Normalizers and permutative endomorphisms of the $2$-adic ring $C^{\ast }$-algebra

Abstract

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra ${\mathcal{D}}_2$ in the $2$-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion ${\mathcal{O}}_2\subset {\mathcal{Q}}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of ${\mathcal{Q}}_2$ are exhibited with prescribed images of the generator $U$.

arXiv commentTo appear in J. Math. Anal. Appl

References in corpus (6)

• A look at the inner structure of the $2$-adic ring $C^{\ast }$-algebra and its automorphism groups
• The inner structure of boundary quotients of right LCM semigroups
• Diagonal automorphisms of the $2$-adic ring $C^{\ast }$-algebra
• Noncommutative topological entropy of endomorphisms of Cuntz algebras
• Symmetry in the Cuntz Algebra on two generators
• Permutative representations of the $2$-adic ring $C^{\ast }$-algebra

Cited by in corpus (3)

• On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra
• The Motzkin subproduct system
• On the entropy and index of the winding endomorphisms of p-adic ring C${}^{\ast }$-algebras