A Look at the Inner Structure of the 2-adic Ring $C^*$‡-algebra and Its Automorphism Groups

Abstract

We undertake a systematic study of the so-called 2-adic ring C * -algebra Q 2 . This is the universal C * -algebra generated by a unitary U and an isometry S 2 such that S 2 U = U 2 S 2 and S 2 S * 2 + U S 2 S * 2 U * = 1. Notably, it contains a copy of the Cuntz algebrathrough the injective homomorphism mapping S 1 to U S 2 . Among the main results, the relative commutant C * (S 2 ) ′ ∩ Q 2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion O 2 ⊂ Q 2 , namely the endomorphisms of Q 2 that restrict to the identity on O 2 are actually the identity on the whole Q 2 . Moreover, there is no conditional expectation from Q 2 onto O 2 . As for the inner structure of Q 2 , the diagonal subalgebra D 2 and C * (U ) are both proved to be maximal abelian in Q 2 . The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of Q 2 . In particular, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian subgroup of Aut(Q 2 ) topologically isomorphic with C(T, T). Finally, it is shown by an explicit construction that Out(Q 2 ) is uncountable and non-abelian.

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

“…It immediately follows from the analysis carried out in [10] that such restrictions are automorphisms of O 2 induced by unitaries in D 2 , henceforth referred to as diagonal automorphisms for short. The results here obtained lend further support to the idea, already expressed in [1], that the group of automorphisms of Q 2 is, in a sense, considerably smaller than that of O 2 , thus making it reasonable to ask the challenging question whether this group may be computed explicitly up to inner automorphisms. Indeed, we show that any extendible localized diagonal automorphism of O 2 is necessarily the product of a gauge automorphism and a localized inner diagonal automorphism.…”

“….u that contains a copy of the Cuntz algebra O 2 in a canonical way. Inter alia, it was proved in [1] the useful fact that the canonical diagonal D 2 maintains the property of being a maximal abelian subalgebra (MASA) in Q 2 also. Actually, more is known, for D 2 is even a Cartan subalgebra of both O 2 and Q 2 [13,Prop.…”

Permutative representations of the 2-adic ring C∗-algebra

“…The quadratic permutative endomorphisms include the canonical endomorphism ϕ of O 2 and the flip-flop λ f . Both of them extend to Q 2 , as proved in [ACR18]. More interestingly, this family of endomorphisms offers a bunch of novel examples of endomorphisms that do extend to Q 2 , although the list of the extendible endomorphisms is still rather limited.…”

“…This section aims to intertwine the analysis carried out in [ACR18], where we addressed the problem of extending endomorphisms of O 2 to Q 2 , with the present analysis. Although a satisfactory answer to the problem is yet to come and might be elusive to get to, particular classes of automorphisms, such as Bogolubov automorphism and localized diagonal automorphisms, have been examined thoroughly.…”

Normalizers and permutative endomorphisms of the 2-adic ring C⁎-algebra

“…The present brief section aims to refine some results concerning the C * -subalgebra of Q 2 generated by U . To begin with, in [ACR18a] the commutative subalgebra C * (U ) was proved to be maximal Abelian in Q 2 , and it was also seen to be the image of a unique conditional expectation from Q 2 . However, our subalgebra fails to be a Cartan subalgebra, as shown by the following result.…”