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

作者：Aiello Valeriano^*; Conti Roberto; Rossi Stefano

来源：Publications of the Research Institute for Mathematical Sciences, 2018, 54(1): 45-87.

DOI：10.4171/PRIMS/54-1-2

• 2-adic ring C*-algebra
• C*-algebra
• dyadic integers
• maximal abelian self-adjoint subalgebra
• Cartan subalgebra
• gauge action
• Bogoljubov automorphism
• automorphism group
• outer automorphism

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 S2U = (US2)-S-2 and S2S*(2) + US2S*U-2* = 1. Notably, it contains a copy of the Cuntz algebra O-2 = C* (S-1, S-2) through the injective homomorphism mapping S-1 to US2. Among the main results, the relative commutant C* (S-2)' boolean AND Q(2) is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion O-2 subset of 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.

• 出版日期2018