1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the function

is a norm on J, denoted by || • ||p.

The width of a lattice L is the maximum number of pairwise noncomparable elements in L.

It has been known for some time ([5] ; see also [4]) that there is just one subdirectly irreducible lattice of width twro, namely the five-element nonmodular lattice N5. It follows that every lattice of width two is in the variety of N5, and that every finitely generated lattice of width two is finite.

Beginning a study of lattices of width three, W. Poguntke [6] showed that there are infinitely many finite simple lattices of width three. Further studies on width three lattices were made in [3], where it was asked whether every finitely generated simple lattice of width three is finite. In this paper we will show that, in fact, more is true:

THEOREM 1.1. Every finitely generated subdirectly irreducible lattice of width three is finite.

A subset A of a topological ring R is bounded if given any neighborhood U of zero, there exists a neighborhood V of 0 such that AV ⊆ U and VA ⊆ U. The topology on R is locally bounded if there exists a bounded neighborhood of 0.

We recall that a seminorm ‖··‖ on a ring R is a function from R into the non-negative real numbers satisfying ‖x‖ = 0 if x = 0, ‖–x‖ = ‖x‖, ‖x + y‖ ≦ ‖x‖ + ‖y‖ and ‖xy‖ ≦ ‖x‖ ‖y‖ for all x, y in R. A seminorm ‖··‖ on R is a norm on R if ‖x‖ = 0 implies x = 0. We note that a seminorm ‖··‖ on R defines a locally bounded topology on R, and a norm on R defines a Hausdorff, locally bounded topology on R. Two norms on R are said to be equivalent if they define the same topology on R.

We denote by d = (d12, …, d1n, d23, …, dn-1,n) a vector of distances between n points. Such a vector d is called a metric if it satisfies the triangle inequalities

(1)

The set of all metrics on n points forms a convex polyhedral cone, the extremal properties of which are discussed in [4]. We will be concerned with a sub-cone that is spanned by metrics of the form

(2)

where t ≧ 0, V is a proper subset of {1, 2, …, n}; and the symbol ⊥ is used for “exclusive or”: i ⊥ j ∈ V means i ∈ V, j ∉ V or i ∉ V,j∉ V.

Consider the following formula due to Young [7] for the calculation of the homogeneous product sum, hλ, in terms of Schur functions;

where the operation Srs is defined as follows:

Y1: Srs, where r < s, “represents the operation of moving one letter from the s-th row up to the r-th row; and the resulting term is regarded as zero, when any row becomes less than a row below it, or when letters from the same row overlap; as, for instance, happens when λ1 = λ2 in the case of S13S23.“

The following example of the above is given by Robinson [4].

Calculation by other means shows that the above analysis of h(3,2,1) is correct; however, it will be noticed that the operator S123S23 does not appear in the above yet it is not specifically excluded by the rule Y1.

Let R be a ring with identity and let H = End (E(RR)) and Q = Dou(E(RR)) = End(HE(RR)). Then Lambek [11] showed that Q is always isomorphic to Qm(R), the maximal right quotient ring of R. And Johnson [10] and Wong-Johnson [26] proved that Qm(R) is regular and right self-injective if and only if R is right non-singular, and then H is isomorphic to Qm(R), too. Moreover, Sandomierski [18] showed that Qm(R) is semi-simple Artinian if and only if R is right finite dimensional and right non-singular. And it is well known that Qm(R) is a quasi-Frobenius ring if and only if E(RR) is a rational extension of RR and the ACC holds on right annihilators of subsets of E(RR).

The purpose of this paper is to give some module-theoretic generalizations of these results. Let PR be an E(PR)-torsionless generator, and let S = End(PR), H = End(E(PR)) and Q = Dou(E(PR)).

Kan and Miller have shown in [9] that the homotopy type of a finite simplicial set K can be recovered from its R-algebra of 0-forms A0K, when R is a unique factorization domain. More precisely, if is the category of simplicial sets and is the category of R-algebras there is a contravariant functor

with

the simplicial set homomorphisms from X to the simplicial R-algebra ∇, where

and the faces and degeneracies of ∇ are induced by

and

respectively.

Soit K un compact du plan complexe C, d'intérieur non vide et connexe. A(K) désigne l'algèbre uniforme des fonctions continues sur et analytiques dans . Pour chaque , on note λz la mesure harmonique sur la frontière de du point z, et on pose λ = λz0, z0 étant un point fixé de . On dit que A(K) est une algèbre de Dirichlet faible* sur (— désigne le conjugué) est faiblement* dense dans L∞(λ) (la topologie faible* sur L∞(λ) est la topologie faible pour la dualité entre L1(λ) et L∞(X)); vu les inégalités de Harnack pour les fonctions harmoniques cette propriété est indépendante de la mesure harmonique du point z0 fixé dans .

Let {pn(z)}; be a sequence of functions analytic in a region D. A problem in analysis which has received much attention is the following: describe the sets Z ⊂ D for which

(1)

implies hn is 0 for all n, (To make the problem interesting, only those situations are studied where finite subsets of the pn(z) are linearly independent in D.) Another way of phrasing this is: Characterize the uniqueness sets of pn(z), a uniqueness set Z being a set in D such that the restriction of {pn(z)}; to Z is linearly independent. If Z is not a uniqueness set then for some {hn}; not all 0, we have

(2)

This formula is called a non-trivial representation of 0 (on Z).

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.

Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

Let p be an integer and let H(p) be the class-number of the field

where ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).

In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).

We designate by h(p) the class-number of the real quadratic field

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by

(1.1)

where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is

(1.2)

Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).

Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

Let L be a finitely generated extension of a field K of characteristic p ≠ 0. If L/K is algebraic, then there is a unique intermediate field S such that

S is just the maximal separable extension of K in L. If L/K is not algebraic, then Dieudonne [4] showed there exist maximal separable extensions D of K in L such that L ⊆ Kp–∞⊗KD. In general, not every maximal separable extension of K in L has the property. Those which do have the property are called distinguished. Kraft [7] established that a maximal separable extension D of K in L is distinguished if and only if [L:D] is as small as possible. If the minimum of the [L:D] is pr, r is called the order of inseparability of L/K, denoted inor (L/K).

Let L1 be an intermediate field of L/K. If L/K is algebraic, then the maximal separable extension S1 of K in L1 is contained in the maximal separable extension S of K in L, and moreover S is separable over S1.

G. R. Allan and A. M. Sinclair proved in [1] that if a commutative radical Banach algebra possesses bounded approximate identities then for every sequence (αn) of real numbers such that limn→∞αn = 0 there exists such that

In the other direction it is shown in [6] that if is separable and if the nilpotents are dense in then for every sequence (βn) of positive reals there exists such that

(This result was given in [2] for the Volterra algebra.)

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.

In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

Suppose is a C*-algebra and H is a Hilbert space. Let denote the set of completely positive maps from into the set B(H) of (bounded linear) operators on H. This paper studies the vector space spanned by , i.e., the linear maps that are finite linear combinations of completely positive maps. From another viewpoint, a map ϕ is in precisely when it has a decomposition ϕ = (ϕ1 – ϕ2) + i(ϕ3 – ϕ4) with ϕ1, ϕ2, ϕ3, ϕ4 in CP ; this decomposition is analogous to the Hahn decomposition for measures [8, 111.4.10] (see also Theorem 20). The analogous class of maps with “completely positive” replaced by “positive” was studied by R. I. Loebl [11] and S.-K. Tsui [17], and when is commutative, this latter class coincides withi , since every positive linear map on a commutative C*-algebra is completely positive [16].

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.

In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which

(1.1)

In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.