[Paper Review] Quantum lens spaces and principal actions on graph C*-algebras

Wojciech Szymański|ArXiv.org|Sep 29, 2002

Advanced Operator Algebra Research7 references4 citations

TL;DR

This paper establishes the principality of $ζ_p$- and $×$-actions on quantum odd-dimensional spheres via their realization as gauge actions on graph $C^*$-algebras, proving that fixed-point algebras yield quantum lens spaces and quantum complex projective spaces. The key contribution is a general criterion for principality of gauge actions on graph $C^*$-algebras, enabling K-theoretic and ideal structure computations through graph-theoretic data.

ABSTRACT

We study certain principal actions on noncommutative C*-algebras. Our main examples are the Z_p- and T-actions on the odd-dimensional quantum spheres, yielding as fixed-point algebras quantum lens spaces and quantum complex projective spaces, respectively. The key tool in our analysis is the relation of the ambient C*-algebras with the Cuntz-Krieger algebras of directed graphs. A general result about the principality of the gauge action on graph algebras is given.

Motivation & Objective

• To establish the principality of $ζ_p$- and $×$-actions on noncommutative $C^*$-algebras associated with odd-dimensional quantum spheres.
• To show that these actions yield quantum lens spaces and quantum complex projective spaces as fixed-point algebras.
• To utilize the framework of Cuntz-Krieger algebras of directed graphs to simplify the analysis of $C^*$-algebraic structures and actions.
• To provide a general criterion for principality of gauge actions on arbitrary graph $C^*$-algebras.

Proposed method

• Realize the $C^*$-algebras of quantum spheres and lens spaces as graph $C^*$-algebras using the Cuntz-Krieger construction.
• Define the $ζ_p$- and $×$-actions on the quantum sphere $C(S^{2n-1}_q)$ as automorphisms that scale generators $z_i$ by roots of unity or unitary scalars.
• Transport these actions to the corresponding graph $C^*$-algebra via isomorphisms, showing they correspond to canonical gauge actions.
• Use the graph algebra framework to compute $K$-theory via the cokernel and kernel of the matrix $A_E$ defined by vertex and edge relations.
• Prove principality by showing the associated map $\Phi$ is surjective onto $P_v \otimes z^k$, using path decompositions in the graph.
• Apply the general principality criterion to specific cases, including $C(SU_q(2))$ and quantum complex projective spaces.

Experimental results

Research questions

• RQ1 Are the $ζ_p$-actions on quantum odd-dimensional spheres principal in the sense of Ellwood?
• RQ2 Do the $×$-actions on quantum spheres yield quantum complex projective spaces as fixed-point algebras?
• RQ3 Can the principality of gauge actions on graph $C^*$-algebras be characterized in terms of graph-theoretic data?
• RQ4 How does the $K$-theory of quantum lens spaces and quantum projective spaces relate to the structure of the underlying graph?
• RQ5 Is the gauge action on $C^*(L_3)$, isomorphic to $C(SU_q(2))$, principal, and what is its fixed-point algebra?

Key findings

• The $ζ_p$-action on $C(S^{2n-1}_q)$ is principal, with fixed-point algebra isomorphic to the $C^*$-algebra of the quantum lens space $C(L_q(p;m_1,\dots,m_n))$.
• The $×$-action on $C(S^{2n-1}_q)$ is principal, yielding the quantum complex projective space $C(\mathbb{C}P^{n-1}_q)$ as the fixed-point algebra.
• The gauge action on the graph algebra $C^*(L_3)$, corresponding to $C(SU_q(2))$, is principal, with fixed-point algebra isomorphic to the minimal unitization of the compact operators.
• The $K$-theory of graph $C^*$-algebras is computed via $K_0 = \operatorname{coker}(A_E)$ and $K_1 = \ker(A_E)$, where $A_E$ is the vertex-edge incidence matrix.
• The principality of the gauge action on a graph $C^*$-algebra holds if and only if every vertex emits at least one edge, ensuring surjectivity of the map $\Phi$.
• The construction via graph algebras simplifies the analysis of ideal structure and $K$-invariants, reducing noncommutative $C^*$-algebra problems to combinatorial graph data.