Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

Hypercyclic Pairs of Coanalytic Toeplitz Operators

A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A ⁿ B ^k x : n, k ≥ 0} is dense in . If f, g ∈H ^∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M ^* _f , M ^* _g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H ^∞(G) such that the pair (M ^* _f , M ^* _g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.     

Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact

We prove that every closed normal subgroupH of a locally compact amenable groupG is a Ditkin set with respect to the Herz-Figà-Talamanca algebraA _p (G) (p>1). Let be the canonical map ofG ontoG/H andF a closed subset ofG/H. We show thatF is a Ditkin set if and only if everyuA _p (G), which vanishes on ^–1, lies on the norm closure of the subspace ofA _p (G) generated by {u _h |hH, vA _p (G)C ₀₀(G)} whereu _h (x)=u(x h). As far as we know, this result seems to be new even forG abelian andp=2.     

Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions

Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (^*) -orientation D ^*(G) ofG is an orientation ofG such that the outdegree of each vertex on G is 1 and the outdegrees of other vertices are all 2, where G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (^*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.A research fellow of the Japan Society for the Promotion of Science.     

p-Group actions on homology spheres

When an arbitraryp-groupG acts on a _n -homologyn-sphereX, it is proved here that the dimension functionn:S(G)(S(G) is the set of subgroups ofG), defined byn(H)=dimX ^H, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.     

Separating convex sets in the plane

Given a setA inR ² and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane.We prove that, for any collectionF ofn disjoint disks inR ², there is a lineL that separates a disk inF from a subcollection ofF with at least (n–7)/4 disks. We produce configurationsH _n andG _n , withn and 2n disks, respectively, such that no pair of disks inH _n can be simultaneously separated from any set with more than one disk ofH _n , and no disk inG _n can be separated from any subset ofG _n with more thann disks.We also present a setJ _m with 3m line segments inR ², such that no segment inJ _m can be separated from a subset ofJ _m with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least n/3+1 elements ofF.     

On bending of a convex surface to a convex surface with prescribed spherical image

We prove the following theorem. LetF be a regular convex surface homeomorphic to the disk. Suppose the Gaussian curvature ofF is positive and the geodesic curvature of its boundary is positive as well. LetG be a convex domain on the unit sphere bounded by a smooth curve and strictly contained in a hemisphere. LetP be an arbitrary point on the boundary ofF andP ^* be an arbitrary point on the boundary ofG. If the area ofG is equal to the integral curvature of the surfaceF, then there exists a continuous bending of the surfaceF to a convex surfaceF such that the spherical image ofF coincides withG andP ^* is the image of the point inF corresponding to the pointP F under the isometry.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 295–300, August, 1995.     

Complementing lattice-ordered groups: The finite-valued case

A complementedl-groupG is one in which to eacha G there exists ab G so that¦a¦ ¦b¦=0, while¦a¦ ¦b¦ is a unit ofG. IfG is anl-subgroup ofH, and the latter is complemented, then we say thatH is a complementation ofG ifG ^c , the convex hull ofG inH, is a strongly rigid extension ofG andG ^c is az-subgroup ofH. This article presents necessary and sufficient conditions for a finite-valuedl-subgroup to admit a complementation.Presented by M. Henriksen.     

