Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

ALPS: Matrices with nonpositive off-diagonal entries

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices having nonpositive off-diagonal entries. The major classes considered are the sets of diagonally stable, stable, and semipositive matrices, denoted respectively by $\text{A}$, $\text{L}$, and $\text{S}$. The conditions defining these classes are weakened, and the resulting classes are examined. Their relationship to the classes of real matrices $\text{P}$ and $\text{P}$₀, whose off-diagonal entries are nonpositive and whose principal minors are respectively all positive and all nonnegative, is also included.     

George Baron and the Mathematical Correspondent

George Baron immigrated to the United States from England at the end of the eighteenth century. He edited the first mathematical journal in the United States, the Mathematical Correspondent. This journal was strongly influenced by the popular English journals that contained mathematical problems and their solutions.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Yang Hui's commentary on the Ying Nu chapter of the Chiu Chang Suan Shu

One of the many commentators of the Chiu chang suan shu (Nine Chapters on the Mathematical Art), written approximately in the first century, is Yang Hui, who wrote Hsiang chieh chiu chang suan fa (A Detailed Analysis of the Mathematical Methods in the “Nine Chapters”) and the Chiu chang suan fa tsuan lei (Reclassification of the Mathematical Methods in the “Nine Chapters”) in 1261.The seventh chapter of the Chiu chang suan shu is called ying nu or ying pu tsu meaning “too much and not enough” or “surplus and deficiency.” It has been pointed out by historians of mathematics that the method known in Europe as the Rule of False Position is the same as the Chinese ying pu tsu method used in this chapter. Yang has made a detailed and thorough study of the twenty problems in this chapter. He has not only explained and expanded the method of ying pu tsu, but he has also suggested alternative and simpler solutions for some of the problems.This paper endeavours to translate the twenty problems, to analyse the ying pu tsu method, and above all to present a critical study of Yang's alternative methods.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Rickart C1 algebras,II

The theory of inner-outer factorization in the Hardy spaces H^p in the unit disc $\text{D}$ is well known and has many applications. It does not carry over to the spaces H^p on the polydisc $\text{D}$ⁿ or the ball $\text{B}$ⁿ when n > 1. However, for Lumer's Hardy spaces (LH)^p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)^p has a factorization f = I_ε × E_ε, where I_ε is internal and E_ε is external, and E_ε? (LH)^p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when $\text{p =}\text{2}\text{m}$, where m is an integer.     

ALPS: Classes of stable and semipositive matrices

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices. The standard conditions for three major classes of matrices are made both stronger and weaker, and the resulting classes are examined. The major classes are diagonally stable, stable, and semipositive matrices, denoted by $\text{A,L,}$ and $\text{S,}$ respectively. Their relationship to the classes of matrices whose principal minors are positive (denoted by $\text{P}$ and non-negative (denoted by $\text{P}$₀) is also presented.     

An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

On the sources of my book Moderne algebra

In December 1971, Garrett Birkhoff asked me to give my view on the main sources for my book. I wrote him a seven-page letter with two supplements. He intended to publish an edited version of my letter, with some commentary of his own, but in the course of our correspondence it turned out that both versions were unsatisfactory. I shall now present an extended record, explaining more fully how I came to write the book and what was the general situation in algebra at that time.     

Compact G-Fuzzy topological spaces

Considering complete Boolean algebras $\text{G}$ as sets of truth values a new concept of compactness—so-called probabilistic compactness — is introduced to $\text{G}$-fuzzy topological spaces. The aim of this paper is to show that the most important theorems of the theory of ordinary compact spaces remain true; e.g. probabilistic compactness is preserved under projective limits, every probabilistic compact space has an unique $\text{G}$-fuzzy uniformity being compatible with the underlying $\text{G}$-fuzzy topology, etc. Finally using the selection theorem due to Kuratowski and Ryll-Nardzewski a non-trivial example of a probabilistic compact space is given.     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

A decomposition of R into two homeomorphic rigid parts

Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D₁, D₂, h) of two countable dense subsets D₁ and D₂ and a homeomorphism h: X?D₁ → X?D₂, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for $\text{X =}\text{R}$, such atriple is shown to exist.     

The class A(R,S) of (0, 1)-matrices

Let R and S be two vectors whose components are m and n non-negative integers, respectively. Let $\text{A}$(R, S) be the class consisting of all (0, 1)-matrices of size m by n with row sum vector R and column sum vector S. In this paper we derive a lower bound to the cardinality of the class $\text{A}$(R, S), which can be computed readily.     

The contraction of K to NM

By studying the contraction of K to $\text{N}\text{M}$ inside the Iwasawa decomposition of a semisimple Lie group of real rank 1, we realize the representations of $\text{NM}$ as limits of representations of K, obtaining in particular, limiting formulas for matrix entries.