Softcover ISBN:  9783985470662 
Product Code:  EMSMEM/8 
List Price:  $75.00 
AMS Member Price:  $60.00 
Softcover ISBN:  9783985470662 
Product Code:  EMSMEM/8 
List Price:  $75.00 
AMS Member Price:  $60.00 

Book DetailsMemoirs of the European Mathematical SocietyVolume: 8; 2024; 108 ppMSC: Primary 19; Secondary 46
A \(C^*\)algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's \(K\)theory to a commutative \(C^*\)algebra. This book 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.
The authors introduce the idea of a \(C^*\)algebra that "decomposes" over a class \(\mathcal{C}\) 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 wellbehaved intersection. The authors 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 MayerVietoris principle in the framework of controlled \(K\)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.
The authors 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; their 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 ordinalnumber invariant measuring the complexity level. They 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. The authors 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.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
ReadershipResearchers interested in \(C^* \)algebras.

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A \(C^*\)algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's \(K\)theory to a commutative \(C^*\)algebra. This book 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.
The authors introduce the idea of a \(C^*\)algebra that "decomposes" over a class \(\mathcal{C}\) 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 wellbehaved intersection. The authors 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 MayerVietoris principle in the framework of controlled \(K\)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.
The authors 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; their 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 ordinalnumber invariant measuring the complexity level. They 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. The authors 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.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Researchers interested in \(C^* \)algebras.