[Paper Review] Endomorphisms of O_n which preserve the canonical UHF-subalgebra

Roberto Conti, Mikael Rørdam+1 more|arXiv (Cornell University)|Oct 7, 2009

Advanced Operator Algebra Research3 references6 citations

TL;DR

This paper constructs unital endomorphisms of the Cuntz algebra 𝒪ₙ that preserve the canonical UHF-subalgebra 𝒻ₙ, demonstrating that such endomorphisms can arise from unitaries not in 𝒻ₙ—contradicting a previously assumed converse. The key contribution is a counterexample in 𝒪₂ where a unitary outside 𝒻₂ induces an endomorphism preserving 𝒻₂, and the image's relative commutant contains a copy of 𝒪₂, enabling an embedding of 𝒪₂⊗𝒪₂ into 𝒪₂ preserving the UHF structure.

ABSTRACT

Unital endomorphisms of the Cuntz algebra O_n which preserve the canonical UHF-subalgebra F_n of O_n are investigated. We give examples of such endomorphisms for which the associated unitary element in O_n does not belong to F_n. One such example, in the case where n=2, arises from a construction of a unital endomorphism on O_2 which preserves the canonical UHF-subalgebra and where the relative commutant of the image in O_2 contains a copy of O_2.

Motivation & Objective

• To investigate unital endomorphisms of the Cuntz algebra 𝒪ₙ that preserve the canonical UHF-subalgebra 𝒻ₙ.
• To determine whether every such endomorphism arises from a unitary in 𝒻ₙ, challenging a previously assumed converse.
• To construct explicit examples of endomorphisms preserving 𝒻ₙ that are induced by unitaries not in 𝒻ₙ.
• To explore the structure of the relative commutant of the image of such endomorphisms, particularly in 𝒪₂.
• To establish a systematic framework for generating and analyzing endomorphisms preserving the UHF-subalgebra.

Proposed method

• Utilizes the one-to-one correspondence between unitaries in 𝒪ₙ and unital endomorphisms λᵤ defined by λᵤ(Sⱼ) = uSⱼ for generators Sⱼ.
• Constructs a specific unitary u in 𝒪₂ not in 𝒻₂ such that λᵤ preserves 𝒻₂, using combinatorial properties of multi-indices and word structures.
• Analyzes the cocycle structure associated with the endomorphism via the gauge action, particularly the unitary cocycle zₜ⁽¹⁾ and its higher-order analogues zₜ⁽ᵏ⁾.
• Applies the condition that λᵤ(𝒻ₙ) ⊆ 𝒻ₙ if and only if the cocycle zₜ⁽¹⁾ is unitary and the algebra Aᵤ generated by cocycles is finite-dimensional.
• Employs the Higman-Thompson group structure of the group ℒₙ of finite sums of words in 𝒪ₙ to generate candidate unitaries.
• Demonstrates that if the relative commutant of λᵤ(𝒪₂) contains a copy of 𝒪₂, then a unitary v ∉ 𝒻₂ can be found such that λᵥ agrees with λᵤ on 𝒻₂.

Experimental results

Research questions

• RQ1 Does every unital endomorphism of 𝒪ₙ that preserves the canonical UHF-subalgebra 𝒻ₙ necessarily arise from a unitary in 𝒻ₙ?
• RQ2 Can a unitary u ∈ 𝒪ₙ ∖ 𝒻ₙ induce an endomorphism λᵤ that still preserves 𝒻ₙ?
• RQ3 What structural conditions on the unitary u ensure that λᵤ(𝒻ₙ) ⊆ 𝒻ₙ?
• RQ4 Under what conditions is the relative commutant of λᵤ(𝒪ₙ) in 𝒪ₙ large enough to contain a copy of 𝒪ₙ?
• RQ5 Is the algebra Aᵤ generated by the cocycles zₜ⁽ᵏ⁾ finite-dimensional if and only if λᵤ preserves 𝒻ₙ?

Key findings

• A unital endomorphism λᵤ of 𝒪₂ preserving the canonical UHF-subalgebra 𝒻₂ exists even when the inducing unitary u is not in 𝒻₂, providing a counterexample to a previously assumed converse.
• The relative commutant of the image of such an endomorphism λᵤ in 𝒪₂ contains a copy of 𝒪₂, indicating a rich internal structure.
• An explicit unitary u ∈ 𝒪₂ ∖ 𝒻₂ is constructed such that λᵤ(𝒻₂) ⊆ 𝒻₂, using combinatorial data from multi-indices of length 1 and 2.
• The existence of such an endomorphism implies the existence of a unital endomorphism λᵥ with v ∉ 𝒻₂ that agrees with λᵤ on 𝒻₂, thus preserving the UHF-subalgebra.
• An embedding of 𝒪₂⊗𝒪₂ into 𝒪₂ is constructed that maps 𝒻₂⊗𝒻₂ into 𝒻₂, demonstrating a non-trivial tensor product structure within 𝒪₂.
• The condition λᵤ(𝒻ₙ) ⊆ 𝒻ₙ holds if and only if the unitary cocycle zₜ⁽¹⁾ is unitary and the algebra Aᵤ generated by all cocycles zₜ⁽ᵏ⁾ is finite-dimensional.