The inner structure of boundary quotients of right LCM semigroups

Abstract

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta )$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2\le p<\infty$ generators to the $p$-adic ring $C^{\ast }$-algebra. For the case where $P$ is abelian and $C^{\ast }(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^{\ast }(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.

29 pages. To appear in Indiana Univ. Math. J

References in corpus (2)

• The Regular C*-algebra of an Integral Domain
• Cartan subalgebras in C*-algebras. Existence and uniqueness

Cited by in corpus (5)

• Normalizers and permutative endomorphisms of the $2$-adic ring $C^{\ast }$-algebra
• Permutative representations of the $2$-adic ring $C^{\ast }$-algebra
• $C^{\ast }$-algebras of right LCM monoids and their equilibrium states
• On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra
• On the entropy and index of the winding endomorphisms of p-adic ring C${}^{\ast }$-algebras