The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Let G be a finite Abelian group acting by tensor-product automorphisms on a UHF-C¹-algebra $\text{D}$. Extending a result of A. Kishimoto it is shown that the number of extremal traces on the fixed-point algebra $\text{D}$^G equals the cardinality of the subgroup K of automorphisms in G which are weakly inner in the trace representation of $\text{D}$.     

Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group $\text{G}\text{F}$ are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and $\text{G}\text{F}$ a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if $\text{G}\text{F}$ is finite dimensional or if F is one dimensional and $\text{G}\text{F}$ a separable Hilbert group.     

Factorial representations of path groups

The reduction of the energy representation of the group of mappings from I = [0, 1], S¹1, $\text{R}$₊ or $\text{R}$ into a compact semisimple Lie group G is given. For G = SU(2), the factoriality of the representation, which is of type III in the case I=$\text{R}$, is proved.     

Some discontinuous translation-invariant linear forms

The author proved in [3] that every translation-invariant linear form on $\text{D}$(Rⁿ), as well as on other spaces of test functions and distributions, is necessarily continuous. The same result has also been proved for the Hilbert space L²(G) where G is a compact connected Abelian group. In contrast to this it is proved here that there do exist discontinuous translation-invariant linear forms on the Banach spaces l₁(Z) and L¹(R), and on the Hibert spaces L²(D) and L²(R). Here Z denotes the additive group of the integers, D denotes the totally disconnected compact Abelian Cantor discontinuum group, and R denotes the additive group of the real numbers. The proofs divide into two parts: A general criterion (Theorem 1) and proofs that the spaces l₁(Z), L²(D), L²(R), and L¹(R) satisfy this criterion (Theorems 2, 3, 4, and 5, respectively).     

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.     

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.     

Compact abelian groups of automorphisms of von Neumann algebras

Let G be a compact abelian group, acting σ-weakly continuously as a group of 1-automorphisms α on a von Neumann algebra $\text{R}$. We give necessary and sufficient conditions for α to be inner, based on the structure of the lattice of projections in the center of the fixed-point algebra. As an application, we show that if α satisfies a spectrum condition with respect to a suitably chosen positive semigroup in the dual of G, then α is inner, and the implementing unitary representation can be chosen with positive spectrum.     

Hamiltonicity in (0–1)-polyhedra

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? $\text{R}$ⁿ is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? $\text{R}$ⁿ has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? $\text{R}$^d having (0–1)-valued vertices such that $\text{G(P) ? G(P′)}$. Some combinatorial consequences of these results are also discussed.     

Complex dynamical variables for multiparticle systems with analytic interactions. I

The n-body problem is formulated as a problem of functional analysis on a Hilbert space $\text{G}$ whose elements are analytic functions of complex dynamical variables. It is assumed that the two-body interaction is local and spherically symmetric, and belongs to the two-particle space $\text{G}$. The n-body resolvent R(λ) is constructed with the help of Fredholm methods. The operator R(λ) on $\text{G}$ is associated with a family of operators R(λ, ?) on $\text{L}$² which are resolvents of closed linear operators H(?), the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) contains a set of parallel half-lines starting at the thresholds of scattering channels and making an angle 2? with the positive real axis. The half-lines are branch cuts of R(λ, ?), but matrix elements of R(λ, ?) can be continued analytically across these. The operator R(λ, ?) may have isolated poles. The location of these does not depend on ?. Each pole is associated with one or more eigenvectors of H(?) belonging to spaces $\text{G}$. There may be poles off the real axis, the location of a pole determining for which values of ? it is on the physical sheet of H(?). It is shown how poles off the real axis give rise to resonances in the scattering cross section, the shape of a resonance being as one would expect on the basis of a model in which the scattering takes place via a decaying compound state having an eigenvector of H(?) with complex energy as its wave function.     

On a duality for crossed products of C1-algebras

Let $\text{U}$ be a C¹-algebra, and G be a locally compact abelian group. Suppose α is a continuous action of G on $\text{U}$. Then there exists a continuous action $\text{}\text{?}$ga of the dual group $\text{G}\text{?}$ of G on the C¹-crossed product by α such that the C¹-crossed product is isomorphic to the tensor product and the C¹-algebra of all compact operators on L²(G).     

Certain quotient spaces are countably separated,II

Let $\text{A}$ be a Type I C¹-algebra, $\text{B}$ a C¹-subalgebra, and G a group of ¹-automorphisms of $\text{A}$ which leave $\text{B}$ invariant. If $\text{A}\text{?}$/G is countably separated, then $\text{B}\text{?}$/G is also countably separated. This result has several applications in group representations.     

Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

We consider nonlinear elliptic eigenvalue problems on unbounded domains G?$\text{R}$ⁿ. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L²(G). Moreover, we establish that the infimum λ1 of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ1 from the trivial line of solutions if λ1 belongs to the essential spectrum of L.     

Caractères des groupes de Lie

Let G be a real Lie group with Lie algebra $\text{G}$. M. Duflo has constructed irreducible unitary representations of G associated to some G-orbits Ω in the dual $\text{G}$¹ of $\text{G}$. We prove a character formula when Ω is tempered, closed, and of maximal dimension.     

The KMS condition and passive states

Any state which is passive for a C¹-dynamical system (A, $\text{R}$, α), and ergodic for a system (A, G, γ), where α and γ commute, is KMS at some temperature. Any state which is passive for (A, $\text{R}$, α) and central for (A, G, γ) is approximable by convex combinations of KMS states at different temperatures.     

On the existence of a free subgroup of finite index

Let G be a finitely generated accessible group. We will study the homology of G with coefficients in the left G-module H¹(G;$\text{Z}$[G]). This G-module may be identified with the G-module of continuous functions with values in $\text{Z}$ on the G-space of ends of G, quotiented by the constant functions. The main result is as follows: Suppose G is infinite, then the abelian group H₁(G;H¹(G;$\text{Z}$[G])) has rank 1 if G has a free subgroup of finite index and it has rank 0 if G has not.     

Non-archimedean group algebras

The group ring R(G) of a group G over a coefficient ring R is well known and so is the L¹ group algebra

$\text{l}{}_1\text{(G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α∈C,∑|α}{}_{\mathrm{g>}}\text{|<∞}$

. We study in this note

$\text{l(R,G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α}{}_{\mathrm{g}}\text{∈R,g∈G,limα}{}_{\mathrm{g}}\text{=0}$

. where R is Z_p(Q_p) the ring (field) of p-adic integers (numbers) equipped with the p-adic valuation. Analogues of certain well known results for group rings and l₁(G) are obtained for l(R,G).     

Unbounded derivations tangential to compact groups of automorphisms,II

Let G be a compact abelian group, and τ an action of G on a C¹-algebra $\text{U}$, such that $\text{U}$^τ(γ)$\text{U}$^τ(γ)¹ = $\text{U}$^τ(0) $\text{U}$^τ for all $\text{γ ?}\text{G}\text{?}$, where $\text{U}$^τ(γ) is the spectral subspace of $\text{U}$ corresponding to the character γ on G. Derivations δ which are defined on the algebra $\text{U}$_F of G-finite elements are considered. In the special case δ¦_{$\text{U}$^τ} = 0 these derivations are characterized by a cocycle on $\text{G}\text{?}$ with values in the relative commutant of $\text{U}$^τ in the multiplier algebra of $\text{U}$, and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded. Under various restrictions on G and τ properties of the cocycle are deduced which again give characterizations of δ in terms of decompositions into generators of one-parameter subgroups of τ(G) and approximately inner derivations. Finally, a perturbation technique is devised to reduce the case δ($\text{U}$_F) ? $\text{U}$_F to the case δ($\text{U}$_F) ? $\text{U}$_F and δ¦_{$\text{U}$^τ} = 0. This is used to show that any derivation δ with D(δ) = $\text{U}$_F is wellbehaved and, if furthermore G = T¹ and δ($\text{U}$_F) ? $\text{U}$_F the closure of δ generates a one-parameter group of ¹-automorphisms of $\text{U}$. In the case G = T^d, d = 2, 3,… (finite), and δ($\text{U}$_F) ? $\text{U}$_F it is shown that δ extends to a generator of a group of ¹-automorphisms of the σ-weak closure of $\text{U}$ in any G-covariant representation.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.     

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Smoothness and absolute convergence of Fourier series in compact totally disconnected groups

In this paper we study in the context of compact totally disconnected groups the relationship between the smoothness of a function and its membership in the Fourier algebra $\text{G}$G. Specifically, we define a notion of smoothness which is natural for totally disconnected groups. This in turn leads to the notions of Lipshitz condition and bounded variation. We then give a condition on α which if satisfied implies Lip_α(G) ? $\text{R}$(G). On certain groups this condition becomes: $\text{α >}\text{1}\text{2}$ (Bernstein's theorem). We then give a similar condition on α which if satisfied implies that Lip_α(G) ∈ BV(G) ? $\text{R}$(G). On certain groups this condition becomes: α > 0 (Zygmund's theorem). Moreover we show that $\text{α >}\text{1}\text{2}$ is best possible by showing that $\text{Lip}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(G)}$ ? $\text{R}$(G).