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The C*-algebra of the Boidol group

Ying-Fen Lin, Jean Ludwig

TL;DR

This paper determines the C*-algebra of Boidol's 4D exponential Lie group by describing it as an algebra of operator fields over the group's spectrum. It partitions the coadjoint-orbit space into strata $\Gamma_3,\Gamma_2,\Gamma_1,\Gamma_0$ and analyzes the corresponding irreducible representations, providing explicit kernel realizations and equivalences. The authors introduce the operator-field algebra $D^*(G)$, impose norm-convergence and dual-limit conditions to capture the behavior of dual limits, and prove that the Fourier transform identifies $C^*(G)$ with $D^*(G)$, yielding a complete, explicit description of Boidol's group C*-algebra. This extends the operator-field framework used for other dimensionally small exponential Lie groups to the remaining 4D non-$\ast$-regular case and supplies a detailed, practically usable model for representation-theoretic and C*-algebraic analysis.

Abstract

The Boidol group is the smallest non-*-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.

The C*-algebra of the Boidol group

TL;DR

This paper determines the C*-algebra of Boidol's 4D exponential Lie group by describing it as an algebra of operator fields over the group's spectrum. It partitions the coadjoint-orbit space into strata and analyzes the corresponding irreducible representations, providing explicit kernel realizations and equivalences. The authors introduce the operator-field algebra , impose norm-convergence and dual-limit conditions to capture the behavior of dual limits, and prove that the Fourier transform identifies with , yielding a complete, explicit description of Boidol's group C*-algebra. This extends the operator-field framework used for other dimensionally small exponential Lie groups to the remaining 4D non--regular case and supplies a detailed, practically usable model for representation-theoretic and C*-algebraic analysis.

Abstract

The Boidol group is the smallest non-*-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.

Paper Structure

This paper contains 11 sections, 16 theorems, 119 equations.

Table of Contents

  1. Introduction and notations
  2. The Boidol group
  3. The coadjoint orbit space
  4. The Fourier transform
  5. The irreducible representations
  6. Norm control of dual limits.
  7. Norm convergence
  8. An approximation of $\pi_{\rho_k,\lambda_k}(F)$
  9. Convergence in operator norm: $\omega\ne 0$
  10. Convergence in operator norm: $\omega=0$
  11. The final theorem

Key Result

Proposition 2.2

The relative topology on $\Gamma_3$ is Hausdorff. In $\Gamma_2$ the subsets $\Gamma_{2,\varepsilon,\sigma}$, $\varepsilon, \sigma\in \{+1, -1\}$, are open and Hausdorff, $\Gamma_1$ is discrete and $\Gamma_0$ is homeomorphic to ${\mathbb R}$.

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Definition 3.1
  • ...and 38 more