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The Quarterly Journal of Mathematics Advance Access





Published: Tue, 09 Jan 2018 00:00:00 GMT

Last Build Date: Tue, 09 Jan 2018 03:47:13 GMT

 



Homological finiteness properties of fibre products

Tue, 09 Jan 2018 00:00:00 GMT

Abstract
We study the homological finiteness property FPn of fibre products. A homological version of the n−(n+1)−(n+2) Conjecture is suggested and solved in some cases. Though the Homological 1–2–3 Conjecture is still open, we prove a homological version of the Virtual Surjection Conjecture in the case of virtual surjection on pairs.



Diagonal automorphisms of the 2-adic ring C*-algebra

Thu, 04 Jan 2018 00:00:00 GMT

Abstract
The 2-adic ring C*-algebra Q2 naturally contains a copy of the Cuntz algebra O2 and, a fortiori, also of its diagonal subalgebra D2 with Cantor spectrum. This paper is aimed at studying the group AutD2(Q2) of the automorphisms of Q2 fixing D2 pointwise. It turns out that any such automorphism leaves O2 globally invariant. Furthermore, the subgroup AutD2(Q2) is shown to be maximal abelian in Aut(Q2). Saying exactly what the group is amounts to understanding when an automorphism of O2 that fixes D2 pointwise extends to Q2. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.