Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Über die Riemannsche Einbettungsgeometrie der Veronesemannigfaltigkeiten

Let E be a n-dimensional euclidean vector space. The subset V _k ⁿ ={x ... x | x E} of ^kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on ^kE. Passing to the corresponding projective space , one may consider as a riemannian submanifold of the space form . In this paper we study properties of the pair of riemannian manifolds.     

Abstract Bott periodicity inKK-theory

We show that the universalC*-algebras KqA and K²A are homotopy equivalent and define abstract analogues of the Bott elements inKK-theory.     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

Normality preserving multiplication operators

We show that a multiplication operator (T)=ATB is normality preserving if and only if it is hyponormality preserving, if and only if it is either of the formA=fg,B=h f, orA=D,B=D* for someC andD* D=I. Also we show that is (semi-) Fredholmness prserving if and only ifA andB are (semi-) Fredholm operators.Supported by the Science Foundation of Zhejiang Province and NSF.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

The algebraic theory of torsion. II: Products

The algebraic K-theory product K ₀(A) K ₁ B K ₁(A B) for rings A, B is given a chain complex interpretation, using the absolute torsion invariant introduced in Part I. Given a finitely dominated A-module chain complex C and a round finite B-module chain complex D, it is shown that the A B-module chain complex C D has a round finite chain homotopy structure. Thus, if X is a finitely dominated CW complex and Y is a round finite CW complex, the product X × Y is a CW complex with a round finite homotopy structure.     

On a construction related to the non-abelian tensor square of a group

LetG andG be isomorphic groups. We introduce and study a quotient (G) of the free productG *G which is a group extention of the non-abelian tensor squareG G. This seems to bring computational advantages to calculate this last group. Looking over as an operator in the class of groups we prove that it preserves properties of the argumentG such as finiteness, set of prime divisors, nilpotency and solvability. For a finitep-groupG we find a good polynomial bound for the order of (G).     

K-Theory of the Commutator Ideal

We consider for unital c ^*-algebras the short exact sequence 0 I A * _C B A _max B 0and give an explicit formula for the boundary map for separable C ^*-algebras for which the Künneth formula for tensor products holds without Tor-terms. In the course of the article, we study the similar case of an ideal concerning one-relator groups.     

Lifting-Probleme für Vektorfunktionen und ⊗-Sequenzen

In this article the topologically exact sequences of locally convex spaces are characterized for which for every locally convex space F the map id : FE F Q is a homomorphism, or equivalently, the map id L : FK F E is a topological injection. This is motivated by the problem of lifting Q-valued functions with certain given properties to E-valued functions with the same or slightly weaker properties, which may also be considered as the investigation of parameter dependences of solutions of linear (differential) equations. Applications to partial differential equations and to Fredholm functions are given.     

Discrete decomposability of the restriction ofA q(λ) with respect to reductive subgroups and its applications

Summary LetGG be real reductive Lie groups and q a -stable parabolic subalgebra of Lie (G) . This paper offers a sufficient condition on (G, G, q) that the irreducible unitary representation ofG with non-zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupG, each of finite multiplicity. As an application to purely analytic problems, new results on discrete series are also obtained for some pseudo-Riemannian (non-symmetric) spherical homogeneous spaces, which fit nicely into this framework. Some explicit examples of a decomposition formula are also found in the cases whereA _q is not necessarily a highest weight module.Oblatum 3-IV-1993The author is supported by the NSF grant DMS-9100383.     

Einige Aspekte in der Zuordnungstheorie

Zusammenfassung Gegeben seien endliche MengenX, Y undZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y) Z}.Man nenntA X (bzw.B Y)zuordenbar, wenn es eine Injektion:A Y (bzw.: B X) mit(x) Z _x (bzw.(y) Z _y ) gibt, und (A, B) mit #A=#B > 0 einZuordnungspaar, wenn eine Bijektionf:A B mitf(x)Z _x B (bzw.f ^–1 (y) Z _y A) existiert. Die Bijektionf heißtZuordnungsplan fürA, B.In der vorliegenden Arbeit werden Fragen nach der Existenz von optimal zuordenbaren Mengen und optimalen Zuordnungspaaren behandelt, wenn man auf den MengenX undY Ordnungen vorgibt, wobei auch Nebenbedingungen berücksichtigt werden. In manchen Fällen lassen sich anhand der Beweise Zuordnungspläne oder ihre Berechnungsvorschrift explizit angeben.Zum Schluß werden die Aussagen an konkreten, dem Bereich der Wirtschaftswissenschaften entnommenen Beispielen erläutert.

