[Paper Review] Continuous perturbations of noncommutative Euclidean spaces and tori

Li Gao|arXiv (Cornell University)|May 22, 2016

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes continuous Moyal deformations of noncommutative Euclidean spaces and tori by constructing Lip^{1/2} continuous paths of unitary generators in the C*-algebra of bounded operators on a Hilbert space. Using a tensor product construction based on Haagerup and Rørdam's rotation algebra paths, it generalizes their result to higher-dimensional noncommutative tori and noncompact noncommutative Euclidean spaces, proving that the deformation is uniformly continuous with Hölder regularity of order 1/2.

ABSTRACT

We prove a noncompact version of Haagerup and Rørdam's result about continuous paths of the rotation $C^*$-algebras. It gives a continuous Moyal deformation of Euclidean plane. Moveover, the construction is generalized to noncommutative Euclidean spaces of dimension $d\ge 2$. As a corollary, we obtain Lip$^{\frac12}$ continuous maps for the generators of noncommutative $d$-tori.

Motivation & Objective

• To extend Haagerup and Rørdam's continuous path construction for rotation C*-algebras to higher-dimensional noncommutative tori and noncommutative Euclidean spaces.
• To establish a continuous Moyal deformation of the noncommutative plane and higher-dimensional noncommutative Euclidean spaces.
• To prove that the generators of noncommutative d-tori admit Lip^{1/2} continuous paths in the C*-algebra of bounded operators on a Hilbert space.
• To generalize the bounded perturbation technique from the Heisenberg commutation relation to construct continuous deformations in the noncompact setting.

Proposed method

• Constructs continuous unitary paths for noncommutative d-tori using tensor products of Haagerup-Rørdam paths in U(H) for d=2.
• Applies the tensor product construction to d=3 by defining u1(θ) = u(θ12)⊗u(θ13)⊗I, u2(θ) = v(θ12)⊗I⊗u(θ23), u3(θ) = I⊗v(θ13)⊗v(θ23), ensuring correct commutation relations.
• Uses the triangle inequality and Hölder continuity of the original paths to derive Lip^{1/2} continuity for the higher-dimensional generators.
• Adapts the bounded perturbation method from the Heisenberg relation to construct continuous Moyal deformations of noncommutative Euclidean spaces of dimension d ≥ 2.
• Employs Voiculescu’s noncommutative Weyl-von Neumann theorem to establish approximate unitary equivalence of representations, enabling the triangle inequality for the metric ρ.
• Leverages the isomorphism O2⊗O2 ≅ O2 to extend continuous embeddings into the Cuntz algebra O2.

Experimental results

Research questions

• RQ1 Can continuous paths of unitary generators be constructed for noncommutative d-tori with d > 2, preserving the commutation relations and exhibiting Hölder continuity of order 1/2?
• RQ2 Does a continuous Moyal deformation of noncommutative Euclidean space exist for dimension d ≥ 2, analogous to the d=2 case?
• RQ3 Can the bounded perturbation technique used in the compact case (rotation algebras) be adapted to the noncompact setting of noncommutative Euclidean spaces?
• RQ4 Is there a uniform Lip^{1/2} continuity bound for the generator paths in terms of the skew-symmetric matrix parameter θ ∈ R^{d(d−1)/2}?
• RQ5 Can the continuous embedding of noncommutative d-tori into B(H) or O2 be established with explicit Hölder regularity of the path maps?

Key findings

• The paper constructs d continuous maps u1,…,ud from the parameter space A[d] ≡ [0,1]^{d(d−1)/2} to U(H) such that the generators satisfy the standard commutation relations for noncommutative d-tori.
• The constructed paths satisfy the Lip^{1/2} continuity estimate: ||uj(θ) − uj(θ')|| ≤ C(∑k |θjk − θ'jk|^{1/2}) for all j and θ,θ' ∈ A[d].
• The construction generalizes to noncommutative Euclidean spaces of dimension d ≥ 2 via a similar tensor product method applied to the Heisenberg commutation relations.
• A continuous Moyal deformation of the noncommutative plane is established, extending the Haagerup-Rørdam result to the noncompact setting.
• The metric ρ on the space of skew-symmetric matrices satisfies the triangle inequality, enabling the construction of a well-defined continuous field of C*-algebras.
• The results imply a continuous embedding of noncommutative d-tori into the Cuntz algebra O2 with Lip^{1/2} regularity, due to the isomorphism O2⊗O2 ≅ O2.