2021/04/21 by Rufus Willett, Guoliang Yu, Willett, Rufus +1
Computer Science · Mathematics · #19K35 #46L80 #46L85 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2104.10766
openalex publication_date 2021/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 controlled KK-theory, as introduced by the authors in 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.