Induced representations of $C\sp{\ast} $-algebras and complete positivity
James G.
Bennett
1-36
Abstract: It is shown that $ ^{*}$representations may be induced from one $ {C^{\ast}}$-algebra B to another $ {C^{\ast}}$-algebra A via a vector space equipped with a completely positive B-valued inner product and a $ ^{*}$representation of A. Theorems are proved on induction in stages, on continuity of the inducing process and on completely positive linear maps of finite dimensional ${C^{\ast}}$-algebras and of group algebras.
Lexicographic partial order
Henry
Crapo
37-51
Abstract: Given a (partially) ordered set P with the descending chain condition, and an ordered set Q, the set $ {Q^P}$ of functions from P to Q has a natural lexicographic order, given by $f \leqslant g$ if and only if $f(y) < g(y)$ for all minimal elements of the set $ \{ x;f(x) \ne g(x)\}$ where the functions differ. We show that if Q is a complete lattice, so also is the set ${Q^P}$, in the lexicographic order. The same holds for the set $ {\operatorname{Hom}}(P,Q)$ of order-preserving functions, and for the set $ {\text{Op}}(P)$ of increasing order-preserving functions on the set P. However, the set $ {\text{Cl}}(P)$ of closure operators on P is not necessarily a lattice even if P is a complete lattice.
Vector valued eigenfunctions of ergodic transformations
E.
Flytzanis
53-60
Abstract: We study the solutions X, T, of the eigenoperator equation $\displaystyle X(h( \cdot ))\,= \,TX( \cdot )\,{\text{a}}{\text{.e}}{\text{.}}$ , where h is a measurable transformation in a $ \sigma$-finite measure space $(S,\Sigma ,m)$, T is a bounded linear operator in a separable Hilbert space H and $X:S \to H$ is Borel measurable. We solve the equation for some classes of measure preserving transformations. For the general case we obtain necessary conditions concerning the eigenoperators, in terms of operators induced by h in the scalar function spaces over the measure space. Finally we investigate integrability properties of the eigenfunctions.
Square integrable representations and a Plancherel theorem for parabolic subgroups
Frederick W.
Keene
61-73
Abstract: Let G be a semisimple Lie group with Iwasawa decomposition $G\, = \,KAN$. Let ${\mathfrak{g}_0} \,=\, {\mathcal{f}_0} \,+\, \mathfrak{a} \,+\, \mathfrak{n}$ be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type ${A_1}$ or ${A_2}$. The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the ${A_1}$ case for the solvable subgroup NA by combining Mackey's ``Little Group'' method with an idea due to C. C. Moore: we find an operator D, defined on the $ {C^\infty }$ functions on NA with compact support, such that $\displaystyle \phi (e) = \int_{{{(NA)}^\wedge}} {{\text{tr}}} (D\pi (\phi ))d\mu (\pi )$ where ${(NA)^ \wedge }$ is the unitary dual, e is the identity, and $\mu$ is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the $ {A_1}$ case, D involves fractional powers of the Laplace operator and hence is not a differential operator.
Nullity and generalized characteristic classes of differential manifolds
Sin Leng
Tan
75-88
Abstract: Using the Kamber-Tondeur construction of characteristic classes for foliated bundes, the author has given a method for constructing generalized characteristic classes for a differentiable manifold M without imposing conditions on M. In particular a vanishing theorem on the manifold M is obtained. The construction is particularly useful if the ordinary characteristic ring Pont*(M) of the manifold M vanishes much below the dimension of M.
Functions operating on positive definite matrices and a theorem of Schoenberg
Jens Peter Reus
Christensen;
Paul
Ressel
89-95
Abstract: We prove that the set of all functions $f:\,[ - 1,\,1] \to [ - 1,\,1]$ operating on real positive definite matrices and normalized such that $f(1)\, = \,1$, is a Bauer simplex, and we identify its extreme points. As an application we obtain Schoenberg's theorem characterising positive definite kernels on the infinite dimensional Hilbert sphere.
Invariant differential equations on certain semisimple Lie groups
F.
