Exel's crossed product and crossed products by completely positive maps

We introduce crossed products of a $C^*$-algebra $A$ by a completely positive map $\varrho:A\to A$ relative to an ideal in $A$. They generalize various crossed products by endomorphisms when $\varrho$ is multiplicative. When $A$ is commutative they include $C^*$-algebras associated to Markov operators by Ionescu, Muhly, Vega, and to topological relations by Brenken, but in general they are not modeled by topological quivers popularized by Muhly and Tomforde. We show that Exel's crossed product $A\rtimes_{\alpha,L} \mathbb{N}$, generalized to the case where $A$ is not necessarily unital, is the crossed product of $A$ by the transfer operator $L$ relative to the ideal generated by $\alpha(A)$. We give natural conditions under which $\alpha(A)$ is uniquely determined by $L$, and hence $A\rtimes_{\alpha,L} \mathbb{N}$ depends only on $L$. Moreover, the $C^*$-algebra $\mathcal{O}(A,\alpha,L)$ associated to $(A,\alpha,L)$ by Exel and Royer always coincides with our unrelative crossed product by $L$. As another non-trivial application of our construction we extend a result of Brownlowe, Raeburn and Vittadello, by showing that the $C^*$-algebra of an arbitrary infinite graph $E$ can be realized as a crossed product of the diagonal algebra $\mathcal{D}_E$ by a `Perron-Frobenious' operator $L$. The important difference is that in general there is no endomorphism $\alpha$ of $\mathcal{D}_E$ making $(\mathcal{D}_E,\alpha,L)$ an Exel system.Comment: 45 pages, this is the version accepted to Houston J. Math. Univ (subsections on universal representations and relative Cuntz-Pimsner algebras are added

Extensions of C*-dynamical systems to systems with complete transfer operators

Starting from an arbitrary endomorphism $\alpha$ of a unital C*-algebra $A$ we construct a bigger C*-algebra $B$ and extend $\alpha$ onto $B$ in such a way that the extended endomorphism $\alpha$ has a unital kernel and a hereditary range, i.e. there exists a unique non-degenerate transfer operator for $(B,\alpha)$, called the complete transfer operator. The pair $(B,\alpha)$ is universal with respect to a suitable notion of a covariant representation and depends on a choice of an ideal in $A$. The construction enables a natural definition of the crossed product for arbitrary $\alpha$.Comment: Compressed and submitted version, 9 page

Topologically free actions and ideals in twisted Banach algebra crossed products

We generalize the well known $C^*$-algebraic result of Kawamura-Tomiyama and Archbold-Spielberg for crossed products of discrete transformation groups to the realm of Banach algebras and twisted actions. Namely we show that topological freeneess is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for some (and hence any) $p\in [1,\infty]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.Comment: 25 page

Postglacial expansion of the arctic keystone copepod calanus glacialis

Calanus glacialis, a major contributor to zooplankton biomass in the Arctic shelf seas, is a key link between primary production and higher trophic levels that may be sensitive to climate warming. The aim of this study was to explore genetic variation in contemporary populations of this species to infer possible changes during the Quaternary period, and to assess its population structure in both space and time. Calanus glacialis was sampled in the fjords of Spitsbergen (Hornsund and Kongsfjorden) in 2003, 2004, 2006, 2009 and 2012. The sequence of a mitochondrial marker, belonging to the ND5 gene, selected for the study was 1249 base pairs long and distinguished 75 unique haplotypes among 140 individuals that formed three main clades. There was no detectable pattern in the distribution of haplotypes by geographic distance or over time. Interestingly, a Bayesian skyline plot suggested that a 1000-fold increase in population size occurred approximately 10,000 years before present, suggesting a species expansion after the Last Glacial Maximum.GAME from the National Science Centre, the Polish Ministry of Science and Higher Education Iuventus Plus [IP2014 050573]; FCT-PT [CCMAR/Multi/04326/2013]; [2011/03/B/NZ8/02876

Crossed Product of a C*-Algebra by a Semigroup of Interactions

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev’s crossed product by an endomorphism, and is related to Exel’s interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms