. The converse is proved for ${w^ \ast }$-closed left ideals. Let G be further abelian and let I be a closed ideal in ${L^1}(G)$. The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group.
On the regularity of the Riemann function for hyperbolic equations
William L.
Goodhue
483-490
Abstract: In an earlier paper, A. Friedman demonstrated that the Riemann function for a strictly hyperbolic system with Gevrey coefficients was locally Gevrey of some higher order except along the bicharacteristics. By representing the Riemann function in terms of a progressing wave expansion, this result is extended beyond caustics.
An approximation theorem for biholomorphic functions on $D\sp{n}$
Joseph A.
Cima
491-497
Abstract: Let F be a biholomorphic mapping of the polydisk ${D^n}$ into ${{\mathbf{C}}^n}$. We construct a sequence of polynomial mappings $ \{ {P_j}\}$ such that each ${P_j}$ is subordinate to $ {P_{j + 1}}$, each $ {P_j}$ is subordinate to F and the ${P_j}$ converge uniformly on compacta to F. The polynomials ${P_j}$ are biholomorphic.
Lower semicontinuity of parametric integrals
Edward
Silverman
499-508
Abstract: It has been known for a long time that the usual two-dimensional parametric integrals in three-space are lower semicontinuous with respect to uniform convergence. In an earlier paper we saw that an easy argument extends this result to all parametric integrals generated by simply-convex integrands, with no restrictions on the dimension of the surfaces or the containing space. By using these techniques again, and generalizing to surfaces a result concerning convergent sequences of closed curves we show that a parametric integral generated by a parametric integrand which is convex in the Jacobians is lower semicontinuous with respect to uniform convergence provided all of the functions lie in a bounded subset of the Sobolev space $H_s^1$ where $s + 1$ exceeds the dimension of the parametric integral.