[Paper Review] Quantum projective space from Toeplitz cubes

Piotr M. Hajac, Atabey Kaygun+1 more|arXiv (Cornell University)|Aug 3, 2010

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper constructs a noncommutative deformation of complex projective space $\mathbb{P}^N(\mathbb{C})$ using $N$-fold tensor products of the Toeplitz algebra, forming a multipullback C*-algebra. The key result is that the lattice of kernels of canonical projections onto components is free, preserving the combinatorial structure of the affine covering under noncommutative deformation.

ABSTRACT

From N-tensor powers of the Toeplitz algebra, we construct a multipullback C*-algebra that is a noncommutative deformation of the complex projective space CP(N). Using Birkhoff's Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multipullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.

Motivation & Objective

• To develop a noncommutative deformation of complex projective space $\mathbb{P}^N(\mathbb{C})$ by deforming the standard pieces (unit discs) rather than the global space.
• To preserve the topological and combinatorial structure of the affine covering of $\mathbb{P}^N(\mathbb{C})$ under noncommutative deformation.
• To establish that the lattice of ideals generated by the kernels of canonical projections is free, ensuring the deformation respects the original set-theoretic gluing data.
• To provide a new framework for constructing fiber products of spectral triples in noncommutative geometry by gluing along boundaries.
• To explore K-theoretic and index-theoretic properties of the resulting quantum projective space, particularly its non-triviality and potential as a $U(1)$-principal bundle.

Proposed method

• Construct the Toeplitz quantum projective space $C(\mathbb{P}^N(\mathcal{T}))$ as a multipullback C*-algebra inside $\prod_{i=0}^N \mathcal{T}^{\otimes N}$, where $\mathcal{T}$ is the Toeplitz algebra.
• Use the canonical projections $\pi_i: C(\mathbb{P}^N(\mathcal{T})) \to \mathcal{T}^{\otimes N}$ to define ideals $\ker \pi_i$, representing the quantum analogues of affine patches.
• Apply Birkhoff’s Representation Theorem to analyze the lattice generated by the ideals $\{\ker \pi_i\}_{i=0}^N$.
• Leverage the nuclearity and flatness of the C*-completed tensor product to ensure well-behaved gluing and algebraic structure.
• Utilize the fact that the lattice of kernels is isomorphic to the lattice of subsets generated by the affine covering of $\mathbb{P}^N(\mathbb{C})$ to prove freeness.
• Compare the resulting C*-algebra with quantum group deformations (e.g., Podleś spheres) to show non-isomorphism for $N \geq 2$.

Experimental results

Research questions

• RQ1 Can the affine covering of $\mathbb{P}^N(\mathbb{C})$ be preserved under a noncommutative deformation by deforming the standard pieces (unit discs) via the Toeplitz algebra?
• RQ2 Does the lattice of ideals generated by the kernels of canonical projections on the components of the multipullback C*-algebra remain free in the quantum case, as it is in the classical case?
• RQ3 How does this construction relate to existing quantum group deformations of projective spaces, such as those of Podleś or quantum Grassmannians?
• RQ4 Can this construction serve as a foundation for defining fiber products of spectral triples in noncommutative geometry?
• RQ5 What are the K-theoretic and index-theoretic properties of the resulting quantum projective space, particularly regarding the existence of non-trivial $U(1)$-principal bundles?

Key findings

• The C*-algebra $C(\mathbb{P}^N(\mathcal{T}))$ is constructed as a multipullback of $N+1$ copies of $\mathcal{T}^{\otimes N}$, forming a noncommutative deformation of $\mathbb{P}^N(\mathbb{C})$.
• The lattice of ideals $\{\ker \pi_i\}_{i=0}^N$ generates a free distributive lattice, as proven via Birkhoff’s Representation Theorem.
• This freeness confirms that the quantum deformation preserves the combinatorial structure of the affine covering of $\mathbb{P}^N(\mathbb{C})$.
• For $N \geq 2$, the C*-algebra $C(\mathbb{P}^N(\mathcal{T}))$ is not isomorphic to the quantum group deformation $C(\mathbb{C}P_q^N)$, as the latter has only one character while the former contains the $N$-torus.
• The space of characters of $C(\mathbb{P}^N(\mathcal{T}))$ is homeomorphic to the $N$-torus, indicating richer topological structure than the quantum group version.
• The construction suggests the existence of non-crossed product $U(1)$-C*-algebras, with evidence already established for $N=1$ via index theory on the mirror quantum sphere.