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The Universal CoeffIcient Theorem for $C^*$-Algebras with Finite Complexity
 
Rufus Willett University of Hawaii at Manoa, Manoa, HI
Guoliang Yu Texas A & M University, College Station, TX
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-066-2
Product Code:  EMSMEM/8
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
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The Universal CoeffIcient Theorem for $C^*$-Algebras with Finite Complexity
Rufus Willett University of Hawaii at Manoa, Manoa, HI
Guoliang Yu Texas A & M University, College Station, TX
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-066-2
Product Code:  EMSMEM/8
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: May 06, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Memoirs of the European Mathematical Society
    Volume: 82024; 108 pp
    MSC: 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 \(C^*\)-sub-algebras from \(C\) that have well-behaved 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 Mayer-Vietoris 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 finite-dimensional \(C^*\)-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 ordinal-number 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.

    Readership

    Researchers interested in \(C^* \)-algebras.

  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 82024; 108 pp
MSC: 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 \(C^*\)-sub-algebras from \(C\) that have well-behaved 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 Mayer-Vietoris 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 finite-dimensional \(C^*\)-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 ordinal-number 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.

Readership

Researchers interested in \(C^* \)-algebras.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.