Rouvière
97-114
Abstract: If G is a semisimple Lie group with one conjugacy class of Cartan subalgebras (e.g. a complex semisimple Lie group), a bi-invariant differential equation on G can be reduced by means of the Radon transform to one on the subgroup MA. In particular, all polynomials of the Casimir operator have a central fundamental solution, and are solvable in $ {C^\infty }(G)$; but, for G complex, the ``imaginary'' Casimir operator is not.
The minimum norm projection on $C\sp{2}$-manifolds in ${\bf R}\sp{n}$
Theagenis J.
Abatzoglou
115-122
Abstract: We study the notion of best approximation from a point $x \in {R^n}$ to a ${C^2}$-manifold. Using the concept of radius of curvature, introduced by J. R. Rice, we obtain a formula for the Fréchet derivative of the minimum norm projection (best approximation) of $x \in {R^n}$ into the manifold. We also compute the norm of this derivative in terms of the radius of curvature.
Branch point structure of covering maps onto nonorientable surfaces
Cloyd L.
Ezell
123-133
Abstract: Let $f:M\, \to \,N$ be a degree n branched cover onto a compact, connected nonorientable surface with branch points ${y_1},\,{y_2},\, \ldots ,\,{y_m}$ in N, and let the multiplicities at points in ${f^{ - 1}}({y_i})$ be $ {\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}}$. The branching array of f, designated by B, is the following array of numbers: \begin{displaymath}\begin{gathered}{\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _... ...m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end{gathered} \end{displaymath} We show that the numbers in the branching array must always satisfy the following conditions: (1) $\displaystyle \sum {\{ {\mu _{ij}} \,+ \,1\vert j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n}$ , (2) $ \sum {\{ {\mu _{ij}}\vert i\,=\, 1,\,2, \ldots ,m;j\,=\, 1,\,2, \ldots ,{k_i}\} }$ is even. Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.
Pure weak mixing
R.
Ellis;
S.
Glasner
135-146
Abstract: We show that for the group of integers Z, the maximal common factor of the maximal weakly mixing minimal flows coincides with the universal purely weakly mixing flow. We show that the maximal weakly mixing minimal flows are not all isomorphic to each other. None of the maximal w.m. flows are regular.
A note on the operator $X\rightarrow AX-XB$
L.
Fialkow
147-168
Abstract: If A and B are bounded linear operators on an infinite dimensional complex Hilbert space $\mathcal{H}$, let $\tau (X)\, = \,AX\, - \,XB$ (X in $ \mathcal{L}(\mathcal{H})$). It is proved that $\sigma (\tau )\, = \,\sigma (\tau \vert{C_p})\,(1\, \leqslant \,p\, \leqslant \infty )$, where, for $ 1\, \leqslant p\, < \,\infty$, ${C_p}$ is the Schatten p-ideal, and ${C_\infty }$ is the ideal of all compact operators in $ \mathcal{L}(\mathcal{H})$. Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for $\tau$ to be injective. It is also proved that if A and B are quasisimilar, then the right essential spectrum of A intersects the left essential spectrum of B.
Absolute continuity in the dual of a Banach algebra
Stephen Jay
Berman
169-194
Abstract: If A is a Banach algebra, G is in the dual space ${A^{\ast}}$, and I is a closed ideal in A, then let ${\left\Vert G \right\Vert _{{I^{\ast}}}}$ denote the norm of the restriction of G to I. We define a relation $\ll$ in $ {A^{\ast}}$ as follows: $G \ll L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if I is a closed ideal in A and ${\left\Vert L \right\Vert _{{I^{\ast}}}} < \delta$ then ${\left\Vert G \right\Vert _{{I^{\ast}}}} < \varepsilon$. We explore this relation (which coincides with absolute continuity of measures when A is the algebra of continuous functions on a compact space) and related concepts in the context of several Banach algebras, particularly the algebra ${C^1}[0,1]$ of differentiable functions and the algebra of continuous functions on the disc with holomorphic extensions to the interior. We also consider generalizations to noncommutative algebras and Banach modules.
Maxima of random algebraic curves
M.
