In this paper, R ⊂ T will be commutative domains having a common identity.

Definition. Suppose that R is a subdomain of T.

(i) If P is a prime ideal of R and Q is a prime ideal of T, we say that Q lies over P if Q ∩ R = P.

(ii) If every prime of R has a prime of T lying over it, we say that R ⊂ T has lying over.

(iii) If there is a unique prime of T lying over P in R, we say that P is unibranched in T.

(iv) If every prime of R is unibranched in T we say that R ⊂ T is unibranched.

One of the authors published in 1965 a paper with identical title (1), in which the following result was proved:

Theorem A. Let ϕ(r) be increasing and convex in log r, with

Following [10] an ergodic measure-preserving transformation is called rank one if it admits a sequence of approximating stacks. Rank one transformations have been studied in [1] and [2] where it was shown that any rank one transformation has simple spectrum. More generally it has been shown by Chacon [4] that a transformation of rank n has spectral multiplicity at most n. M. A. Akcoglu and J. R. Baxter have asked whether the converse is true. In particular: does simple spectrum imply rank one? In this paper we give a negative answer to this question.

Projectors, spectral functions, carriers, and collections of these objects are some of the tools which have been used to study vector lattices. One of our objectives in this paper is to show that these various approaches are not essentially different. We do this by proving that each of the above-mentioned objects can be identified with a collection of prime ideals.

A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.

A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

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.

In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug–1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class.

Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined by

if u ≠ 1, and by EG(u, U) = U if u = 1.

A conference matrix of order n is a square matrix C with zeros on the diagonal and ±1 elsewhere, which satisfies the orthogonality condition CCT = (n — 1)I. If in addition C is symmetric, C =CT, then its order n is congruent to 2 modulo 4 (see [5]). Symmetric conference matrices (C) are related to several important combinatorial configurations such as regular two-graphs, equiangular lines, Hadamard matrices and balanced incomplete block designs [1; 5; and 7, pp. 293-400]. We shall require several definitions.

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.

The object of this paper is to develop a method of constructing balanced incomplete block designs. It consists in utilizing the existence of two balanced incomplete block designs to obtain another such design by what may be called the method of composition.

1. Preliminary results on orthogonal arrays and balanced incomplete block designs. Consider a matrix A = (aij) of k rows and N columns, where each aij represents one of the integers 1, 2, … , s. Consider all t-rowed submatrices of N columns, which can be formed from this array, t ≤ k. Each column of any Crowed submatrix can be regarded as an ordered t-plet.

In this paper we make explicit all $L$-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\operatorname{Re}\left( s \right)\,\ge \,1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions.

We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

Suppose that G is a reductive algebraic group defined over Q. There occurs in the trace formula a remarkable distribution on G(A)1 which is supported on the unipotent set. It is defined quite concretely in terms of a certain integral over G(Q)\G(A)1. Despite its explicit description, however, this distribution is not easily expressed locally, in terms of integrals on the groups G(QV). For many applications of the trace formula, it will be essential to do this. In the present paper we shall solve the problem up to some undetermined constants.

The distribution, which we shall denote by Junip, was defined in [1] and [3] as one of a family {Jo} of distributions. It is the value at T = T0 of a certain polynomial . We shall recall the precise definition in Section 1.

Let be a complex Banach space and the algebra of bounded operators on . M. Rosenblum's theorem [13; 12] (also discovered by M. G. Kreĭn, cf. [9]) states that (if A, B are fixed bounded operators) the spectrum of the operator on defined by = AX – XB is contained in σ (A) – σ(B) = {α – β : α∊σ(A), β∊σ(B)}. In particular, the condition σ(A) ∩ σ(B) = Ø implies that for each Y ∊ there is a unique X ∊ such that AX – XB = Y. This does not completely settle the question of solvability of the equation AX – XB = Y: for example, if A is the backward unilateral shift and B = 0, then the equation has a solution (for any Y) even though σ(B) ⊆ σ(A).

In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn(α, β)(x), α, β > –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α + β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β.

The word ring is used to mean commutative ring. Just as valuations on fields are used to study domains, so valuations on rings can be used to study rings; these rings need not have units [12]. We introduce slightly weaker conditions than having identity in order to get a more general theory. A Prufer ring A is one in which every finitely generated regular ideal is invertible. If we replace invertibility in the total quotient ring K, by invertibility in a ring R where A ⊆ R ⊆ K we get an R-Prufer ring. These rings do occur, for example the Witt ring of a non-Pythagorean field or a ring of bounded continuous functions.

We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

In discussing an isomorphism between two Banach algebras, one will ordinarily tacitly assume that the mapping is linear (i.e., preserves the complex scalars as well as the ring operations). In general this cannot be avoided; for instance if the two Banach algebras are just the field of complex numbers, then the isomorphism is unrestricted, and could be given by any one of the myriads of discontinuous automorphisms of the complex numbers.