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[ DevCourseWeb.com ] The Universal Coefficient Theorem for C*-Algebras with Finite Complexity
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English | 2024 | 9783985470662 | 110 pages| PDF | 1 MB
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 Mayerr–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 finite-dimensional 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.
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English | 2024 | 9783985470662 | 110 pages| PDF | 1 MB
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 Mayerr–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 finite-dimensional 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.
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Get Latest Tips and Tricks and Support Us -->> https://FreeCourseWeb.com
We upload these learning materials for the people from all over the world, who have the talent and motivation to sharpen their skills/ knowledge but do not have the financial support to afford the materials. If you like this content and if you are truly in a position that you can actually buy the materials, then Please, we repeat, Please, Support Authors. They Deserve it! Because always remember, without "Them", you and we won't be here having this conversation. Think about it! Peace...