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Summary LetX, Y be finite sets andZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y)Z}. A X (resp.B Y) is calledassignable if there is an injection: A Y (resp.: B X) with (x) Z _x (resp.(y) Z _y ), (A, B) with #A=#B > 0 anassigned pair if there is a bijection f:A B withf (x) Z _x B (resp.f ^–1(y) Z _y A). The bijectionf is called aplan forA andB.In this paper problems are discussed concerning the existence of optimal assignable sets and optimal assigned pairs ifX andY are totally ordered, additional constraints are also considered. In some cases the proofs give explicit constructions of plans. The results are illustrated by application to problems occurring in Operations Research.

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Diese Arbeit ist mit Unterstützung des Sonderforschungsbereiches 72 an der Universität Bonn entstanden.     

Geometry and the Jones projection of a state

LetA be a von Neumann algebra and a faithful normal state. ThenO = { ºAd(g ¹) :g G _A }andU = { ºAd(u ^*) :u U _A are homogeneous reductive spaces. IfA is aC ^* algebra,e the Jones projection of the faithful state viewed as a conditional expectation, then we prove that the similarity orbit ofe by invertible elements ofA can be imbedded inAA in such a way thate is carried to 1 1 and the orbit ofe to a homogeneous reductive space and an analytic submanifold ofAA.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Caractérisation par l'uniformité des fibrés universels sur la Grassmanienne

Conclusion SoitE un fibré uniforme surG(d,n). Nous avons étudié exhaustivement les cas où rangEd. montrant queE est soit somme directe de fibrés ou droites, soit, sir=d, indécomposable et isomorphe àH _d O _G () où àH _d ^* O _G ().Si rangE=(n-d), nous avons étudié les cas oùE est décomposable en fibrés en droites ou indécomposable et isomorphe àQ _{n d} O _G () où àQ _n ^* _d O _G (). L'étude des fibrés indécomposables de rangd, et (n-d) nous a montré que les quatre fibrés universels surG(d,n), sont caractérisés par leur rang, leur polynôme de Chern et al propriété d'uniformité.     

Tame and wild subspace problems

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B _d =Spec k[(B _d ] is the affine algebraic scheme whoseR-points are theB _k k[B_d]-module structures onR ^d, and M_d is a canonical B_k k[B_d]-module supported by k[B_d]^d. Further, say that an affine subscheme of B_d isclass true if the functor F_gn X M_{d k[B]} X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[] andB. If B_d contains a class-true plane for somed, then the schemes B_e contain class-true subschemes of arbitrary dimensions. Otherwise, each B_d contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF _{L(d, i)} (X); furthermore,F _{L(d, i)} (X) is not isomorphic toF _{L(l, j)} (Y) if(d, i) (l, j) andX 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 313–352, March, 1993.     

Quantum Poisson processes and dilations of dynamical semigroups

Summary The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM _n with stationary faithful state . If (, ) is the quantum measure space in question ( a von Neumann algebra and a faithful normal weight), then the semigroupe ^tL of transition operators on (M _n , ) has generator whereu is an arbitrary unitary element of the centraliser of (M _n ,).Supported by the Netherlands Organization for Scientific Research NWO     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Induced product representations of extended Cuntz algebras

Let _H be the extended Cuntz algebra over the Hilbert space H. Since its zero grade part _H⁰ is the C^*-inductive limit of B(H^r), we look for some family of representations on an inductive limit of H^r as r. When such construction is shaped according to the structure of _H⁰, von Neumanns notion of a reference sequence of unit vectors for Hilbert infinite tensor products emerges; after a further Rieffel induction step, a class IPR[H] of representations of _H arises. For any two such representations, we describe explicitly their associated intertwiners. Any two representations in IPR[H] are either disjoint or unitarily equivalent. Actions of the group by translation on sequences of unit vectors are involved, as well as the ideals of .