reflexive subspace
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2014 ◽  
Vol 90 (1) ◽  
pp. 134-140
Author(s):  
W. E. LONGSTAFF

AbstractA definition of the reflexive index of a family of (closed) subspaces of a complex, separable Hilbert space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ is given, analogous to one given by D. Zhao for a family of subsets of a set. Following some observations, some examples are given, including: (a) a subspace lattice on $H$ with precisely five nontrivial elements with infinite reflexive index; (b) a reflexive subspace lattice on $H$ with infinite reflexive index; (c) for each positive integer $n$ satisfying dim $H\ge n+1$, a reflexive subspace lattice on $H$ with reflexive index $n$. If $H$ is infinite-dimensional and ${\mathcal{B}}$ is an atomic Boolean algebra subspace lattice on $H$ with $n$ equidimensional atoms and with the property that the vector sum $K+L$ is closed, for every $K,L\in {\mathcal{B}}$, then ${\mathcal{B}}$ has reflexive index at most $n$.


2012 ◽  
Vol 42 (4) ◽  
pp. 321-328 ◽  
Author(s):  
Wei YUAN ◽  
ChengJun HOU ◽  
GuangFeng CHEN ◽  
AiJu DONG

2008 ◽  
Vol 62 (4) ◽  
pp. 595-599 ◽  
Author(s):  
Kamila Kliś-Garlicka ◽  
Vladimir Müller

2002 ◽  
Vol 132 (1) ◽  
pp. 137-154 ◽  
Author(s):  
NARCISSE RANDRIANANTOANINA

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.


2001 ◽  
Vol 64 (2) ◽  
pp. 307-314
Author(s):  
Jiankui Li

Let  be either a reflexive subspace or a bimodule of a reflexive algebra in B (H), the set of bounded operators on a Hilbert space H. We find some conditions such that a finite rank T ∈  has a rank one summand in  and  has strong decomposability. Let (ℒ) be the set of all operators on H that annihilate all the operators of rank at most one in alg ℒ. We construct an atomic Boolean subspace lattice ℒ on H such that there is a finite rank operator T in (ℒ) such that T does not have a rank one summand in (ℒ). We obtain some lattice-theoretic conditions on a subspace lattice ℒ which imply alg ℒ is strongly decomposable.


1995 ◽  
Vol 38 (3) ◽  
pp. 308-316 ◽  
Author(s):  
K. J. Harrison

AbstractWe give a characterisation of where and are subspace lattices with commutative and either completely distributive or complemented. We use it to show that Lat is a CSL algebra with a completely distributive or complemented lattice and is any operator algebra.


Author(s):  
G. Pisier

SynopsisIn this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.


1991 ◽  
Vol 14 (2) ◽  
pp. 245-252 ◽  
Author(s):  
H. Al-Minawi ◽  
S. Ayesh

LetXbe a real Banach space and(Ω,μ)be a finite measure space andϕbe a strictly icreasing convex continuous function on[0,∞)withϕ(0)=0. The spaceLϕ(μ,X)is the set of all measurable functionsfwith values inXsuch that∫Ωϕ(‖c−1f(t)‖)dμ(t)<∞for somec>0. One of the main results of this paper is: “For a closed subspaceYofX,Lϕ(μ,Y)is proximinal inLϕ(μ,X)if and only ifL1(μ,Y)is proximinal inL1(μ,X)′​′. As a result ifYis reflexive subspace ofX, thenLϕ(ϕ,Y)is proximinal inLϕ(μ,X). Other results on proximinality of subspaces ofLϕ(μ,X)are proved.


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