Kaplansky introduced the notions of CCR and GCR $C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.

Disconjugacy criteria have been established for linear selfadjoint differential equations of order 2n by Sternberg [4] and Ahlbrandt [1]. Such differential equations can be written in the form

1.1

where it is assumed that the coefficients are real and that Pn(x) ≠ 0. We shall be interested in nontrivial solutions v(x) of (1.1), which satisfy

1.2

for distinct points α and β. The smallest β> α such that (1.2) is satisfied nontrivially by a solution of (1.1), is denoted by μ1(α) and called the first conjugate point of x = α with respect to (1.1). If no such conjugate point exists we write μ1(α) = ∞, and say that (1.1) is disconjugate on [α, ∞).

The principal purpose of this paper is to generalize these disconjugacy criteria to the general linear nonselfadjoint differential equation of the form

1.3

We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.

Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent:

(i) every derivation of A is inner;

(ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit.

In a paper with the above title H. Heilbronn [4] proved that, given any real θ, there exist infinitely many positive integers n such that the distance ║θn2║ from θn2 to its nearest integral neighbour satisfies the bound

He actually proved a somewhat stronger statement which shows that the integers n occur with some regularity and he suggested that perhaps the exponent may be replaced by . This theorem has attracted considerable attention and spawned a number of generalizations (see [1], [7] and the references therein), yet no essential improvement has been given for the original problem (but see [6], [8]).

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.

(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.

We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

The Heisenberg group is considered as the boundary of a manifold. A class of hypersurfaces in this manifold can be regarded as copies of the Heisenberg group. The properties of geodesics in the interior and on the hypersurfaces are worked out in detail. These properties are strongly related to those of the Heisenberg group.

A normal Moore space is non-metrizable only if it fails to be ƛ-collectionwise normal for some uncountable cardinal ƛ [1].

For each uncountable cardinal X we present a class of normal, locally metrizable Moore spaces and a particular space Sλ in . If there is any space of class which is not X-collectionwise normal, then Sλ is such a space. The conditions for membership in make a space in behave like a subset of a product of a Moore space with a metric space. The class is sufficiently large to allow us to prove the following. Suppose F is a locally compact, 0-dimensional Moore space (not necessarily normal) with a basis of cardinality X and M is a metric space which is O-dimensional in the covering sense. If there is a normal, not X-collectionwise normal Moore space X where X ⊂ Y × M, then Sx is a normal, not λ-collectionwise normal Moore space.

Let denote an arbitrary Boolean algebra. Let Latin letters a, b, … denote general elements of while the symbols 0, 1 denote the special smallest and largest elements. Let Greek letters α, β, … denote various prime dual ideals of elements of . It is recalled that a prime dual ideal of is a proper subset of closed under finite intersections of its elements and maximal with respect to those properties. Every prime dual ideal includes the element 1 and for each element a of includes either a or ā (complement of ain ) but not both. Occasional reference will be made to principal dual ideals of . These are subsets of composed of all elements of majorizing some fixed non-zero element of . Finally, let X() denote the collection of all prime dual ideals of .

A finite group G is a spherical space form group if it admits a fixed point free representation τ:G → U(k) for some k; for the remainder of this paper, we assume G is such a group. The eta invariant of Atiyah et al [2] defines Q/Z valued invariants of equivariant bordism. In [6], we showed the eta invariant completely detects the reduced equivariant unitary bordism groups and completely detects all but the 2-primary part of the reduced equivariant SpinC bordism groups . The coefficient ring is without torsion; all the torsion in is of order 2. The prime 2 plays a distinguished role in the discussion of equivariant SpinC bordism and is quite different from at the prime 2. Let ker*(η, G) denote the kernel of all eta invariants and let ker*(SW, G) denote the kernel of the Z2-equivariant Stiefel-Whitney numbers (see Section 1 for details). Then:

THEOREM 0.1. Let. If M = ker*(η, G) ∩ ker*(SW, G), M = 0.

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup ${{\left( {{\mu }_{t}} \right)}_{t>0}}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if ${{\left( {{\mu }_{t}} \right)}_{t>0}}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto {{\mu }_{t}}\left( x \right),t>0$, such that ${{\lim }_{t\to 0}}t\log {{\mu }_{t}}\left( e \right)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega $ can be represented by a continuous function in $\Omega $. Examples are discussed.

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

This paper investigates special conditions under which the tensor product of two von Neumann algebras will be non-isomorphic to the tensor product of two others. The main tools are the algebraic invariants property Λx (x ≧ 0) (first defined by Powers [18]) and the r∞ and ρ sets (defined by Araki and Woods [3]).

It is well-known that an infinite loop space is an H-space whose multiplication enjoys nice properties concerning associativity and commutativity. A practical way of identifying infinite loop spaces is the utilization of May's recognition principle [3; 4]. To apply this principle, one requires an E∞-operad action on a space X; this action gives rise to various multiplications on X. In this note, it is shown that such multiplications enjoy an operad action up to homotopy that encodes the associativity and commutativity information, and that May's delooping theorem may be applied to them. We refer to [3] for the terminology of operads and monads.

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

L'ensemble des nombres ∈ Z/n, premiers avec l'entier n, forme un groupe (le groupe géométrique) G, par rapport à la multiplication. Etant donné un ensemble de nombres réels, M, un groupoïde Q, formé d'éléments quelconques, x, est automorphe par le groupe géométrique si (i) pour tout élément x ∈ Q et tout nombre m ∈ M, la multiplication xm est définie; (ii) l'application (x→xm) est un automorphisme de Q.

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or ∞. The proof uses Mourre theory.

Recently I (1) gave some new basic hypergeometric identities of the Cayley and Orr type with the help of a certain basic differential operator. The present paper deals with some bilateral generalizations of those identities together with certain new identities of the same type. In § 4 is indicated how the generalizations of Orr's identities given recently by Shukla (8, Theorems I, II) may be connected with each other. Later in § 5 certain general expansions of hypergeometric functions are deduced.

Let $G$ be a finite group and $\chi $ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi H$ has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi $ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi H$ has a linear constituent with multiplicity 1.

We show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$. This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as

$$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$

$\boldsymbol {A}_0$

$\Theta $

$\mathbb {C}^d$

$\boldsymbol {A}_0$

$\mathcal {H}_{-1}(\boldsymbol {A}_0)$

$\Theta $

The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.

As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.