Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

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.     

Noncritical maps on geodesically complete spaces with curvature bounded above

Annals of Global Analysis and Geometry - We define and study the regularity of distance maps on geodesically complete spaces with curvature bounded above. We prove that such a regular map is...     

Completely positive completion of partial matrices whose entries are completely bounded maps

We study completion problems of partial matrices associated with a graph where entries are completely bounded maps on aC ^*-algebra. We characterize a graph for which every -partial completely positive matrix has a completely positive completion. As a special case we study -partial functional matrices. We give a necessary and sufficient condition for a -partial functional matrix to have a positive completion and a representation for such matrices. These generalize some results on inflated Schur product maps due to Paulsen, Power and Smith. As an application, we study completely positive completions of partial matrices whose entries are completely bounded multipliers of the Fourier algebra of a locally compact group.     

Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above

The hypothesis of local compactness of the target is removed from an earlier result about interior Hölder continuity of locally energy minimizing maps ? from a Riemannian polyhedron (X, g) to a suitable ball B of radius R <  π/2 (best possible) in a geodesic space with curvature ≤ 1. Furthermore, the variational Dirichlet problem for harmonic maps from an open set \(\Omega \Subset X\) to B is shown to be uniquely solvable, and the solution is continuous up to the boundary ?Ω at any regular point of ?Ω at which the prescribed boundary map is continuous.     

Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.     

Nilpotent completely positive maps

We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely positive maps.     

Representations of completely bounded multilinear operators

A definition of a completely bounded multilinear operator from one C¹-algebra into another is introduced. Each completely bounded multilinear operator from a C¹-algebra into the algebra of bounded linear operators on a Hilbert space is shown to be representable in terms of ¹-representations of the C¹-algebra and interlacing operators. This result extends Wittstock's Theorem that decomposes a completely bounded linear operator from a C¹-algebra into an injective C¹-algebra into completely positive linear operators.     

On Banach spaces with few spreading models

If the set of spreading models of a Banach space is countable (up to equivalence), then it cannot contain a strictly increasing infinite chain of spreading models generated by normalized weakly null sequences. Moreover, such a space must have a spreading model which is `close' to or for some .

     

Operator maps of Jensen-type

Let \(\mathbb {B}_J({\mathcal {H}})\) denote the set of self-adjoint operators acting on a Hilbert space \(\mathcal {H}\) with spectra contained in an open interval J. A map \(\Phi :\mathbb {B}_J({\mathcal {H}})\rightarrow {{\mathbb {B}}}({\mathcal {H}})_\text {sa} \) is said to be of Jensen-type if

$$\begin{aligned} \Phi (C^*AC+D^*BD)\le C^*\Phi (A)C+D^*\Phi (B)D \end{aligned}$$

for all \( A, B \in \mathbb {B}_J({\mathcal {H}})\) and bounded linear operators C, D acting on \( \mathcal {H} \) with \( C^*C+D^*D=I\), where I denotes the identity operator. We show that a Jensen-type map on an infinite dimensional Hilbert space is of the form \(\Phi (A)=f(A)\) for some operator convex function f defined in J.

     

Linear pursuit-evasion games with bounded state spaces

The problem of linear pursuit-evasion games with bounded state spaces is considered. Some sufficient conditions for optimality are established, and an example is given.This research was carried out while the author was a Visiting Associate Research Engineer at the University of California at Berkeley. The research was supported by the Office of Naval Research, Grant No. N00014-69-A-0200-1012. The author would like to express his gratitude to Professor G. Leitmann for discussions and for making possible his visit at Berkeley.     

Completely positive module maps and completely positive extreme maps

Let be unital -algebras and be the set of all completely positive linear maps of into . In this article we characterize the extreme elements in , for all , and pure elements in in terms of a self-dual Hilbert module structure induced by each in . Let be the subset of consisting of -module maps for a von Neumann algebra . We characterize normal elements in to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.

     

Products of Michael spaces and completely metrizable spaces

For disjoint subsets of the Michael space has the topology obtained by isolating the points in and letting the points in retain the neighborhoods inherited from . We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space , of minimal weight , with Lindelöf but with not normal. ( denotes the countable product of a discrete space of cardinality .) If denotes , the normality of implies the normality of for any complete metric space (of arbitrary weight). However, the statement `` normal implies normal' is axiom sensitive.