Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.     

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.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

Quasi-free derivations on the Canonical Anticommutation Relation algebra

An unbounded 1-derivation δ on a C¹-algebra $\text{U}$ is called approximately bounded if there is an increasing sequence of full matrix subalgebras {$\text{U}$_n} whose union is dense in the domain of $\text{U}$ and a sequence {h_n} of self-adjoint elements of $\text{U}$ such that h_n implements δ on $\text{U}$_n for every n, and {∥h_n ? Q_n(h_n)∥} is a bounded sequence where Q_n is the canonical conditional expectation of $\text{U}$ onto $\text{U}$_n. We prove that a quasi-free derivation on the Canonical Anticommutation Relation algebra is approximately bounded if the self-adjoint operator from which it arises is of finite multiplicity and bounded. We conjecture that all quasi-free derivations are approximately bounded. We also prove that a quasi-free derivation is bounded if and only if the self-adjoint operator from which it arises is of the trace class.     

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.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

On the density of various classes of groups

Let $\text{T}$ be a subset of the set of all isomorphism classes of finite groups. We consider the number F_$\text{g}$(x) of positive integers n≤x such that all groups of order n lie in $\text{T}$. When $\text{T}$ consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for F_$\text{g}$(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable groups. In the course of the arguments, we also obtain, for almost all n, a lower bound for the number of groups of a given order n.     

The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

The asymptotic distribution of orbits for discrete subgroups of motions in Euclidean and non-Euclidean spaces are found; our principal tool is the wave equation. The results are new for the crystallographic groups in Euclidean space and for those groups in non-Euclidean spaces which have fundamental domains of infinite volume. In the latter case we show that the only point spectrum of the Laplace-Beltrami operator lies in the interval ($\text{?(}\text{(m ? 1)}\text{2}\text{)}{}^2\text{,0}$]; furthermore we show that when the subgroup is nonelementary and the fundamental domain has a cusp, then there is at least one eigenvalue in this interval.     

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).     

The construction of SL(2, 3)-polynomials

Octic polynomials over $\text{Z}$ with Galois group SL(2, 3) are constructed. This is done via suited quartic totally real polynomials with group A₄ over $\text{Q}$. A table of the cycle patterns of the imprimitive transitive permutation groups of degree 8 is included.     

Polyhedral groups and graph amalgamation products

A polyhedral group G is defined to be the orientation-preserving subgroup of a discrete reflection group acting on hyperbolic 3-space $\text{H}$³, and having a fundamental polyhedron of finite volume. A special presentation for G is obtained from the geometry of the polyhedron. This gives G the structure of a graph amalgamation product, and which, in some cases, splits as a free product with amalgamation. The simplest examples of polyhedral groups are the so-called tetrahedral groups. Other examples are given amongst the the groups PGL(2,O_m), where O_m is the ring of algebraic integers in the quadratic imaginary field $\text{Q(}\text{-m}\text{)}$, m>0.     

A Van Kampen theorem for π2

If the path connected topological space X has a countable open cover $\text{U}$ with path connected elements, then π₂(X,1) is computed as a colimit determined by the second homotopy groups of the intersection of elements of $\text{U}$ and the indices of the fundamental group injections of these intersections into the fundamental group of X. Aside from assuming that the inclusions induce such monomorphisms, certain other inclusions are also required to induce monomorphisms of fundamental groups and restrictions are placed on the arrangement of the elements of $\text{U}$.     

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$\text{G}$_$\text{c}$.     

Arbitrary vs. regular semigroups

The notion of regularity for semigroups is studied, and it is shown that an unambiguous semigroup (i.e., whose $\text{L}$ and $\text{R}$ orders are respectively unions of disjoint trees) can be embedded in a regular semigroup with the same subgroups and the same ideal structure (except that a zero is added to the regular semigroup).In a previous paper [1] it was shown that any semigroup is the homomorphic image of an unambiguous semigroup with the same groups and a similar ideal structure.Together these two papers thus prove that an arbitrary semigroup divides a regular semigroup with a similar structure.The resulting regular semigroup is finite (resp. torsion, or bounded torsion) if the given semigroup has that property.     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Harmonic analysis on unitary groups

A theory of harmonic analysis on a metric group (G, d) is developed with the model of U$\text{U}$, the unitary group of a C¹-algebra $\text{U}$, in mind. Essential in this development is the set $\text{G}\text{?}{}_{\mathrm{d}}$ of contractive, irreducible representations of G, and its concomitant set P_d(G) of positive-definite functions. It is shown that $\text{G}\text{?}{}_{\mathrm{d}}$ is compact and closed in $\text{G}\text{?}$. The set $\text{G}\text{?}{}_{\mathrm{d}}$ is determined in a number of cases, in particular when G = U($\text{U}$) with $\text{U}$ abelian. If $\text{U}$ is an AW¹-algebra, it is shown that $\text{G}\text{?}$_d is essentially the same as $\text{U}\text{?}$. Unitary groups are characterised in terms of a certain Lie algebra $\text{g}$_u and several characterisations of G = U($\text{U}$) when $\text{U}$ is abelian are given.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

A rapidly convergent iteration method and Gâteaux differentiable operators

Let {F_r}_0?r?p be a family of Banach spaces satisfying, if 0?r₁?r₂?p, (i)F_r1 ? F_r2; (ii)$\text{¦f¦}{}_{\mathrm{r1}}\text{? ¦f¦}{}_{\mathrm{r2}}\text{(f ? F}{}_{\mathrm{r1}}\text{)}$; and (iii)$\text{?(r) = ln(¦f¦}{}_{\mathrm{r}}\text{)}$ is a convex function. Let G₀ be a Banach space and. $\text{F}$ be a Gâteaux differentiate mapping, and suppose that $\text{F}$′(x)(F_p) is dense in G₀. Under appropriate assumptions, the equation $\text{F}$(x)=0 has a solution in F_r for 0?r?p. The results extend the Inverse Function Theorem of J. Moser to the class of Gâteaux differentiable operators.