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Finitely generated weakly monotone C*-algebra

Maria Elena Griseta, Janusz Wysoczański

TL;DR

The paper characterizes the C*-algebra generated by finitely many annihilation operators on the weakly monotone Fock space as a quotient of a Cuntz-Krieger algebra $O_A$ associated with a lower-triangular incidence matrix $A$, via the ideal generated by $a_0 - a_0^*a_0$ where $a_0$ is the vacuum projection. It then identifies and analyzes a maximal abelian subalgebra (MASA) given by the diagonal, describing its spectrum as a subset of $[0,1]^n$ with a discrete part and a boundary. The main contributions include a faithful representation of the weakly monotone algebra, a precise quotient construction, and a detailed, geometric description of the MASA spectrum, including explicit cases for $n=2$ and $n=3$. These results connect weakly monotone operator algebras to graph/Cuntz-Krieger frameworks and provide structural data valuable for representation theory and potential applications in quantum probability.

Abstract

We consider the $C^*$-algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone $C^*$-algebra. We give an abstract representation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz-Krieger $C^*$-algebra $\mathcal{O}_A$ corresponding to a suitable matrix $A$. Furthermore, we show that the diagonal subalgebra of the weakly monotone $C^*$-algebra is a MASA and we give the detailed description of its Gelfand spectrum.

Finitely generated weakly monotone C*-algebra

TL;DR

The paper characterizes the C*-algebra generated by finitely many annihilation operators on the weakly monotone Fock space as a quotient of a Cuntz-Krieger algebra associated with a lower-triangular incidence matrix , via the ideal generated by where is the vacuum projection. It then identifies and analyzes a maximal abelian subalgebra (MASA) given by the diagonal, describing its spectrum as a subset of with a discrete part and a boundary. The main contributions include a faithful representation of the weakly monotone algebra, a precise quotient construction, and a detailed, geometric description of the MASA spectrum, including explicit cases for and . These results connect weakly monotone operator algebras to graph/Cuntz-Krieger frameworks and provide structural data valuable for representation theory and potential applications in quantum probability.

Abstract

We consider the -algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone -algebra. We give an abstract representation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz-Krieger -algebra corresponding to a suitable matrix . Furthermore, we show that the diagonal subalgebra of the weakly monotone -algebra is a MASA and we give the detailed description of its Gelfand spectrum.
Paper Structure (7 sections, 11 theorems, 78 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 78 equations, 2 figures.

Table of Contents

  1. introduction
  2. preliminaries
  3. The weakly monotone $C^*$-algebra
  4. A MASA for the weakly-monotone $C^*$-algebra
  5. A MASA for the weakly-monotone $C^*$-algebra $C^*(A_1,A_2)$
  6. A MASA for the weakly-monotone $C^*$-algebra $C^*(A_1,\ldots,A_n)$
  7. Spectrum of the MASA

Key Result

Lemma 3.3

The following relations are satisfied:

Figures (2)

  • Figure 1: The identification of the spectrum $\Delta(2)$ with $\Delta_2\subset[0,1]^2$.
  • Figure 2: The identification of the spectrum $\Delta(3)$ with $\Delta_3\subset[0,1]^3$.

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Remark 3.7
  • ...and 18 more