Square-integrable factor representations of locally compact groups
Jonathan
Rosenberg
1-33
Abstract: The well-known theory of square-integrable representations is generalized to the case of primary representations (not necessarily type I) quasi-contained in either the regular representation or the representation induced from a character of the center of a (not necessarily unimodular) locally compact group, and relations with the topology of the primitive ideal space of the group $ {C^\ast}$-algebra are obtained. The cases of discrete and almost connected groups are examined in more detail, and it is shown that for such groups, square-integrable factor representations must be traceable. For connected Lie groups, these representations can (in principle) be determined up to quasi-equivalence using a complicated construction of L. Pukanszky-for type I simply connected solvable Lie groups, the characterization reduces to that conjectured by C. C. Moore and J. Wolf. In the case of unimodular exponential groups, essentially everything is as in the nilpotent case (including a result on multiplicities in the decomposition of $ {L^2}(G/\Gamma )$, $ \Gamma$ a discrete uniform subgroup of G). Finally, it is shown that the same criterion as for type I solvable Lie groups characterizes the squareintegrable representations of certain solvable $ \mathfrak{p}$-adic groups studied by R. Howe.
Free states of the gauge invariant canonical anticommutation relations
B. M.
Baker
35-61
Abstract: The gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is viewed as an inductive limit of finitedimensional ${C^\ast}$-algebras, and a study is made of a simple class of its representations. In particular, representations induced by restricting the wellknown gauge invariant generalized free states from the entire canonical anticommutation relations (henceforth CAR) are considered. Denoting (a) a state of the CAR by $\omega$ and its restriction to the GICAR by ${\omega ^ \circ }$, (b) the unique gauge invariant generalized free state of the CAR such that $\omega (a{(f)^\ast}a(g)) = (f,Ag)$ by ${\omega _A}$, it is shown that $(1)\;\omega _A^ \circ$ induces (an impure) factor representation of the GICAR if and only if ${\text{Tr}}\;A(I - A) = \infty$, (2) two (impure) GICAR factor representations $\omega _A^ \circ$ and $\omega_B^\circ$ are quasi-equivalent if and only if $ {A^{1/2}} - {B^{1/2}}$ and $ {(I - A)^{1/2}} - {(I - B)^{1/2}}$ are Hilbert-Schmidt class operators.
The $\alpha $-union theorem and generalized primitive recursion
Barry E.
Jacobs
63-81
Abstract: A generalization to $ \alpha$-recursion theory of the McCreight-Meyer Union Theorem is proved. Theorem. Let $\Phi$ be an $\alpha $-computational complexity measure and $\{ {f_\varepsilon }\vert\varepsilon < \alpha \}$ an $\alpha$-r.e. strictly increasing sequence of $\alpha$-recursive functions. Then there exists an $\alpha$-recursive function k such that $C_k^\Phi = { \cup _{\varepsilon < \alpha }}C_{{f_\varepsilon }}^\Phi $. The proof entails a no-injury cancellation atop a finite-injury priority construction and necessitates a blocking strategy to insure proper convergence. Two infinite analogues to ($ \omega$-) primitive recursive functions are studied. Although these generalizations coincide at $\omega$, they diverge on all admissible $\alpha > \omega$. Several well-known complexity properties of primitive recursive functions hold for one class but fail for the other. It is seen that the Jensen-Karp ordinally primitive recursive functions restricted to admissible $\alpha > \omega$ cannot possess natural analogues to Grzegorczyk's hierarchy.
Necessary and sufficient conditions for the ${\rm GHS}$ inequality with applications to analysis and probability
Richard S.
Ellis;
Charles M.
Newman
83-99
Abstract: The GHS inequality is an important tool in the study of the Ising model of ferromagnetism (a model in equilibrium statistical mechanics) and in Euclidean quantum field theory. This paper derives necessary and sufficient conditions on an Ising spin system for the GHS inequality to be valid. Applications to convexity-preserving properties of certain differential equations and diffusion processes are given.
Application of the dual-process method to the study of a certain singular diffusion
David
Williams
101-110
Abstract: This paper should be regarded as a sequel to a paper by Holley, Stroock and the author. Its primary purpose is to provide further illustration of the application of the dual-process method. The main result is that if $d \geqslant 2$ and $\varphi$ is the characteristic function of an aperiodic random walk on ${{\mathbf{Z}}^d}$, then there is precisely one Feller semigroup on the d-dimensional torus with generator extending $A = \{ 1 - \varphi (\theta )\} \Delta$. A necessary and sufficient condition for the associated Feller process to leave the singular point 0 is determined. This condition provides a criterion for uniqueness in law of a stochastic differential equation which is naturally associated with the process.
