If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

By "positive definite matrices" or, briefly, definite matrices, we mean in this note self-adjoint matrices all the characteristic values of which are positive. Alternatively, they can be defined as matrices, all the hermitian quadratic forms of which are real and positive. It is well known that these two definitions are equivalent and it is the existence of two equivalent definitions which renders the subject particularly interesting. Thus, it follows from the first definition that A-1 is positive definite for positive definite A ; it follows from the second definition that the sum of two positive definite matrices is also positive definite. Similarly, we shall call a matrix positive semidefinite or, briefly, semidefinite if it is self-adjoint and none of its characteristic values are negative, or if all of its hermitian quadratic forms are non-negative. A matrix which is definite is also semidefinite but a semidefinite matrix is, of course, not necessarily definite.

All finite dimensional indecomposable representations of Ut(Sl (2)) at roots of 1 are determined.

In this paper we are concerned with the fourth order nonlinear differential equation

where the following conditions are always assumed to hold:

(a) r(t) is continuous and positive for t ≠ 0, and

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.

Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.

As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

In this paper we shall give an elementary proof of the theorem

(1.1)

where φ(k) denotes Euler's function, and

(1.2)

where p denotes the prime, and and are integers with (,) = 1, positive.

It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]).

Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups.

For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.

Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

A space X is in-expandable, where m is an infinite cardinal, if for every locally finite collection {Hα| α ∈ A} of subsets of X with |A| ≦ m(cardinality of A ≦ m) there exists a locally finite collection of open subsets {Gα| α ∈ A} such that Hα ⊆ Gα for every α ∈ A. X is expandable if it is m-expandable for every cardinal m. The notion of expandability is closely related to that of collection wise normality introduced by Bing [1], X is collectionwise normal if for every discrete collection of subsets {Hα|α ∈ A} there is a discrete collection of open subsets {Gα|α ∈ A} such that Hα ⊆ Gα for every α ∈ A. Expandable spaces share many of the properties possessed by collectionwise normal spaces. For example, an expandable developable space is metrizable and an expandable metacompact space is paracompact.

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius

1

of a Hilbert space operator T. Among the many interesting developments, we may mention:

(a) C. Berger's proof of the “power inequality”

2

(b) R. Bouldin's result that

3

for any isometry V commuting with T;

(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;

(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

Let

denote the Nth Fermat curve. The period lattice of F(N) is contained with finite index in the product of certain lattices Lr,s (see [6]), and to this inclusion of lattices there corresponds an isogeny of the Jacobian of F(N) onto a product of abelian varieties. The purpose of this paper is to determine when two factors in this product are isogenous over C, and whether they are absolutely simple.

Let denote the q-shifted factorial, and set Recently, Gasper [7] showed that some bibasic summation formulas derived by Carlitz [5], Al-Salam and Verma [1], and Wm. Gosper could be extended to the indefinite bibasic summation formula where p and q are independent bases and a, b, c are arbitrary parameters.

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.

We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called ${{\Gamma }_{2}}$-calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.

Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.

Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.

A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.

We show that certain ${{C}^{*}}$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

It has been shown by Diederichsen [2] that for integral representations of a finite group, the irreducible constituents in any complete reduction are not necessarily unique up to order and unimodular equivalence. In this same article, it is shown that for certain finite groups, such as the cyclic group of order 4, there are infinitely many classes of indecomposable representations under unimodular equivalence.