The uniqueness-of-norm problem for Calkin algebras

We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989)

Detecting word substitutions in text

Searching for words on a watchlist is one way in which large-scale surveillance of communication can be done, for example in intelligence and counterterrorism settings. One obvious defense is to replace words that might attract attention to a message with other, more innocuous, words. For example, the sentence the attack will be tomorrow" might be altered to the complex will be tomorrow", since 'complex' is a word whose frequency is close to that of 'attack'. Such substitutions are readily detectable by humans since they do not make sense. We address the problem of detecting such substitutions automatically, by looking for discrepancies between words and their contexts, and using only syntactic information. We define a set of measures, each of which is quite weak, but which together produce per-sentence detection rates around 90% with false positive rates around 10%. Rules for combining persentence detection into per-message detection can reduce the false positive and false negative rates for messages to practical levels. We test the approach using sentences from the Enron email and Brown corpora, representing informal and formal text respectively

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

Extensions and the weak Calkin algebra of Read's Banach space admitting discontinuous derivations

Read produced the first example of a Banach space E such that the associated Banach algebra B(E) of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalize Read's main theorem about B(E) from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence {0}→W(E)→B(E)→l2~→{0}, W(E) where W(E) denotes the ideal of weakly compact operators on E, while l2~ is the unitization of the Hilbert space l2, endowed with the zero product

Signed directed social network analysis applied to group conflict

Real-world social networks contain relationships of multiple different types, but this richness is often ignored in graph-theoretic modelling. We show how two recently developed spectral embedding techniques, for directed graphs (relationships are asymmetric) and for signed graphs (relationships are bothpositive and negative), can be combined. This combination is particularly appropriate for intelligence, terrorism, and law enforcement applications. We illustrate by applying the novel embedding technique to datasets describing conflict in North-West Africa, and show how unusual interactions can be identified