K-theory for algebras of operators on Banach spaces

It is proved that, for each pair (m,n) of non-negative integers, there is a Banach space X for which the group K_0(B(X)) is isomorphic to m copies of the integers and the group K_1(B(X)) is isomorphic to n copies of the integers. Along the way we compute the K-groups of all closed ideals of operators contained in the ideal of strictly singular operators, and we derive some results about the existence of splittings of certain short exact sequences

Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space

We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).Comment: to appear in C.R. Math. Acad. Sci. Pari

General polygamy inequality of multi-party quantum entanglement

Using entanglement of assistance, we establish a general polygamy inequality of multi-party entanglement in arbitrary dimensional quantum systems. For multi-party closed quantum systems, we relate our result with the monogamy of entanglement to show that the entropy of entanglement is an universal entanglement measure that bounds both monogamy and polygamy of multi-party quantum entanglement.Comment: 4 pages, 1 figur

Random bipartite entanglement from W and W-like states

We describe a protocol for distilling maximally entangled bipartite states between random pairs of parties from those sharing a tripartite W state, and show that, rather surprisingly, the total distillation rate (the total number of EPR pairs distilled per W, irrespective of who shares them) may be done at a higher rate than distillation of bipartite entanglement between specified pairs of parties. Specifically, the optimal distillation rate for specified entanglement for the W has been previously shown to be the asymptotic entanglement of assistance of 0.92 EPR pairs per W, while our protocol can asymptotically distill 1 EPR pair per W between random pairs of parties, which we conjecture to be optimal. We thus demonstrate a tradeoff between the overall asymptotic rate of EPR distillation and the distribution of final EPR pairs between parties. We further show that by increasing the number of parties in the protocol that there exist states with fixed lower-bounded distillable entanglement for random parties but arbitrarily small distillable entanglement for specified parties.Comment: 5 pages, 1 figure, RevTeX. v2 - upper bound on random distillation is expressed more generally and corollaries to the bound added. Minor notation changes. v3 - further notation changes (Ernd now designated Et), discussion of finite distillation rounds and single-copy bound on Et added. Theorem added - relative entropy is shown to be an upper bound to Et for all pure states. Discussion of W formation from EPRs (previously shown in others' work) removed. Some addition, removal and reordering of reference

A weak*-topological dichotomy with applications in operator theory

Denote by $[0,\omega_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,\omega_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,\omega_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,\omega_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,\omega_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,\omega_1)$ can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of $C_0[0,\omega_1)$ and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy-Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on $C_0[0,\omega_1)$. As a consequence, we find that this ideal has a bounded left approximate identity, thus resolving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space $C_0[0,\omega_1)$ and the operators acting on it.Comment: accepted to Transactions of the London Mathematical Societ