On the -theory of -algebras arising from integral dynamics

Abstract

We investigate the K-theory of unital UCT Kirchberg algebras Q S arising from families S of relatively prime numbers. It is shown that K * (Q S ) is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct C * -algebra naturally associated to S. The C * -algebra representing the torsion part is identified with a natural subalgebra A S of Q S . For the K-theory of Q S , the cardinality of S determines the free part and is also relevant for the torsion part, for which the greatest common divisor g S of {p − 1 : p ∈ S} plays a central role as well. In the case where |S| ≤ 2 or g S = 1 we obtain a complete classification for Q S . Our results support the conjecture that A S coincides with ⊗ p∈S Op. This would lead to a complete classification of Q S , and is related to a conjecture about k-graphs.Datethen we get a complete classification for Q S with the rule that Q S and Q T are isomorphic if and only if |S| = |T | and g S = g T , see Conjecture 6.5.At a later stage, the authors learned that Li and Norling obtained interesting results for the multiplicative boundary quotient for N H + by using completely different methods, see [LN16, Subsection 6.5]. Briefly speaking, the multiplicative boundary quotient related to Q S is obtained by replacing the unitary u by an isometry v, see Subsection 2.2 for details. As a consequence, the K-theory of the multiplicative boundary quotient does not feature a non-trivial free part. It seems that A S is the key to reveal a deeper connection between the K-theoretical structure of these two C * -algebras. As this is beyond the scope of the present work, we only note that the inclusion map from A S into Q S factors through the multiplicative boundary quotient as an embedding of A S and the natural quotient map. The results of [LN16] together with our findings indicate that this embedding might be an isomorphism in K-theory. This idea is explored further in [Sta, Section 5].The paper is organized as follows: In Section 2, we set up the relevant notation and list some useful known results in Subsection 2.1. We then link Q S to boundary quotients of right LCM semigroups, see Subsection 2.2, and a-adic algebras, see Subsection 2.3. These parts explain the central motivation behind our interest in the K-theory of Q S . In addition, the connection to a-adic algebras allows us to apply a duality theorem from [KOQ14], see Theorem 3.1, making it possible to invoke real dynamics. This leads to a decomposition result for K * (Q S ) presented in Section 4, which essentially reduces the problem to determining the K-theory of A S . The structure of the torsion subalgebra A S is discussed in Section 5. Finally, the progress on the classification of Q S we obtain via a spectral sequence argument for the K-theory of A S is presented in Section 6.Remark 5.9. Similar to Λ S,θ , we can also consider the row-finite k-graph Λ S,σ with σ p,q being the flip, i.e. σ p,q (m, n) := (n, m). That is to say, we keep the skeleton of Λ S,θ , but replace θ by σ. In thi...

Boundary Quotient -algebras of Products of Odometers

“…After we finished this paper, Nicolai Stammeier informed us that, using a completely different approach, it was shown in [BOS15] that Q(F + θ ⊲⊳ Z) is nuclear, simple and purely infinite if {n i } k i=1 ⊂ N \ {0, 1} is a pairwise coprime set. So Theorem 5.13 generalizes their results.…”

“…We should mention that F + θ here is not necessary to be right LCM, and so F + θ ⊲⊳ G is not right LCM in general. Therefore, one can not apply the results in the recent works on right LCM semigroups, such as [ABLS16,BOS15,BLS16,BRRW14,Stam16,Star15], to our cases.…”

“…Since then, there has been attracting a lot of attention to the study of semigroup C*-algebras. See, for example, [ABLS16,BOS15,BLS16,BRRW14,Stam16,Star15] and the references therein. In [BRRW14], Brownlowe-Ramagge-Robertson-Whittaker defined a quotient C*-algebra Q(P ) of C * (P ).…”

A boundary quotient diagram for right LCM semigroups

“…S = N ⋊ P , where P is the free abelian submonoid of N × generated by a family P of relatively prime numbers. For such semigroups, nontrivial partial results emerged in the recent past [LN16,BOS16]. These results lead to intriguing questions and relate to conjectures about C * -algebras of k-graphs, see [BOS16, Conjecture 5.11].…”

“…With regards to the associated C * -algebras, we provide evidence that S ci plays an important role in the quest for the K-theory of Q(S) and Q p (S). This is discussed in Section 5 in the context of integral dynamics [BOS16] and Baumslag-Solitar monoids [Spi12].…”