On generalised Putnam—Fuglede theorems

LetB(H) denote the algebra of operators on the Hilbert spaceH, and letP denote the class ofAB(H) which are such that the restriction ofA to an invariant subspace is inP wheneverAP and which satisfy the property, henceforth called property (P 2), that if the restriction ofA to an invariant subspace is normal, then the subspace reducesA. GivenP-classesP ₁ andP ₂, the pair (P ₁,P ₂) is said to satisfy the (PF)-property if givenAP ₁ andB ^* P ₂ such thatAB=XB for someXB(H), we haveA ^* X=XB ^* . Generalising the (classical) Putnam—Fuglede theorem, it is shown here that the pair (P ₁,P ₂) has the (PF)-property if and only if, givenAP ₁ andB ^*P ₂ such thatAX=XB for some quasi-affinityXB(H), the following conditions hold: (i)B ^* is normal impliesA is normal; (ii)A has a normal direct summand impliesB ^* has a normal direct summand; (iii)A andB ^* pure impliesX is non-existent. An interestingP-class is the classC ₀ of contractions withC ₀ completely non-unitary parts which satisfy property (P 2). AssumingH to be separable, it is shown that ifC ₁ denotes thoseA C ₀ for which the defect operatorsD _A =(1–A^*A)^1/2 is of Hilbert—Schmidt class and for which either the pure part ofA has empty point spectrum or the eigen-values ofA are all simple, then the pair (C ₀,C ₁) has the (PF)-property. The classC ₁ defines aP-class; a crucial role in the proof of this statement is played by the interesting result that aC ₀ contraction with spectrum on the unit circle can not satisfy property (P 2). Applications of these results are considered, amongst them that ifA andB are quasi-similar hyponormal contractions such that the pure part ofA has finite multiplicity andD _B is of Hilbert —Schmidt class, then their normal parts are unitarily equivalent and their pure parts are quasi-similar.     

The primitive ideal space of solvable Lie groups

LetG be a connected and simply connected solvable Lie group. In a previous paper (cf.[22]) we associated withG a familyM of geometrical objects (generalized orbits), and with each elementO ofM a unitary equivalence classF(O) of factor representations. IfG is nilpotent,M coincides with the orbit space of the coadjoint representation, and the mapOF(O) reproduces essentially the Kirillov isomorphism betweenM and the dual ofG. As a fargoing extension of this, the principal result of the present paper asserts, that upon assigning to 0M the kernel of the representation, associated with some element ofF(O), of the groupC ^* algebraC ^*(G), we obtain a bijection betweenM and the primitive ideal space ofC ^*(G).This work was supported by a grant from the National Science Foundation     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

Natural <Emphasis Type="Italic">T</Emphasis>-Functions on the Cotangent Bundle of a Weil Bundle

A natural T-function on a natural bundle F is a natural operator transforming vector fields on a manifold M into functions on FM. For any Weil algebra A satisfying dim M width(A) + 1 we determine all natural T-functions on T ^* T ^A M, the cotangent bundle to a Weil bundle T ^A M.     

The Effros–Ruan conjecture for bilinear forms on C<Superscript>*</Superscript>-algebras

In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C^*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact. Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C^*-algebras, of which at least one is exact. In this paper we prove that the Effros–Ruan conjecture holds for general C^*-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C^*-algebras A and B, there exist states f ₁, f ₂ on A and g ₁, g ₂ on B such that for all a∈A and b∈B,

Translation-invariant subspaces of functions on the group of linear transformations on the real line

The characterization of right translation-invariant subspaces ofL _∞(G ^*), where , is studied. We introduce the class of multiplier functions which, in the semisimple case, play a role similar to that played by the exponentials for the real line. However, it is proved that multiplier functions ofG ^* with respect toR fail to characterize right translation-invariant subspaces ofL _∞(G ^*). That is, we construct a right translation-invariant, w^*-closed subspace ofL _∞(G ^*) which contains no multiplier function. This paper is a part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor H. Furstenberg, to whom the author wishes to express his thanks for his helpful guidance, and valuable remarks.     

Signatures andS-discrete lattice-ordered groups

The central theme of this article is the approximation of lattice-ordered groups (l-groups) first by Specker groups and, subsequently, by the so-calledS-discretel-groups. The sense of these approximations is made precise via the notion of a signature, defined below, and by that ofa ^*-subgroups. Sample result: ifG is a projectablel-group then it has anl-subgroupH which is Specker and for which the mapPPH defines a boolean isomorphism between the algebras of polars ofG andH.Presented by L. Fuchs.This article was written while this author was a Stouffer Visiting Professor at the University of Kansas. He wishes to thank the members of the Mathematics Department of that institution for their hospitality.     

Inner amenable groups and the inducedC *-algebra properties

A discrete group can generate aC ^*-algebra, denoted byC ^*(G), by considering the so called conjugation regular representation of the groupG. Based on this treatment, the inner amenability ofG shall be characterized by the existence of the states onC ^* (G) satisfying some certain conditions.