The algebraic structure of a topological algebra influences its topological structure in a way which is profound but not well understood. (See § 7 below for various examples.) Here we examine this influence rather generally, and give a fairly complete analysis of one of the many forms it can take, namely, the influence of the identities of on the group identities obeyed by the homotopy group (or groups of the components) of .

In [5] C. Akemann and G. Pedersen defined four concepts of semicontinuity for elements of A**, the enveloping W*-algebra of a C*-algebra A. For three of these the associated classes of lower semicontinuous elements are , and (notation explained in Section 2), and we will call these the classes of strongly lsc, middle lsc, and weakly lsc elements, respectively. There are three corresponding concepts of continuity: The strongly continuous elements are the elements of A itself, the middle continuous elements are the multipliers of A, and the weakly continuous elements are the quasi-multipliers of A. It is natural to ask the following questions, each of which is three-fold.

(Q1) Is every lsc element the limit of a monotone increasing net of continuous elements?

(Q2) Is every positive lsc element the limit of an increasing net of positive continuous elements?

(Q3) If h ≧ k, where h is lsc and k is usc, does there exist a continuous x such that h ≧ x ≧ k?

In this paper we prove a certain ${{L}^{2}}$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a consequence, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

In this paper we consider finite graphs without loops and multiple edges. A graph is considered to be an ordered pair 〈G, *〉 where G is a finite set the elements of which are called the vertices of while * is a subset of [G]2 (where [G]2 is the set of all subsets of two elements of G). The elements of * are called the edges of . If {P, Q} ∊ *, we say that Q is adjacent to P. The degree of a vertex is the number of vertices adjacent to it. Let k be an integer. We say that is the complete k-graph if G has k elements and * = [G]2.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by

1.1

where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.

Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show that

where

The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet.

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

We consider the C*-algebras constructed from certain hyperbolic dynamical systems. The construction, due to Ruelle, generalizes the C*-algebras of Cuntz and Krieger. We discuss relations between the C*-algebras, show the existence of natural asymptotically abelian systems and investigate the K-theory and E-theory of these C*-algebras.

A subset S of a topological space X is z-embedded in X in case each zero-set of S is the restriction to S of a zero-set of X. (A zero-set is the set of zeros of a real-valued continuous function.) For the basic theory of z-embedding, see [3] and [4] (and see [4] for a comprehensive bibliography of relevant papers).

A graph G is called complete if any two of its vertices are connected by an edge; a set of vertices of G are said to be independent if no two of them are connected by an edge. It follows from a well-known theorem of Ramsay (1) that for each pair of positive integers k, l there is an integer f(k, l), which we take to be minimal, such that every graph with f(k, l) vertices either contains a complete graph of k vertices or a set of l independent points.

A difference set (,D) is defined in (2) as a subset D of k elements in a group of order υ with the following properties :

(1) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (d1d2) of elements of D such that x = 1d2;

(2) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (3d4) of elements of D such that x = 3d4−1.

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular representation (GRR) of a given abstract group G if

(I) G ≅ A(X) , and

(II) A(X) acts on V(X) as a regular permutation group; that is, given u, v ∈ V(X), there exists a unique φ ∈ A(X) for which φ(u) = v.

That for any abstract group G there exists a graph X satisfying (I) is well-known (cf. [3]).

Negative definite functions (all definitions are given in § 1 below) on a locally compact, σ-compact group G have been used in several different contexts recently [2, 5, 7, 11]. In this paper we show how such functions relate to other properties such a group may have. Here are six properties which G might have. They are grouped into three pairs with one property of each pair involving negative definite functions. We show that the paired properties are equivalent and, where possible, give counter-examples to other equivalences. We assume throughout that G is not compact.

(1A) G does not have property T.

(IB) There is a continuous, negative definite function on G which is unbounded.

(2A) G has the (weak and/or strong) dual R-L property.

(2B) For every closed, non-compact set Q ⊂ G there is a continuous, negative definite function on G which is unbounded on Q.

In 1883 Poincaré [13] recognized that the discrete subgroups of PSL(2, C) could be extended from their natural action on the complex plane to acting on hyperbolic 3-space and he attempted to analyze these groups in an analogous manner to his classical treatment of Fuchsian groups, with fundamental polyhedra playing the role of the fundamental polygons for Fuchsian groups. This approach, however, did not lead very far, perhaps not surprisingly when one appreciates the close connection between the geometry of these groups and the topology of 3-manifolds. Since that time the state of knowledge remained essentially unchanged until 1964 when work by Ahlfors [1] and soon afterwards by Bers [3] revitalized the subject of Kleinian groups. The modern approach tends to use analytic methods, although recently Marden [11] has had considerable success in carrying forward Poincaré's geometric approach.

Let be a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes.

Brainerd and Lambek (2, Corollary 4) have proved recently that any complete Boolean ring is self-injective. It is easy to see that every complete Boolean ring is a continuous regular ring, that is, a regular ring of which the lattice of principal left ideals is continuous. This suggests that in a continuous regular ring it might be possible to prove the injectivity. However, a simple example (Example 3) shows that the conjecture is not true in general. Our main theorem is the following. Every continuous regular ring with no ideals of index 1 is (both left and right) self-injective (Theorem 3).

It is known to Wolfson (13, Theorem 5.1) and Zelinsky (15) that the ring S of all linear transformations of a vector space of dimension ≥ 2 over a division ring is generated by idempotents and also by non-singular elements.

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by

(1)

We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form

For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Suppose 1 ≦ P < ∞, μ is real, and denote by Lμ,p the collection of functions f, measurable on (0, ∞ ), and which satisfy

1.1

Also denote by [X] the collection of bounded operators from a Banach space X to itself. For v > 0, Re α > 0, Re β > 0, let

1.2

and

1.3

where ξ and η are complex numbers. Iv,α,ξ and Jv,β,η, are generalizations of the Riemann-Liouville and Weyl fractional integrals respectively, and consequently we shall refer to them as fractional integrals.