Das;
S. S.
Bhatt
195-212
Abstract: Let ${X_1},{X_2}, \ldots ,{X_n}$ be a sequence of independent and identically distributed random variables with common characteristic function ${\exp}( - {\left\vert Z \right\vert^\alpha })$ where $0 < \alpha \leqslant 2$, and $P(x) = \sum\nolimits_1^n {{X_k}{x^k}}$. Then we show that the numbers ${M_n}$ of maxima of the curves $y = P(x)$ have expectation $E{M_n} \sim c \log n$, as $n \to \infty$, where $c = c(\alpha ) = {c_1}(\alpha ) + {c_2}(\alpha )$ and \begin{displaymath}\begin{array}{*{20}{c}} {{c_1}(\alpha )\, = \,\frac{1} {{{\pi... ...,\alpha )}}\,\exp \,( - z)\,dz} \right\}} \,dv.} } \end{array} \end{displaymath}
Interpolation by complex splines
J.
Tzimbalario
213-222
Abstract: In this paper we solve the problem of interpolation by certain class of cardinal complex splines. This solution is used to complete the study of cardinal trigonometric splines started in [10] and also to give shorter proofs and to complete the results found for the interpolation problem by complex splines over the unit circle by I. J. Schoenberg [9], J. H. Ahlberg, E. N. Nilson and J. L. Walsh [1].
Absolutely area minimizing singular cones of arbitrary codimension
David
Bindschadler
223-233
Abstract: The examples of area minimizing singular cones of codimension one discovered by Bombieri, DeGiorgi and Guisti are generalized to arbitrary codimension, thus filling a dimensional gap. Previously the only nontrivial examples of singular area minimizing integral currents of codimension other than one were obtained from holomorphic varieties and hence of even codimension. Specifically, let S be the N-fold Cartesian product of p-dimensional spheres and C be the cone over S. We prove that for p sufficiently large, C is absolutely area minimizing. It follows from the technique used that C restricted to the ball of radius ${N^{1/2}}$ is the unique solution to the oriented Plateau problem with boundary S.
Degrees of irreducible characters of $(B,\,N)$-pairs of types $E\sb{6}$ and $E\sb{7}$
David B.
Surowski
235-249
Abstract: Let G be a finite (B, N)-pair whose Coxeter system is of type ${E_6}$ or ${E_7}$. Let $1_B^G$ be the permutation character of the action of G on the left cosets of the Borel subgroup B in G. In this paper we give the character degrees of the irreducible constituents of $ 1_B^G$.
Centers of hypergroups
Kenneth A.
Ross
251-269
Abstract: This paper initiates a study of Z-hypergroups, that is, commutative topological hypergroups K such that $K/Z$ is compact where Z denotes the maximum subgroup (equivalently, the center) of K. The character hypergroup $ {K^\wedge}$ is studied and its connection with the locally compact abelian group ${Z^\wedge}$ is given. Each Z-group is shown to correspond in a natural way to a Z-hypergroup. It is observed that the dual of a Z-group is itself a hypergroup. The basic orthogonality relations on Z-groups due to S. Grosser and M. Moskowitz are shown to hold for most Z-hypergroups. Some results on measure algebras of compact hypergroups due to C. F. Dunkl are extended to a class of noncompact hypergroups.
Amenable pairs of groups and ergodic actions and the associated von Neumann algebras
Robert J.
Zimmer
271-286
Abstract: If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair $(X,Y)$. This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.
On the stable decomposition of $\Omega \sp{2}S\sp{r+2}$
E. H.
Brown;
F. P.
Peterson
287-298
Abstract: In this paper we show that $ {\Omega ^2}{S^{r + 2}}$ is stably homotopy equivalent to a wedge of suspensions of other spaces $C_k^1$, and that $C_k^1$ is homotopy 2-equivalent to the Brown-Gitler spectrum.
On approximation by shifts and a theorem of Wiener
R. A.
Zalik
299-308
Abstract: We study the completeness in ${L_2}(R)$ of sequences of the form $\{ f({c_n} - t)\}$, where $\{ {c_n}\}$ is a sequence of distinct real numbers. A Müntztype theorem is proved, valid for a large class of functions and, in particular, for $f(t) = \exp ( - {t^2})$.
Erratum to: ``Periodic homeomorphisms of $3$-manifolds fibered over $S\sp{1}$'' (Trans. Amer. Math. Soc. {\bf 223} (1976), 223--234)
Jeffrey L.
Tollefson
309-310