[Paper Review] Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups

Pekka Salmi|arXiv (Cornell University)|Apr 23, 2010

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group 𝔾 and certain left-invariant C*-subalgebras of 𝒞₀(𝔾), generalizing classical results of Lau and Losert. It further proves that every compact quantum subgroup of a co-amenable quantum group is itself co-amenable, and dualizes the correspondence to show a similar link between open subgroups of amenable locally compact groups and invariant C*-subalgebras of their group C*-algebras.

ABSTRACT

We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $\mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(\mathbb{G})$. We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group $G$ and non-zero, invariant C*-subalgebras of the group C*-algebra $C^*(G)$.

Motivation & Objective

• To generalize the classical correspondence between compact subgroups of locally compact groups and left-invariant C*-subalgebras of 𝒞₀(G) to the setting of locally compact quantum groups.
• To establish a one-to-one correspondence between compact quantum subgroups of a co-amenable quantum group 𝔾 and specific left-invariant C*-subalgebras of 𝒞₀(𝔾).
• To prove that every compact quantum subgroup of a co-amenable quantum group is itself co-amenable, extending a classical group-theoretic fact.
• To dualize the correspondence in the co-commutative, co-amenable case, showing a one-to-one link between open subgroups of an amenable locally compact group G and non-zero, invariant C*-subalgebras of 𝒞*(G).

Proposed method

• Uses the reduced C*-algebraic framework of locally compact quantum groups as defined by Kustermans and Vaes, with co-multiplication Γ and left/right Haar weights.
• Defines left-invariant C*-subalgebras via the condition that right translations preserve elements, analogous to group-invariant functions.
• Applies the theory of multiplier algebras and strict topologies to handle unbounded multipliers and implement the duality between quantum groups and their duals.
• Utilizes the Fourier–Stieltjes algebra and representation-theoretic tools, including X-trivial representations, to analyze invariance and duality.
• Employs the Pontryagin duality framework in the co-commutative case, identifying 𝔾 = Ĝ and relating subgroups of G to subalgebras of 𝒞*(G).
• Leverages results from Takesaki–Tatsuuma duality and Enock’s work on von Neumann algebraic quantum groups to support the duality arguments.

Experimental results

Research questions

• RQ1 Is there a quantum analog of the classical Lau–Losert correspondence between compact subgroups and left-invariant C*-subalgebras in locally compact groups?
• RQ2 Can the co-amenability of a compact quantum subgroup be deduced from the co-amenability of the ambient quantum group?
• RQ3 How does the duality between quantum groups and their duals manifest in the C*-algebraic setting for open subgroups and invariant subalgebras?
• RQ4 What conditions ensure that a left-invariant C*-subalgebra of 𝒞₀(𝔾) arises from a compact quantum subgroup?
• RQ5 In the co-commutative, co-amenable case, is there a one-to-one correspondence between open subgroups of G and invariant C*-subalgebras of 𝒞*(G)?

Key findings

• There is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group 𝔾 and certain left-invariant C*-subalgebras of 𝒞₀(𝔾).
• Every compact quantum subgroup of a co-amenable quantum group is itself co-amenable, generalizing the classical fact that quotients of amenable groups by normal open subgroups are amenable.
• In the co-commutative, co-amenable case, there is a dual correspondence: open subgroups of an amenable locally compact group G correspond bijectively to non-zero, invariant C*-subalgebras of 𝒞*(G).
• The C*-algebra 𝒞*(H) of an open, normal subgroup H ⊆ G is symmetric if and only if H is normal, and this is equivalent to the annihilator F⊥ being a two-sided ideal in 𝒞*(G).
• The compact quantum subgroup induced by 𝒞*(H) is isomorphic to the dual of the quotient group G/H, i.e., (𝔾, π) ≅ ((G/H)̂, ρ), confirming the duality in the co-commutative case.
• The proof of the implication (1) ⇒ (2) in Theorem 18 is significantly shortened using the X-trivial representation approach, suggested by the referee, which also resolves a subtle technical gap in the original argument.