In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

In [57] (2.12), D. S. Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [56], are precisely (when they are made into topological spaces via the Scott topology) the injective T0-spaces, i.e., the injective objects in the category T0 of T0-spaces and continuous maps with regard to the class of all embeddings. Moreover, the sort of morphisms between continuous lattices Scott considered are precisely the continuous maps with regard to the respective Scott topologies. These are fairly non-Hausdorff topologies. (Indeed, the Scott topology induces the partial order of the lattice L via x ≦ y if and only if x ∊ cl{j}, the “specialization order” of the topology; hence L is Hausdorff in the Scott topology if and only if L has at most one element.) In topological algebra, compact Lawson semilattices (= compact Hausdorff topological ∧-semilattices such that the ∧-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest.

Although many types of rooted planar maps have been enumerated (see [8] for example), not much has been done on enumeration of entirely unrooted planar maps. Yet in virtually all cases of interest, it has appeared that comparatively very few of the maps are symmetric (have non-trivial automorphisms). This suggests that an asymptotic formula for the numbers of unrooted maps of a particular type on n edges can be obtained by dividing the numbers of rooted maps of that type on n edges by 4n, where 4n is the number of potentially distinct rootings of an asymmetric n-edged map. The assertion that almost all maps of a given type are asymmetric has previously been proved in only two non-trivial cases: for 3-connected planar triangulations by Tutte [9] and for all n-edged 3-connected planar maps in [5]. We prove here that it is also true for 3-connected planar maps with a given number of vertices and faces, uniformly as either parameter approaches infinity.

Let 2 ≦ s ∊ N and {n1, …, ns) ⊆ N*. In 1884, J. Sylvester [13] published the following well-known result on the singularity degree S of the monomial curve whose corresponding semigroup is S: = 〈n1, …, ns): If s = 2, then

Let K: = –Z\S and

for all 1 ≦ i ≦ s. We introduce the invariant

of S involving a correction term to the Milnor number 2δ [4] of S. As a modified version and extension of Sylvester's result to all monomial space curves, we prove the following theorem: If s = 3, then

We prove similar formulas for s = 4 if S is symmetric.

Recall BSOn is the classifying space for the special orthogonal group of rank n. It is well known that the mod 2 cohomology ring of BSOn is given as follows:

where wi is the i-th mod 2 Stiefel-Whitney class. For the 3-connective cover of BSOn, BSpinn, Quillen in [4] has determined H*(BSpinn) completely for all n. Let BŜOn〈8〉, BŜOn〈16〉 be the classifying spaces for n-plane spin bundle ξ satisfying w4(ξ) = 0 and w4(ξ) = w8(ξ) = 0 respectively. This note follows the method of A. Borel [2] and gives the mod 2 cohomology ring of BŜOn〈8〉 and BŜOn〈16〉 for small n. In particular we answer the question “when is H*(BŜOn〈8〉; Z2), or H*(BŜOn〈16〉; Z2) a polynomial algebra?”

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.

In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

In this paper we study integral operators with domain a Banach function space Lρ1 and range another Banach function space Lρ2 or the space L0 of all measurable functions. Recall that a linear operator T from Lρ1 into L0 is called an integral operator if there exists a μ × v-measurable function T(x, y) on X × Y such that

Such an integral operator is called a Carleman integral operator if for almost every x ∊ X the function

is an element of the associate space L′ρ1, i.e.,

A character of a finite group G is monomial if it is induced from a linear (degree one) character of a subgroup of G. A group G is an M-group if all its complex irreducible characters (the set Irr(G)) are monomial.

In [1], Dade gave an example of an M-group with a normal subgroup which is itself not an M-group. In his group G, the supersolvable residual N is an extra special 2-group and G/N is supersolvable of even order. Moreover, the prime 2 is used in such a way that no analogous construction is possible in the case that |N| or |G:N| is odd. This led Isaacs in [8] and Dade in [2] to consider the effect of certain “oddness“ hypotheses in the study of monomial characters.

Our main results are in the same spirit. Although our techniques seem to require a restrictive assumption on the supersolvable residual of the groups we consider, our theorems provide more evidence that under fairly general circumstances normal subgroups of M-groups should be M-groups.

For us an “algebra” is a finitary “universal algebra” in the sense of G. Birkhoff [9]. We are concerned in this paper with algebras whose endomorphisms are determined by small subsets. For example, an algebra A is rigid (in the strong sense) if the only endomorphism on A is the identity idA. In this case, the empty set determines the endomorphism set E(A). We place the property of rigidity at the bottom rung of a cardinal-indexed ladder of properties as follows. Given a cardinal number κ, an algebra A is minimally free over a set of cardinality κ (κ-free for short) if there is a subset X ⊂ A of cardinality κ such that every function f:X → A extends to a unique endomorphism ϕ ∊ E(A). (It is clear that A is rigid if and only if A is 0-free.) Members of X will be called counters; and we will be interested in how badly counters can fail to generate the algebra.

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:

(1.1)

where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

Let e be an integer ≧ 2, and p a prime = 1 (mod e). Euler's criterion states that for D ∊ Z,

(1.1)

if and only if D is an e-th power residue (mod p). If D is not an e-th power (mod p), one has

(1.2)

for some e-th root α(≠1) of unity (mod p). Sometimes expressions for roots of unity (mod p) can be given in terms of quadratic partitions of p. For example,

(1.3)

are the four distinct fourth roots of unity (mod p) for a prime p ≡ 1 (mod 4) in terms of a solution (a, b) of the diophantine system

(a, b unique), whereas for p ≡ 1 (mod 3), a solution (L, M) of the system

gives

(1.4)

as the three distinct cuberoots of unity (mod p).