TY - JOUR

T1 - On the Connes–Kasparov isomorphism, I

T2 - The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

AU - Clare, Pierre

AU - Higson, Nigel

AU - Song, Yanli

AU - Tang, Xiang

N1 - Publisher Copyright:
© The Mathematical Society of Japan and Springer Nature Japan KK, part of Springer Nature 2024.

PY - 2024/4

Y1 - 2024/4

N2 - 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.

AB - 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.

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U2 - 10.1007/s11537-024-2220-2

DO - 10.1007/s11537-024-2220-2

M3 - Article

AN - SCOPUS:85187869184

SN - 0289-2316

VL - 19

SP - 67

EP - 109

JO - Japanese Journal of Mathematics

JF - Japanese Journal of Mathematics

IS - 1

ER -