The Universal Coefficient Theorem for C*-Algebras with Finite Complexity
Seiten
A C*-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov’s KK-theory to a commutative C*-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear C*-algebras.
We introduce the idea of a C*-algebra that “decomposes” over a class 𝒞 of C*-algebras. Roughly, this means that locally there are approximately central elements that approximately cut the C*-algebra into two C∗-subalgebras from 𝒞 that have well-behaved intersection. We show that if a C*-algebra decomposes over the class of nuclear, UCT C∗-algebras, then it satisfies the UCT. The argument is based on a Mayer–Vietoris principle in the framework of controlled KK-theory; the latter was introduced by the authors in an earlier work. Nuclearity is used via Kasparov’s Hilbert module version of Voiculescu’s theorem, and Haagerup’s theorem that nuclear C*-algebras are amenable.
We say that a C*-algebra has finite complexity if it is in the smallest class of C*-algebras containing the finitedimensional
C*-algebras, and closed under decomposability; our main result implies that all C*-algebras in this class satisfy the UCT. The class of C*-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a C*-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear C*-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear C*-algebras.
We introduce the idea of a C*-algebra that “decomposes” over a class 𝒞 of C*-algebras. Roughly, this means that locally there are approximately central elements that approximately cut the C*-algebra into two C∗-subalgebras from 𝒞 that have well-behaved intersection. We show that if a C*-algebra decomposes over the class of nuclear, UCT C∗-algebras, then it satisfies the UCT. The argument is based on a Mayer–Vietoris principle in the framework of controlled KK-theory; the latter was introduced by the authors in an earlier work. Nuclearity is used via Kasparov’s Hilbert module version of Voiculescu’s theorem, and Haagerup’s theorem that nuclear C*-algebras are amenable.
We say that a C*-algebra has finite complexity if it is in the smallest class of C*-algebras containing the finitedimensional
C*-algebras, and closed under decomposability; our main result implies that all C*-algebras in this class satisfy the UCT. The class of C*-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a C*-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear C*-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear C*-algebras.
University of Hawaii at Manoa, USA
Texas A&M University, USA
| Erscheinungsdatum | 20.02.2024 |
|---|---|
| Reihe/Serie | MEMOIRS OF THE EUROPEAN MATHEMATICAL SOCIETY ; 8 |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 170 x 240 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik |
| Schlagworte | amenable C*-algebra • controlled KK-theory • decomposition of a C*-algebra • Universal Coefficient Theorem |
| ISBN-13 | 9783985470662 / 9783985470662 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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