We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the Baum-Connes assembly map for other equivariant homology theories. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting.

No abstract

While management of atrial fibrillation (AF) patients is improved by guideline-conform application of anticoagulant therapy, rate control, rhythm control, and therapy of accompanying heart disease, the morbidity and mortality associated with AF remain unacceptably high. This paper describes the proceedings of the 3rd Atrial Fibrillation NETwork (AFNET)/European Heart Rhythm Association (EHRA) consensus conference that convened over 60 scientists and representatives from industry to jointly discuss emerging therapeutic and diagnostic improvements to achieve better management of AF patients. The paper covers four chapters: (i) risk factors and risk markers for AF; (ii) pathophysiological classification of AF; (iii) relevance of monitored AF duration for AF-related outcomes; and (iv) perspectives and needs for implementing better antithrombotic therapy. Relevant published literature for each section is covered, and suggestions for the improvement of management in each area are put forward. Combined, the propositions formulate a perspective to implement comprehensive management in AF.

We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.2000 Mathematics Subject Classification. 18E30, 19K35, 46L80, 55U35.

Let G be a locally compact group. We describe elements of KK G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK G : It is the universal split exact stable homotopy functor.To describe a Kasparov triple (E, φ, F ) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L 2 G) ⊗ A ′ and more generally if the group action on A is proper in the sense of Exel and Rieffel.

At least 30 million people worldwide carry a diagnosis of atrial fibrillation (AF), and many more suffer from undiagnosed, subclinical, or 'silent' AF. Atrial fibrillation-related cardiovascular mortality and morbidity, including cardiovascular deaths, heart failure, stroke, and hospitalizations, remain unacceptably high, even when evidence-based therapies such as anticoagulation and rate control are used. Furthermore, it is still necessary to define how best to prevent AF, largely due to a lack of clinical measures that would allow identification of treatable causes of AF in any given patient. Hence, there are important unmet clinical and research needs in the evaluation and management of AF patients. The ensuing needs and opportunities for improving the quality of AF care were discussed during the fifth Atrial Fibrillation Network/European Heart Rhythm Association consensus conference in Nice, France, on 22 and 23 January 2015. Here, we report the outcome of this conference, with a focus on (i) learning from our 'neighbours' to improve AF care, (ii) patient-centred approaches to AF management, (iii) structured care of AF patients, (iv) improving the quality of AF treatment, and (v) personalization of AF management. This report ends with a list of priorities for research in AF patients.

We define the filtrated K-theory of a C * -algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory.For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification.We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C * -algebras over a space X with four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

Let G be a locally compact group, let X be a universal proper G-space, and letX be a G-equivariant compactification of X that is H -equivariantly contractible for each compact subgroup H ⊆ G. Let ∂X =X \ X. Assuming the Baum-Connes conjecture for G with coefficients C and C(∂X), we construct an exact sequence that computes the map on K-theory induced by the embedding C * r G → C(∂X) r G. This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincaré duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic χ(G\X) is non-zero, then the unit element of C(∂X) r G is a torsion element of order |χ(G\X)|. Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant K-homology. (1991): 19K35, 46L80 Mathematics Subject Classification