Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.

Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

If X is a nonempty topological T1 space then the set of all homeomorphisms whose domains and ranges are closed subsets of X forms a semigroup under partial composition of functions. We call it IF(X). If, in a semigroup, every element a is matched with a unique element b such that aba = a and bab = b then the semigroup is an inverse semigroup (b is called the inverse of a and is denoted by a−1). We have that IF(X) is an inverse semigroup with the algebraic inverse of a map ƒ being just the inverse map ƒ-1. In this paper we examine epimorphisms from IF(X) onto IF(Y). The main theorem gives conditions under which an epimorphism must be an isomorphism.

Throughout this note, let R be a discrete valuation ring with prime element π, residue class field , and quotient field K. Let Λ be an R-order in a finite dimensional K-algebra A. A Λ-lattice is an R-free finitely generated left Λ-module. For k > 0, we set

where M is any Λ-lattice. Obviously, for Λ-lattices M and N,

Maranda [1] and D. G. Higman [3] considered the reverse implication, and Proved

THEOREM. Let Λ be an R-order in a separable K-algebra A. Then there exists a positive integer k (which depends on Λ) with the following property: for each pair of Λ-lattices M and N,

Indeed,m it suffices to choose k so that

Maranda proved this result for the special case where Λ is the integral group ring RG of a finite group G.

Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

Let A be a nonassociative algebra. We let An denote the subalgebra generated by all products of n elements from A. Inductively we define A(0) = A and A(n+1) = (A(n))2. We say that A is nilpotent if, for some n, An = {0}. A is solvable if A(n) = {0} for some n. An algebra is locally nilpotent (locally solvable) if each finitely generated subalgebra is nilpotent (solvable). In this paper will always be some variety of algebras defined by a set of homogeneous identities. We say that local nilpotence is a radical property in if each contains a maximal locally nilpotent ideal L and A/L has no non-zero locally nilpotent ideals. The ideal L is then called the Levitzki radical of A.

In [8], McKenzie introduced the notion of a bounded homomorphism between lattices, and, using this concept, proved several deep results in lattice theory. Some of these results were intimately connected with the work of Jónsson and Kiefer in [6] where an attempt was made to characterize finite sublattices of free lattices. McKenzie's characterization and others that followed (see [7] and [5]) still have not answered the (now) celebrated Jônsson conjecture:

A finite lattice is a sublattice of a free lattice if and only if it satisfies (SD∨), (SD∨) and (W).

(The properties mentioned here are defined in the text.)

α-Recursion theory was invented simultaneously by Kripke [15] and Platek [22] and served to generalize the theories of Takeuti [34], Machover [20], Kreisel and Sacks [14] and others. Kripke (in [16]) derived machinery to construct an analogue to Kleene's T-predicate enabling him to assert that all of unrelativized ordinary recursion theory (as found in Kleene [13]) lifted to α-recursion theory. As a result, we were able to set down in [8] α-analogues to Blum's [1] well-studied axioms, thus, introducing the study of α-computational complexity theory.

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |ϕ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.

For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional ϕ on the C*-algebra satisfies

(†)

then ϕ satisfies also the equivalent conditions

• (i) ϕ(xy) = ϕ(yx) for all x, y in ;

• (ii) ϕ(x*x) = ϕ(xx*) for all x in ;

• (iii) ϕ(x) = ϕ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element.

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limits

exist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.

THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only if

where c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Let A and B be C*-algebras acting on a Hilbert space H, and let

where A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?

Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree.

The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:

PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then

The celebrated result of Lomonosov [6] on the existence of invariant subspaces for operators commuting with a compact operator have been generalized in different directions (for example see [2], [7], [8], [9]). The main result of [9] (see also [7]) is: If is a norm closed algebra of (bounded) operators on an infinite dimensional (complex) Banach space , if K is a nonzero compact operator on , and if then has a non-trivial (closed) invariant subspace. In [7], it is mentioned that the above result holds if instead of compactness for K we assume that K is a non-invertible injective operator with a non-zero eigenvalue belonging to the class of decomposable, hyponormal, or subspectral operators.

In two previous papers [1, 2] we investigated the zeros of certain arithmetic functions. Using units of cubic fields, we also succeeded to construct, almost by accident, and as a by-product so to speak, entirely new and comparatively complicated combinatorial identities. In an interesting paper combinatorialist L. Carlitz [10] proved those identities in an elementary way. In a/m, we had to prove that the units used were fundamental ones.

Encouraged by these results, we took a closer look at this method that had led to the construction of combinatorial identities. Since the latter are such an important tool in mathematics, we thought it would be “einer Messe wert“ to generalize these results and lay the theoretical foundations of a new method for the construction of highly sophisticated combinatorial identities.

All topological spaces under discussion are assumed to be Tychonoff.

For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such that

The extender κ is called a Kn-function (cf. [3]).

Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.

R. N. Ball (unpublished) and G. E. Davis and C. D. Fox [1] established that if Ω is a doubly homogeneous totally ordered set, the l-group A (Ω) of all orderpreserving permutations of Ω endures a compatible tight Riesz order. Specifically T = {g ∈ A(Ω)+ : supp (g) is dense in Ω} is a compatible tight Riesz order for A(Ω). Using this fact, I inserted Theorem 3.7 into [2; MR 53 (1977), #13070] at the galley proof stage. (It was also included in MR 54 (1977), #7350 and [3; p. 472].) Theorem 3.7 stated: Let Ω be homogeneous. Then A(Ω) endures a compatible tight Reisz order if and only if Ω is dense. I stated that it was obvious that if Ω were homogeneous and discrete, A(Ω) could not endure a compatible tight Riesz order. This “obvious” is neither obvious nor true.

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].