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A Pile of Shifts II: Structure and $K$-Theory

Shelley Hebert, Slawomir Klimek, Matt McBride, J. Wilson Peoples

TL;DR

This work analyzes $C^*$-algebras generated by $C({oldsymbol Z}_s)$ and four natural shifts on the $s$-adic tree, framing each algebra as a crossed product by an endomorphism with hereditary range and a gauge action. For each shift Bunce--Deddens, Hensel, Bernoulli, and Serre, the authors identify a minimal coefficient algebra, construct invariant ideals, and compute the corresponding quotient algebras, leading to six-term exact sequences in $K$-theory. The main results determine $K$-theory for all four shifts, with explicit descriptions of $K_0$ and $K_1$ and, in several cases, direct-limit formulations reflecting the hierarchical structure of the shifts; notably $K_1$ vanishes in all cases and $K_0$ involves combinations of function groups on ${oldsymbol Z}_s$ and standard integer lattices. The significance lies in connecting noncommutative geometry of ultrametric Cantor sets to number-theoretic dynamics via crossed products by endomorphisms, providing concrete invariants for a family of shift algebras and deepening the link between $s$-adic arithmetic and operator algebras.

Abstract

We discuss $C^*$-algebras associated with several different natural shifts on the Hilbert space of the $s$-adic tree, continuing the analysis from [Banach J. Math. Anal. 19 (2025), 32, 30 pages, arXiv:2412.00854] and in particular we describe their structure and compute the $K$-theory groups.

A Pile of Shifts II: Structure and $K$-Theory

TL;DR

This work analyzes -algebras generated by and four natural shifts on the -adic tree, framing each algebra as a crossed product by an endomorphism with hereditary range and a gauge action. For each shift Bunce--Deddens, Hensel, Bernoulli, and Serre, the authors identify a minimal coefficient algebra, construct invariant ideals, and compute the corresponding quotient algebras, leading to six-term exact sequences in -theory. The main results determine -theory for all four shifts, with explicit descriptions of and and, in several cases, direct-limit formulations reflecting the hierarchical structure of the shifts; notably vanishes in all cases and involves combinations of function groups on and standard integer lattices. The significance lies in connecting noncommutative geometry of ultrametric Cantor sets to number-theoretic dynamics via crossed products by endomorphisms, providing concrete invariants for a family of shift algebras and deepening the link between -adic arithmetic and operator algebras.

Abstract

We discuss -algebras associated with several different natural shifts on the Hilbert space of the -adic tree, continuing the analysis from [Banach J. Math. Anal. 19 (2025), 32, 30 pages, arXiv:2412.00854] and in particular we describe their structure and compute the -theory groups.
Paper Structure (30 sections, 17 theorems, 74 equations)

This paper contains 30 sections, 17 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\boldsymbol{s}$-adic integers
  4. Hilbert spaces and shifts
  5. Coefficient algebras with a gauge action
  6. Coefficient algebras
  7. Gauge action
  8. O'Donovan condition
  9. Invariant ideals
  10. The Bunce--Deddens shift
  11. The algebra $\boldsymbol{A_U}$
  12. The coefficient algebra of $\boldsymbol{A_U}$
  13. The Structure of $\boldsymbol{A_U}$
  14. $\boldsymbol{K}$-theory of $\boldsymbol{A_U}$
  15. The Hensel shift
  16. ...and 15 more sections

Key Result

Proposition 3.1

The algebra $\mathcal{I}_{\mathcal{J}}$ is an ideal in $A_{\mathcal{J}}$.

Theorems & Definitions (17)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 5.1
  • Proposition 5.2
  • ...and 7 more