Projective modules over Laurent polynomial rings
Richard G.
Swan
111-120
Abstract: Quillen's solution of Serre's problem is extended to Laurent polynomial rings. An example is given of a $A[T,{T^{ - 1}}]$-module P which is not extended even though A is regular and ${P_\mathfrak{m}}$ is extended for all maximal ideals $\mathfrak{m}$ of A.
Cyclic actions on lens spaces
Paik Kee
Kim
121-144
Abstract: A 3-dimensional lens space $L = L(p,q)$ is called symmetric if ${q^2} \equiv \pm 1 \bmod p$. Let h be an orientation-preserving PL homeomorphism of even period $n( > 2)$ on L with nonempty fixed-point set. We show: (1) If n and p are relatively prime, up to weak equivalence (PL), there exists exactly one such h if L is symmetric, and there exist exactly two such h if L is nonsymmetric. (2) $ {\text{Fix}}(h)$ is disconnected only if $ p \equiv 0 \bmod n$, and there exists exactly one such h up to weak equivalence (PL). A ${Z_n}$-action is called nonfree if $ {\text{Fix}}(\phi ) \ne \emptyset$ for some $\phi ( \ne 1) \in {Z_n}$. We also classify all orientation-preserving nonfree $ {Z_4}$-actions (PL) on all lens spaces $L(p,q)$. It follows that each of ${S^3}$ and ${P^3}$ admits exactly three orientation-preserving $ {Z_4}$-actions (PL), up to conjugation.
On the group of automorphisms of affine algebraic groups
Dong Hoon
Lee
145-152
Abstract: We study the conservativeness property of affine algebraic groups over an algebraically closed field of characteristic 0 and of their group of automorphisms. We obtain a certain decomposition of affine algebraic groups, and this, together with the result of Hochschild and Mostow, becomes a major tool in our study of the conservativeness property of the group of automorphisms.
Homotopy operations under a fixed space
D. E.
Kruse;
J. F.
McClendon
153-174
Abstract: The problem of classifying extensions of a function up to relative homotopy leads in a natural way to the homotopy operations of the title. The operations, stable and unstable, primary and higher order, are defined and studied. Some specific applications are worked out.
Relations among characteristic classes
Stavros
Papastavridis
175-187
Abstract: Let M be an n-dimensional, compact, closed, ${C^\infty }$ manifold, and $v:M \to BO$ be the map classifying its stable normal bundle. Let $S \subseteq {H^\ast}(BO;{Z_2})$ be a set of characteristic classes and let q, k, be fixed nonnegative integers. We define $I_n^q(S,k) = \{ x \in {H^q}(B):{v^\ast}(x) \cdot y = 0$ for all $y \in {H^k}(M;{Z_2})$ and for all n-dimensional, $ {C^\infty }$ closed compact manifolds M, which have the propery that ${v^\ast}(S) = \{ 0\} \}$. In this paper we compute $ I_n^q(S,k)$, where all classes of S have dimension greater than $ n/2$. We examine also the case of BSO and BU manifolds.
Filtrations and canonical coordinates on nilpotent Lie groups
Roe
Goodman
189-204
Abstract: Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over a field of characteristic zero. Introducing the notion of a positive, decreasing filtration $\mathcal{F}$ on $\mathfrak{g}$, the paper studies the multiplicative structure of the universal enveloping algebra $U(\mathfrak{g})$, and also transformation laws between $ \mathcal{F}$-canonical coordinates of the first and second kind associated with the Campbell-Hausdorff group structure on $\mathfrak{g}$. The basic technique is to exploit the duality between $ U(\mathfrak{g})$ and $ S({\mathfrak{g}^\ast})$, the symmetric algebra of ${\mathfrak{g}^\ast}$, making use of the filtration $\mathcal{F}$. When the field is the complex numbers, the preceding results, together with the Cauchy estimates, are used to obtain estimates for the structure constants for $ U(\mathfrak{g})$. These estimates are applied to construct a family of completions $ U{(\mathfrak{g})_\mathfrak{M}}$ of $ U(\mathfrak{g})$, on which the corresponding simplyconnected Lie group G acts by an extension of the adjoint representation.
Linear isotopies in $E\sp{2}$
R. H.
Bing;
Michael
Starbird
205-222
Abstract: This paper deals with continuous families of linear embeddings (called linear isotopies) of finite complexes in the Euclidean plane ${E^2}$. Suppose f and g are two linear embeddings of a finite complex P with triangulation T into a simply connected open subset U of ${E^2}$ so that there is an orientation preserving homeomorphism H of ${E^2}$ to itself with $H \circ f = g$. It is shown that there is a continuous family of embeddings ${h_t}:P \to U(t \in [0,1])$ so that ${h_0} = f,{h_1} = g$, and for each t, $ {h_t}$ is linear with respect to T. It is also shown that if P is a PL star-like disk in ${E^2}$ with a triangulation T which has no spanning edges and f is a homeomorphism of P which is the identity on Bd P and is linear with respect to T, then there is a continuous family of homeomorphisms ${h_t}:P \to P(t \in [0,1])$ such that $ {h_0} = {\text{id}},{h_1} = f$, and for each t, ${h_t}$ is linear with respect to T. An example shows the necessity of the ``star-like'' requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theorem-namely, if f and g are two PL embeddings of a disk P into ${E^2}$ so that $f\vert{\text{Bd}}\;P = g\vert{\text{Bd}}\;P$, then there is a linear isotopy with respect to some triangulation of P which starts at f, ends at g, and leaves the boundary fixed throughout.
Spectral properties of tensor products of linear operators. II. The approximate point spectrum and Kato essential spectrum
Takashi
Ichinose
223-254
Abstract: For tensor products of linear operators, their approximate point spectrum, approximate deficiency spectrum and essential spectra in the sense of T. Kato and Gustafson-Weidmann are determined together with explicit formulae for their nullity and deficiency. The theory applies to $A \otimes I + I \otimes B$ and $A \otimes B$.
Logarithmic Sobolev inequalities for the heat-diffusion semigroup
Fred B.
Weissler
255-269
Abstract: An explicit formula relating the Hermite semigroup ${e^{ - tH}}$ on R with Gauss measure and the heat-diffusion semigroup $ {e^{t\Delta }}$ on R with Lebesgue measure is proved. From this formula it follows that Nelson's hypercontractive estimates for ${e^{ - tH}}$ are equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map $ {L^q}(R)$ into ${L^p}(R),1 < q < p < \infty$. Furthermore,the inequality $\displaystyle \frac{d}{{dq}}\log \left\Vert \phi \right\Vert _q^q \leqslant \fr... ...\Vert \phi \right\Vert _q^q}}} \right] + \log {\left\Vert \phi \right\Vert _q},$ where $1 < q < \infty ,{J^q}\phi = (\operatorname{sgn} \phi )\vert\phi {\vert^{q - 1}}$, and the norms and sesquilinear form $\langle ,\rangle$ are taken with respect to Lebesgue measure on ${R^n}$, is shown to be equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map from ${L^q}({R^n})$ into ${L^p}({R^n})$. This inequality is analogous to Gross' logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.
Segal algebras on non-abelian groups
Ernst
Kotzmann;
Harald
Rindler
271-281
Abstract: Let ${S^1}(G)$ be a Segal algebra on a locally compact group. The central functions of ${S^1}(G)$ are dense in the center of $ {L^1}(G)$. ${S^1}(G)$ has central approximate units iff G $G \in [SIN]$. This is a generalization of a result of Reiter on the one hand and of Mosak on the other hand. The proofs depend on the structure theorems of [SIN]- and [IN]-groups. In the second part some new examples of Segal algebras are constructed. A locally compact group is discrete or Abelian iff every Segal algebra is right-invariant. As opposed to the results, the proofs are not quite obvious.
The $\mu $-invariant of $3$-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented $2$-manifold
Joan S.
Birman;
R.
Craggs
283-309
Abstract: Let $\mathcal{H}(n)$ be the group of orientation-preserving selfhomeomorphisms of a closed oriented surface Bd U of genus n, and let $\mathcal{K}(n)$ be the subgroup of those elements which induce the identity on $ {H_1}({\text{Bd}}\;U;{\mathbf{Z}})$. To each element $h \in \mathcal{H}(n)$ we associate a 3-manifold $ M(h)$ which is defined by a Heegaard splitting. It is shown that for each $h \in \mathcal{H}(n)$ there is a representation $ \rho$ of $\mathcal{K}(n)$ into $ {\mathbf{Z}}/2{\mathbf{Z}}$ such that if $ k \in \mathcal{K}(n)$, then the $\mu$-invariant $ \mu (M(h))$ is equal to the $\mu$-invariant $ \mu (M(kh))$ if and only if k $\in$ kernel $\rho$. Thus, properties of the 4-manifolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from $\mathcal{K}(n)$ onto $ {\mathbf{Z}}/2{\mathbf{Z}}$ are studied and are shown to constitute a complete conjugacy class of subgroups of $\mathcal{H}(n)$. The class has nontrivial finite order.
