Applications of the Connes spectrum to C1-dynamical systems,II

We show that for a C¹-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to $\text{G}\text{?}$ if and only if every nonzero closed ideal in G × _αA has a nonzero intersection with A. Denote by G_J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G_J are discrete, then GX_αA is simple if and only if A is G-simple and $\text{Γ(}\text{α}\text{) =}\text{G}\text{?}$. This result is applicable not only when G is discrete but also when G?$\text{R}$ or G?$\text{T}$ provided that A is not primitive. Specializing to single automorphisms (i.e., G=$\text{Z}$) we show that if (the transposed of) α acts freely on a dense set of points in $\text{A}\text{?}$, then Λ(α)=$\text{T}$. The converse is only proved when A is of type I.     

Filtrations,rees rings,and ideal transforms

Let I be an ideal, and let f = {K_n|n ≥ 0 } be a filtration of the Noetherian ring R, such that Iⁿ ⊆ K_n for all n ≥ 0. We study when the Rees ring $\text{R}$(f) is either finite or integral over the Rees ring $\text{R}$(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (Iⁿ : J^{m(n) + 1}), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.     

Oscillatory properties of strongly superlinear differential equations with deviating arguments

We regard a graph G as a set {1,…, v} together with a nonempty set E of two-element subsets of {1,…, v}. Let p = (p₁,…, p_v) be an element of $\text{R}$^nv representing v points in $\text{R}$ⁿ and consider the realization G(p) of G in $\text{R}$ⁿ consisting of the line segments [p_i, p_j] in $\text{R}$ⁿ for {i, j} ?E. The figure G(p) is said to be rigid in $\text{R}$ⁿ if every continuous path in $\text{R}$^nv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ? $\text{R}$^nv which is the image (Tp₁,…, Tp_v) of p under an isometry T of $\text{R}$ⁿ. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in $\text{R}$² is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in $\text{R}$³.     

The functions operating on multiplier algebras

Let 1 < p < ∞ with p ≠ 2. Let G denote one of the groups $\text{T}$ⁿ, $\text{R}$ⁿ, or $\text{Z}$ⁿ. We show that only entire functions operate in certain algebras of multipliers on L_p(G).     

A multiplication of distributions

If Ω denotes an open subset of $\text{R}$ⁿ (n = 1, 2,…), we define an algebra $\text{g}$ (Ω) which contains the space $\text{D}$′(Ω) of all distributions on Ω and such that $\text{C}{}^{\mathrm{\infty }}\text{(Ω)}$ is a subalgebra of $\text{G}$ (Ω). The elements of $\text{G}$ (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in $\text{G}$(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and $\text{?∈O}{}_{\mathrm{M}}\text{(}\text{R}{}^{\mathrm{2q}}\text{)}$—a classical Schwartz notation—for any G₁,…,G_q∈G(σ), we define naturally an element $\text{?G}{}_1\text{,…,G}{}_{\mathrm{q}}\text{∈G(σ)}$. These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.     

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups $\text{H>}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, U)}$ (where U means the multiplicative group) and $\text{H}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, Pic)}$ and a third sequence of groups Hⁿ(J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and Hⁿ(J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs (P, α) with P a rank one, projective Sⁿ-module and α an isomorphism from the coboundary of P (inPicS^{n + 1}) toS^{n + 1}. (Here Sⁿ denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group $\text{B(}\text{S}\text{R}\text{)}$ to H²(J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H²(J) with Ker[H²(R, U)→H²(S, U)], where H²(R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).     

Most primitive groups have messy invariants

Suppose G is a finite group of complex n × n matrices, and let R^G be the ring of invariants of G: i.e., those polynomials fixed by G. Many authors, from Klein to the present day, have described R^G by writing it as a direct sum Σ^δ_j=1 η_j$\text{C}$[θ₁ ,…, θ_n]. For example, if G is a unitary group generated by reflections, δ = 1. In this note we show that in general this approach is hopeless by proving that, for any ? > 0, the smallest possible δ is greater than | G |^n-1-? for almost all primitive groups. Since for any group we can choose δ ? | G |^n-1, this means that most primitive groups are about as bad as they can be. The upper bound on δ follows from Dade's theorem that the θ_i can be chosen to have degrees dividing | G |.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Automorphisms of the canonical anticommutation relations and index theory

Let H be a complex Hilbert space, P₊ an orthogonal projection on H, and P_? the complementary projection. If $\text{G}$ is any symmetrically normed ideal in the ring of bounded operators on H, then we consider the group of unitary operators on H such that P₊UP_?and P_?UP₊ lie in $\text{G}$. When $\text{G}$ is the Hilbert-Schmidt class, these unitaries define automorphisms of the C¹-algebra $\text{b}$ of the canonical anticommutation relations over H which are implementable in the representation of $\text{b}$ determined by P_?. We investigate the structure of the group $\text{U}$, proving in particular that it has infinitely many connected components, $\text{U}$_k, labelled by the Fredholm index of P₊UP₊. The connected component of the identity, $\text{U}$₀, is generated by unitaries of the form exp(iA), with A self-adjoint and P₊AP_? in $\text{G}$. Finally we consider an application of these results to two dimensional field theory, showing in particular that the charge and chiral charge quantum numbers arise as the Fredholm indices of P_±UP_± for certain unitary U on L²($\text{R}$, $\text{C}$²)     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

