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

  • Valeriano Aiello

    Vanderbilt University, Nashville, USA
  • Roberto Conti

    Università di Roma La Sapienza, Italy
  • Stefano Rossi

    Università di Roma Tor Vergata, Italy
A Look at the Inner Structure of the 2-adic Ring $C^*$‡-algebra and Its Automorphism Groups cover
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Abstract

We undertake a systematic study of the so-called 2-adic ring CC^*-algebra Q2\mathcal Q_2. This is the universal CC^*-algebra generated by a unitary UU and an isometry S2S_2 such that S2U=U2S2S_2U=U^2S_2 and S2S2+US2S2U=1S_2S_2^*+US_2S_2^*U^*=1. Notably, it contains a copy of the Cuntz algebra O2=C(S1,S2)\mathcal O_2=C^*(S_1, S_2) through the injective homomorphism mapping S1S_1 to US2US_2. Among the main results, the relative commutant C(S2)Q2C^*(S_2)'\cap \mathcal Q_2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion O2Q2\mathcal O_2\subset\mathcal Q_2, namely the endomorphisms of Q2\mathcal Q_2 that restrict to the identity on O2\mathcal O_2 are actually the identity on the whole Q2\mathcal Q_2. Moreover, there is no conditional expectation from Q2\mathcal Q_2 onto O2\mathcal O_2. As for the inner structure of Q2\mathcal Q_2, the diagonal subalgebra D2\mathcal D_2 and C(U)C^*(U) are both proved to be maximal abelian in Q2\mathcal Q_2. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of Q2\mathcal Q_2. In particular, the semigroup of the endomorphisms fixing UU turns out to be a maximal abelian subgroup of Aut(Q2)(\mathcal Q_2) topologically isomorphic with C(T,T)C(\mathbb{T},\mathbb{T}). Finally, it is shown by an explicit construction that Out(Q2)(\mathcal Q_2) is uncountable and non-abelian.

Cite this article

Valeriano Aiello, Roberto Conti, Stefano Rossi, A Look at the Inner Structure of the 2-adic Ring CC^*‡-algebra and Its Automorphism Groups. Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, pp. 45–87

DOI 10.4171/PRIMS/54-1-2