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This is the first of two papers dedicated to the detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan's approach to the classification of the tempered dual. [ABSTRACT FROM AUTHOR]

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes–Kasparov conjecture in operator K-theory for these groups. In Part I we presented the Morita equivalence and the Connes–Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual. In fact we shall go further by giving a complete representation-theoretic description and parametrization, in Vogan's terms, of the essential components of the tempered dual, which carry the K-theory of the tempered dual. [ABSTRACT FROM AUTHOR]

In this paper, for 1 < p < ∞, the authors show that the coarse ℓp-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu's Property (H). [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616]. [ABSTRACT FROM AUTHOR]

In the setting of non-type II 1 representations, we propose a definition of deformed Fredholm module [ D T | D T | - 1 , · ] T for a modular spectral triple T , where D T is the deformed Dirac operator. D T is assumed to be invertible for the sake of simplicity, and its domain is an “essential” operator system E T . According to such a definition, we obtain [ D T | D T | - 1 , · ] T = | D T | - 1 d T (·) + d T (·) | D T | - 1 , where d T is the deformed derivation associated to D T . Since the “quantum differential” 1 / | D T | appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in Fidaleo and Suriano (J Funct Anal 275:1484–1531, 2018) of non-type II 1 representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator D T in terms of the periodic solutions of a particular class of eigenvalue Hill equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material. We consider tight-binding magnetic models associated to generic magnetic multi-interfaces and describe the K -theoretical setting in which a bulk-interface duality can be derived. Then, the (trivial) case of a localized magnetic field and the (non trivial) case of the Iwatsuka magnetic field are considered in full detail. This is a pedagogical preparatory work that aims to anticipate the study of more complicated multi-interface magnetic systems. [ABSTRACT FROM AUTHOR]