Duality theory for covariant systems
Magnus B.
Landstad
223-267
Abstract: If $(A,\rho ,G)$ is a covariant system over a locally compact group G, i.e. $\rho$ is a homomorphism from G into the group of $^{\ast}$-automorphisms of an operator algebra A, there is a new operator algebra $\mathfrak{A}$ called the covariance algebra associated with $(A,\rho ,G)$. If A is a von Neumann algebra and $\rho$ is $\sigma$-weakly continuous, $\mathfrak{A}$ is defined such that it is a von Neumann algebra. If A is a $ {C^{\ast}}$-algebra and $ \rho$ is norm-continuous $\mathfrak{A}$ will be a $ {C^{\ast}}$-algebra. The following problems are studied in these two different settings: 1. If $ \mathfrak{A}$ is a covariance algebra, how do we recover A and $ \rho$? 2. When is an operator algebra $ \mathfrak{A}$ the covariance algebra for some covariant system over a given locally compact group G?
Sheaf constructions and their elementary properties
Stanley
Burris;
Heinrich
Werner
269-309
Abstract: We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces.
Some new constructions and estimates in the problem of least area
Harold
Parks
311-346
Abstract: Surfaces of least k dimensional area in ${\textbf{R}^n}$ are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let $1 \,\, \leqslant \,\,k\,\, \leqslant \,n$ be integers and $ B\, \subset \,{\textbf{R}^n}$ be compact and $k\, - \,1$ rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group ${H_{k - 1}}\left( {B;\,\,G} \right)$ (in case $k = \,1$, suppose, additionally, L is contained in the kernel of the usual augmentation map). J. F. Adams has defined what it means for a compact set ${\rm X}\, \subset \,{\textbf{R}^n}$ to span L. Using also a natural notion of what it means for a compact set to be $ \varepsilon$-thick, we show that, for each $ \varepsilon \, > \,0$, there exists an $ \varepsilon$-thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as $ \varepsilon$ approaches 0 a subsequence of the above volume minimizing sets converges in the Hausdorff distance topology to a set, X, which minimizes k dimensional area subject to the requirement that it span L. It follows, of course, from the regularity results of Reifenberg or Almgren that, except for a compact singular set of zero k dimensional measure, X is a real analytic minimal submanifold of ${\textbf{R}^n}$.
Subgroups of classical groups generated by long root elements
William M.
Kantor
347-379
Abstract: All conjugacy classes of subgroups G of classical groups of characteristic p are determined, which are generated by a conjugacy class of long root elements and satisfy
Nonselfadjoint crossed products (invariant subspaces and maximality)
Michael
McAsey;
Paul S.
Muhly;
Kichi-Suke
Saito
381-409
Abstract: Let $\mathcal{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra $ {\mathcal{L}_ + }$ of $\mathcal{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group on $\mathcal{L}$ is nonnegative, and we determine conditions under which ${\mathcal{L}_ + }$ is maximal among the $ \sigma$-weakly closed subalgebras of $ \mathcal{L}$. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2) ${\mathcal{L}_ + }$ is a maximal $\sigma $-weakly closed subalgebra of $\mathcal{L}$; and (3) a version of the Beurling, Lax, Halmos theorem is valid for ${\mathcal{L}_ + }$. In addition, we prove that if $\mathfrak{A}$ is a subdiagonal algebra in a von Neumann algebra $ \mathcal{B}$ and if a form of the Beurling, Lax, Halmos theorem holds for $\mathfrak{A}$, then $ \mathcal{B}$ is isomorphic to a crossed product of the form $\mathcal{L}$ and $ \mathfrak{A}$ is isomorphic to ${\mathcal{L}_ + }$.
Commutativity in series of ordinals: a study of invariants
J. L.
Hickman
411-434
Abstract: It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson's interesting paper.
A class of Schur algebras
M.
Brender
435-444
Abstract: This paper delineates a class of Schur algebras over a finite group G, parametrized by two subgroups $K\, \triangleleft \,H\, \subset \,G$. The constructed Schur algebra ${\text{C}}\left[ G \right]_K^H$ is maximal for the two properties (a) centralizing the elements of H, and (b) containing the elements of K in the identity. Most commonly considered examples of Schur algebras fall into this class. A complete set of characters of ${\text{C}}\left[ G \right]_K^H$ is given in terms of the spherical functions on the group G with respect to the subgroup H. Necessary and sufficient conditions are given for this Schur algebra to be commutative, in terms of a condition on restriction multiplicities of characters. This leads to a second-orthogonality-type relation among a subset of the spherical functions. Finally, as an application, a particular Schur algebra of this class is analyzed, and shown to be a direct sum of centralizer rings.