Operator algebra quantum homogeneous spaces of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), as well as their homogeneous spaces, in the sigma-C*-algebra setting. More precisely, we propose concise definitions of sigma-C*-quantum groups and sigma-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute it for the quantum homogeneous spaces associated to the universal gauge group SU(\infty).

💡 Research Summary

This paper develops a systematic operator‑algebraic framework for quantizing universal gauge groups of infinite rank, such as the unitary group SU(∞), together with their homogeneous spaces. The authors recognize that traditional C*‑quantum group theory is essentially limited to finite‑dimensional compact groups, and therefore introduce σ‑C*‑algebras as a natural enlargement. A σ‑C*‑algebra is a complete ‑algebra that can be written as a countable inductive limit of C‑algebras (or, equivalently, a countable direct sum of C*‑algebras equipped with a suitable topology). This class is closed under the completed tensor product ⊗̂, which allows one to define coproducts and counits in the same way as for ordinary compact quantum groups.

Using this setting, the authors define a σ‑C‑quantum group* (A, Δ, ε) as a σ‑C*‑algebra A together with a coassociative coproduct Δ: A → A⊗̂A and a counit ε: A → ℂ satisfying the usual axioms. The definition mirrors Woronowicz’s C*‑quantum groups but works for algebras that are not themselves C*‑algebras, thus accommodating the function algebra of SU(∞). They then introduce σ‑C‑quantum homogeneous spaces*: given a σ‑C*‑quantum group (A, Δ, ε) and a σ‑C*‑subalgebra B equipped with a coaction ρ: B → B⊗̂A that is free and has trivial fixed‑point algebra (i.e., B^ρ = ℂ), the pair (B, ρ) is called a σ‑C*‑quantum homogeneous space. In the classical picture this corresponds to the quotient G/H, and in the quantum picture B plays the role of the “function algebra” on the non‑commutative quotient.

A substantial part of the work is devoted to the representable K‑theory (RK‑theory) of σ‑C*‑algebras. Because σ‑C*‑algebras are built from countable direct sums, ordinary K‑theory does not behave well; instead, RK‑theory is defined via stable homotopy classes of ‑homomorphisms into the compact operators on a separable Hilbert space, and it respects inductive limits and countable direct sums. The authors prove that RK‑theory satisfies Bott periodicity, Mayer–Vietoris exact sequences, and a Künneth formula in the σ‑C context, thereby providing the necessary computational tools.

Armed with these tools, the paper computes the RK‑groups of quantum homogeneous spaces associated with SU(∞). Choosing a standard finite‑dimensional subgroup H ⊂ SU(∞) (for example, an embedded SU(k)), the quotient space H\SU(∞) is modeled by a σ‑C*‑algebra B = Cσ(H\SU(∞)). By expressing B as an inductive limit of C*‑algebras corresponding to finite‑dimensional quotients, the authors apply the developed exact sequences and Bott periodicity to obtain \