It is shown that simple stably projectionless ${{\text{C}}^{*}}$-algebras which are inductive limits of certain specified building blocks with trivial $\text{K}$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.

Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.

A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces $\left( X,Y \right)$ has the Bishop–Phelps–Bollobás property for every Banach space $Y$ when $X$ is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on ${{\ell }_{p}}\,\times \,{{\ell }_{q}}$$\left( 1\,<p,q\,<\,\infty \right)$.

The good weights for the one-sided Hardy-Littlewood operators have been characterized by conditions . In this paper we introduce a new condition which is analogous to A∞. We show several characterizations of . For example, we prove that the class of weights is the union of classes. We also give a new characterization of weights. Finally, as an application of condition, we characterize the weights for one-sided fractional integrals and one-sided fractional maximal operators.

Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L, that is, the Lie algebra of all derivations of L. Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L). However, there is still some intimate connection between the structure of D(L) and that of L.

Throughout this paper R denotes the set of all real numbers, m(K) the Lebesgue measure of K ⊆ R, H a Hilbert space, L(H) the set of all linear continuous mappings of H into H, endowed with the usual structure of a Banach space.

We consider the mapping F of the set R into L(H) such that

holds for all x, y ∈ R. In (2) we have solved this equation under the assumption that H is of finite dimension. In this paper we prove that a weak measurability of F implies its weak continuity in the case of separable Hilbert space. In Theorem 2 we prove that every weakly continuous solution of (1) in the set of normal transformations has the form F(x) = cos (xN), where the normal transformation N does not depend on x.

Given a positive continuous function $\mu $ on the interval $0\,<\,t\,\le \,1$, we consider the space of so-called $\mu $-Bloch functions on the unit ball. If $\mu \left( t \right)\,=\,t$, these are the classical Bloch functions. For $\mu $, we define a metric $F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of $\mu $-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

In (8, § 3.2) Steinberg proved the following result.

THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1’) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.

For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.

Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice.

The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple.

It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

Denote by Fα,β the multiplicative group generated by the two matrices where α and β are complex numbers. Sanov (3) proved that Fα,β is free, and Brenner (1) showed that Fm,m is free if m ⩾ 2.

In the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

In [11] and [12] we investigated the process of localization of right Noetherian rings R at prime ideals. We shall now extend these investigations to semiprime ideals N of R.

In Section 2 we show that localizing at the injective right R-module E(R/N) is the same as localizing with respect to the multiplicative set

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

Let

(1)

and set

(2)

Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”

(3)

In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases.

The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G

(1.1)

We study the local $L$-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for $\text{G}{{\text{L}}_{2}}\,\times \,\text{G}{{\text{L}}_{3}}$.

The purpose of this note is to characterize those distributive lattices that can be isomorphically embedded in free lattices. If it is known (cf. (2)) that in a free lattice every element is either additively or multiplicatively irreducible, and consequently every sublattice of a free lattice must also have this property. We therefore begin by studying the class of all those distributive lattices in which this condition is satisfied.