On strictly positively invariant cones

For a matrix A ∈ R^{n × n}, it is shown that strict positive invariance of a proper cone $\text{C}$ ? Rⁿ (that is, e^tA$\text{C}\text{/{0}}$ ? int $\text{C}$ ?t 0) implies the existence of a certain direct sum decomposition of Rⁿ into A-invariant subspaces. Our results lead to a characterization of the set of initial points which give rise to solution curves that reach $\text{S}$, under the differential equation ? = Ax. Also given is an application in stability theory.     

On the decomposition of the tensor algebra of the classical Lie algebras

Let $\text{g}$_n be the Lie algebra gl_n(C), let S($\text{g}$_n) be the symmetric algebra of $\text{g}$_n, and let T($\text{g}$_n) be the tensor algebra of $\text{g}$_n. In a recent paper, R. K. Gupta studied certain sequences of representations R_∞ = (R_n)^∞_{n = 1}, where R_n is a representation of $\text{g}$_n. These sequences have the property that every irreducible representation occurring in S($\text{g}$_n) is in exactly one of these sequences. Fixing f, she considers s(R_∞, f) which is the limit on n of the multiplicity of R_n in S^f($\text{g}$_n), the fth-graded piece of S($\text{g}$_n). She and R. P. Stanley independently showed that the limit s(R_∞, f) exists and is given by an amazingly elegant formula. They call s(R_∞, f) the stable multiplicity of R_n in S^f($\text{g}$_n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R_∞ for all of the classical Lie algebras $\text{g}$_n are studied, and a simple formula for the stable multiplicity m(R)_∞, ψ, f, $\text{g}$_∞) of R_n in the ψ-isotypic component of T^f($\text{g}$_n), where ψ is any irreducible character of the symmetric group tS_f, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)_∞, ψ, f, $\text{g}$_∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of $\text{g}$_n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T($\text{g}$_n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S($\text{g}$_n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of T^f($\text{g}$_n) projects onto Kostant's decomposition of S^f($\text{g}$_n).     

Spectral theory of finite volume domains in Rn

Let Ω be an arbitrary open subset of $\text{R}$ⁿ of finite positive measure, and assume the existence of a subset Λ ? $\text{R}$ⁿ such that the exponential functions e_λ = exp i(λ₁x₁ + … + λ_nx_n), λ = (λ₁,…, λ_n) ∈ Λ, form an orthonormal basis for $\text{L}{}_2\text{(Ω)}$ with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of ($\text{R}$ⁿ, +) by K = Λ⁰ = {γ ∈ $\text{R}$ⁿ:γ·λ ∈ 2π$\text{Z}$}, A = {a ∈ $\text{R}$ⁿ:U_a$\text{m}$ U1_a = $\text{m}$}, where U_t is the unitary representation of $\text{R}$ⁿ on $\text{L}{}_2\text{(Ω)}$ given by U_te = e^itλe_λ, t ∈ $\text{R}$ⁿ, λ ∈ Λ, and where $\text{m}$ is the multiplication algebra of $\text{L}{}_{\mathrm{\infty }}\text{(Ω)}$ on L₂. Assume that A is discrete. Then there is a discrete subgroup D ? A of dimension n, a fundamental domain $\text{D}$ for D, and finite sets of representers R_Λ, R_Γ, $\text{R}{}_{\mathrm{\Omega }}$, each containing 0, R_Λ for $\text{A}\text{K}$ in K⁰, and $\text{R}{}_{\mathrm{\Omega }}$ for $\text{A}\text{K}$ in A such that Ω is disjoint union of translates of $\text{D}$: Ω = ∪_a∈RΩ (a + $\text{D}$), neglecting null sets, and Λ = R_Λ ⊕ D⁰. If R_Γ is a set of representers for $\text{D}\text{A}$ in D, then Γ = R_Γ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = $\text{R}$ⁿ, direct sum, (neglecting null sets). The case A = $\text{R}$ⁿ corresponds to Ω = $\text{D}$, Λ = D⁰ and Γ = K. This last case corresponds in turn to a function theoretic assumption of Forelli.     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

On the functional dimension of the space of solutions of partial differential equations

Given a differential polynomial P(D) in Rⁿ with constant coefficients, consider the functional dimension df$\text{N}$ of the space $\text{N}$ = {u∈C(Rⁿ):P(D)u = 0} endowed with the topology of uniform convergence on compact subsets of Rⁿ. If P(D) is elliptic then df$\text{N}$ = n, by a theorem of Y. Kōmura. We prove the converse: If df$\text{N}$ = n then the differential polynomial P(D) must be elliptic.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).