Homotopy Theory of C*-Algebras (Frontiers in Mathematics)

Homotopy theory and C\* algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C\*-algebras. One basic idea of the setup is to merge C\*-algebras and spaces studied in algebraic topology into one category comprising C\*-spaces. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C\*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. It serves a wide audience of graduate students and researchers interested in C\*-algebras, homotopy theory and applications. Homotopy Theory And C*-algebras Are Central Topics In Contemporary Mathematics. This Book Introduces A Modern Homotopy Theory For C*-algebras. One Basic Idea Of The Setup Is To Merge C*-algebras And Spaces Studied In Algebraic Topology Into One Category Comprising C*-spaces. These Objects Are Suitable Fodder For Standard Homotopy Theoretic Moves, Leading To Unstable And Stable Model Structures. With The Foundations In Place One Is Led To Natural Definitions Of Invariants For C*-spaces Such As Homology And Cohomology Theories, K-theory And Zeta-functions. The Text Is Largely Self-contained. It Serves A Wide Audience Of Graduate Students And Researchers Interested In C*-algebras, Homotopy Theory And Applications. 1 Introduction -- 2 Preliminaries -- 2.1 C*-spaces -- 2.2 G – C*-spaces -- 2.3 Model Categories -- 3 Unstable C*-homotopy Theory -- 3.1 Pointwise Model Structures -- 3.2 Exact Model Structures -- 3.3 Matrix Invariant Model Structures -- 3.4 Homotopy Invariant Model Structures -- 3.5 Pointed Model Structures -- 3.6 Base Change -- 4 Stable C*-homotopy Theory -- 4.1 C*-spectra -- 4.2 Bispectra -- 4.3 Triangulated Structure -- 4.4 Brown Representability -- 4.5 C*-symmetric Spectra -- 4.6 C*-functors -- 5 Invariants -- 5.1 Cohomology And Homology Theories -- 5.2 Kk-theory And The Eilenberg-maclane Spectrum -- 5.3 Hl-theory And The Eilenberg-maclane -- 5.4 The Chern-connes-karoubi Character -- 5.5 K-theory Of C*-algebras -- 5.6 Zeta Functions -- 6 The Slice Filtration -- References -- Index. Paul Arne Østvær. Includes Bibliographical References (pages 135-139). Cover......Page 1 Frontiers in Mathematics......Page 3 Homotopy Theory of C*-Algebras......Page 4 9783034605649......Page 5 Contents......Page 6 1 Introduction......Page 8 Acknowledgment......Page 11 2.1 C*-spaces......Page 14 2.2 G-C*-spaces......Page 22 2.3 Model categories......Page 23 3.1 Pointwise model structures......Page 32 3.2 Exact model structures......Page 40 3.3 Matrix invariant model structures......Page 50 3.4 Homotopy invariant model structures......Page 54 3.5 Pointed model structures......Page 66 3.6 Base change......Page 72 4.1 C*-spectra......Page 76 4.2 Bispectra......Page 85 4.3 Triangulated structure......Page 88 4.5 C*-symmetric spectra......Page 93 4.6 C*-functors......Page 106 5.1 Cohomology and homology theories......Page 114 5.2 KK-theory and the Eilenberg-MacLane spectrum......Page 115 5.3 HL-theory and the Eilenberg-MacLane spectrum......Page 118 5.4 The Chern-Connes-Karoubi character......Page 119 5.5 K-theory of C*-algebras......Page 120 5.6 Zeta functions......Page 128 6 The slice filtration......Page 138 Bibliography......Page 142