Cylindricity of isometric immersions between hyperbolic spaces
S.
Alexander;
E.
Portnoy
311-329
Abstract: The motivation for this paper was to prove the following analogue of the Euclidean cylinder theorem: any umbilic-free isometric immersion $ \eta :{H^{n - 1}} \to {H^n}$ between hyperbolic spaces takes the form of a hyperbolic $(n - 2)$-cylinder over a uniquely determined parallelizing curve in $ {\bar H^n}$. Our approach is through the more general study of isometric immersions generated by one-parameter families of hyperbolic k-planes without focal points. A by-product of this study is a natural extension to curves in ${\bar H^n}$ of the notion of a parallel family of k-planes along a curve in ${H^n}$; the extension is based on spherical symmetry of variation fields. Existence and uniqueness properties of this extended notion of parallelism are considered.
Positive cones and focal points for a class of $n$th-order differential equations
M. S.
Keener;
C. C.
Travis
331-351
Abstract: Necessary and sufficient conditions are obtained for both the existence and absence of focal points for a class of nth order linear differential equations. The techniques utilized the theory of ${\mu _0}$-positive operators with respect to a cone in a Banach space.
The $(\varphi , 1)$ rectifiable subsets of Euclidean space
Samir
Kar
353-371
Abstract: In this paper the structure of a subset $E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty$ has been studied by examining its intersection with various translated positions of a smooth hypersurface B. The following result has been established: Let B be a proper $(n - 1)$ dimensional smooth submanifold of $ {{\mathbf{R}}^n}$ with nonzero Gaussian curvature at every point. If $E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty$, then there exists a countably 1-rectifiable Borel subset R of $ {{\mathbf{R}}^n}$ such that $(E \sim R)$ is purely $({{\mathbf{H}}^1},1)$ unrectifiable and $ (E \sim R) \cap (g + B) = \emptyset$ for almost all $g \in {{\mathbf{R}}^n}$. Furthermore, if in addition E is $ {{\mathbf{H}}^1}$ measurable and $ E \cap (g + B) = \emptyset$ for $ {{\mathbf{H}}^n}$ almost all $ g \in {{\mathbf{R}}^n}$ then ${{\mathbf{H}}^1}(E \cap R) = 0$. Consequently, E is purely $ ({{\mathbf{H}}^1},1)$ unrectifiable.
Nonlinear operations and the solution of integral equations
Jon C.
Helton
373-390
Abstract: The letters S, G and H denote a linearly ordered set, a normed complete Abelian group with zero element 0, and the set of functions from G to G that map 0 into 0, respectively. In addition, if $V \in H$ and there exists an additive function $\alpha$ from $S \times S$ to the nonnegative numbers such that $\left\Vert {V(x,y)P - V(x,y)Q} \right\Vert \leqslant \alpha (x,y)\left\Vert {P - Q} \right\Vert$ for each $\{ x,y,P,Q\}$ in $S \times S \times G \times G$, then $V \in \mathcal{O}\mathcal{S}$ only if $\smallint _x^yVP$ exists for each $\{ x,y,P\}$ in $ S \times S \times G$, and $ V \in \mathcal{O}\mathcal{P}$ only if $ _x{\Pi ^y}(1 + V)P$ exists for each $\{ x,y,P\}$ in $S \times S \times G$. It is established that $ V \in \mathcal{O}\mathcal{S}$ if, and only if, $V \in \mathcal{O}\mathcal{P}$. Then, this relationship is used in the solution of integral equations of the form $ f(x) = h(x) + \smallint _c^x[U(u,v)f(u) + V(u,v)f(v)]$, where U and V are in $\mathcal{O}\mathcal{S}$. This research extends known results in that requirements pertaining to the additivity of U and V are weakened.
Replacing homotopy actions by topological actions
George
Cooke
391-406
Abstract: A homotopy action of a group G on a space X is a homomorphism from G to the group of homotopy classes of homotopy equivalences of X. The question studied in this paper is: When is a homotopy action equivalent, in an appropriate sense, to a topological action of G on X?