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+---- Téma: The Universal CoeffIcient Theorem for $C^$-Algebras with Finite Complexity (/showthread.php?tid=325724)
RE: The Universal CoeffIcient Theorem for $C^$-Algebras with Finite Complexity - book24h - 2025-07-02
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Free Download The Universal CoeffIcient Theorem for $C^*$-Algebras with Finite Complexity (Memoirs of the European Mathematical Society) by Rufus Willett, Guoliang Yu
English | February 15, 2024 | ISBN: 3985470669 | 108 pages | PDF | 0.88 Mb
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.
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