Abstract
We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.
Highlights
We consider the semidirect product group Q and its Toeplitz algebras.In previous work [24], we studied the usual Toeplitz algebra T (N N×) associated to the left-invariant partial order with positive cone N N×
The main results there show that there is a natural dynamics on T (N N×) that admits a rich supply of KMS states and exhibits phase transitions like those of the Bost–Connes system
We identify the commutator ideal in the Toeplitz algebra as a corner in a crossed product of a commutative C∗-algebra, and combine the actions to get an action of Q×+ Q on the commutative algebra, which we can study using the well-established theory of transformation group algebras
Summary
We consider the semidirect product group Q and its Toeplitz algebras. We realise ∂multT (N× N) as a crossed product of the usual Toeplitz algebra T (Q+) by an action of the group Q×+ (Proposition 5.4), and analyse the structure of this crossed product C∗-algebra. We identify the commutator ideal in the Toeplitz algebra as a corner in a crossed product of a commutative C∗-algebra, and combine the actions to get an action of Q×+ Q on the commutative algebra, which we can study using the well-established (i.e., relatively old-fashioned!) theory of transformation group algebras. We have been successful for inverse temperatures β > 1, by applying general results from [21] about KMS states on semidirect products in [24, Theorem 7.1]; they are parametrised by probability measures on T (see Theorem 8.1 below). Pimsner algebra analogous to the ones studied in [20]
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