Locally Compact Quantum Groups

• Locally compact quantum groups are noncommutative extensions of locally compact groups, defined via operator algebras equipped with coassociative comultiplications and Haar weights.
• The framework integrates duality, harmonic analysis, and crossed product constructions, enabling applications in noncommutative geometry and quantum physics.
• Their operator-algebraic formulation provides practical insights into amenability, quantum injectivity, and approximation properties crucial for advanced analysis.

A locally compact quantum group (LCQG) is a noncommutative generalization of locally compact groups equipped with a compatible quantum group structure on a $C^*$- or von Neumann algebra, unifying and extending both classical group theory and quantum symmetry. LCQGs support a robust analytic and operator-algebraic framework, encompassing duality, homogeneous spaces, harmonic analysis, crossed products, and approximation properties. This framework serves as the foundation for the study of quantum symmetries in functional analysis, noncommutative geometry, and mathematical physics.

1. Foundational Structure of Locally Compact Quantum Groups

A locally compact quantum group, in the sense of Kustermans–Vaes, is a quadruple $(L^\infty(G),\Delta,\varphi,\psi)$, where $L^\infty(G)$ is a von Neumann algebra, $$\Delta : L^\infty(G) \to L^\infty(G)\bar\otimes L^\infty(G)$$ is a normal unital $*$-homomorphism (the comultiplication) satisfying coassociativity

$$(\Delta\otimes\mathrm{id})\circ\Delta = (\mathrm{id}\otimes\Delta)\circ\Delta,$$

and $\varphi$, $\psi$ are left and right Haar weights—normal, faithful, semi-finite weights satisfying

$$\varphi\left((\omega\otimes\mathrm{id})\Delta(x)\right) = \varphi(x)\omega(1),\quad \psi\left((\mathrm{id}\otimes\omega)\Delta(x)\right)=\psi(x)\omega(1),$$

for $x\in L^\infty(G)_+$ and $\omega \in L^1(G)_+$. The predual $L^1(G) = L^\infty(G)_*$ becomes a completely contractive Banach algebra under the convolution product $f * g = (f \otimes g)\circ\Delta$.

The multiplicative unitary $W \in B(L^2(G)\otimes L^2(G))$ implements the coproduct via $\Delta(x) = W^*(1\otimes x)W$ and encodes the dual structure $\widehat{G}$, with $L^\infty(\widehat{G}) \subset B(L^2(G))$; this provides a Pontryagin-style self-duality extending the group case (Kalantar et al., 2011, Brannan, 2016, Hall et al., 2023).

2. Poisson Boundaries and Harmonic Analysis

For $\mu \in M(G)$ (the quantum probability measures), the associated Markov operator is

$$P_\mu(x) = (\mu \otimes \mathrm{id})\Delta(x), \quad x \in L^\infty(G).$$

An element $x$ is $\mu$-harmonic if $P_\mu(x) = x$; the space of such elements is denoted $\mathcal{H}_\mu$. Ultrafilter-limits of Cesàro averages

$$E_\mu(x) = \lim_{n\to\mathcal{U}} \frac{1}{n}\sum_{k=1}^n P_\mu^k(x)$$

provide a conditional expectation defining the Choi–Effros product on $\mathcal{H}_\mu$:

$x\circ y = E_\mu(xy).$

The von Neumann algebra $(\mathcal{H}_\mu,\circ)$ is termed the Poisson boundary of $(G,\mu)$ (Kalantar et al., 2011).

For compact quantum groups, the noncommutative Choquet–Deny theorem asserts that if $G$ is compact and $\mu$ is nondegenerate, then $\mathcal{H}_\mu = \mathbb{C}1$. The Kaimanovich–Vershik–Rosenblatt theorem generalizes: if $L^1(G)$ is separable, $G$ is amenable if and only if there exists a normal state $\mu$ such that $\mathcal{H}_\mu = \mathbb{C}1$. These results realize deep analogies between quantum and classical boundary theory (Kalantar et al., 2011).

3. Crossed Products and Dynamical Systems

Landstad–Vaes theory extends crossed-product constructions to the quantum case. For an LCQG $(A,\Delta)$, a G-product is $(B,\beta,\eta)$, where $B$ is a $C^*$-algebra, $\beta$ a continuous action of the dual, and $\eta : A\to B$ implements covariance via commutative diagrams with the dual structure. For regular LCQGs, Landstad–Vaes theory classifies weak $G$-dynamical systems, allowing the definition of a crossed product $D\rtimes G$ for weak (i.e., not necessarily continuous in the Podleś sense) actions and characterizes, up to isomorphism, the unique Landstad algebra inside the crossed product (Roy et al., 2016).

Key applications include:

• Kasprzak–Rieffel deformation via unitary 2-cocycles,
• Crossed products for quantum E$_q(2)$ (non-regular LCQG),
• Weakly-implemented actions via unitary representation.

This theoretical framework adapts the classical crossed-product duality and analysis to complex quantum symmetries.

4. Amenability, Quantum Injectivity, and Approximation Properties

Amenability for an LCQG is defined by the existence of a left-invariant mean $m \in L^\infty(G)^*$, satisfying $$m((\omega\otimes\mathrm{id})\Delta(x)) = \omega(1)m(x)$$. Amenability is equivalent to quantum injectivity: there exists a conditional expectation $E : B(L^2(G)) \to L^\infty(\widehat{G})$ mapping $L^\infty(G)$ into the center of $L^\infty(\widehat{G})$, and vice versa. Thus, amenability and quantum injectivity are dual, operator-algebraic properties (Sołtan et al., 2012).

Further, the Haagerup property and weak amenability generalize to LCQGs and are characterized in terms of mixing unitary representations, compact approximants in the Fourier algebra, and categorical/central versions connected with the representation category Rep$(G)$ (Daws et al., 2013, Brannan, 2016, Daws et al., 2023). Approximation properties are preserved under free products of discrete quantum groups, passage to closed quantum subgroups, and Drinfeld double constructions.

5. Duality, Quantum Subgroups, and the Bohr Compactification

Quantum duality appears at multiple levels: the operator algebras $L^\infty(G)$ and $L^\infty(\widehat{G})$, their associated $C^*$-algebras, and the multiplicative unitary. Morphisms (in the reduced picture) correspond to bicharacters $U \in M(C_0(G)\otimes C_0(H))$ satisfying compatibility with coproducts (Daws, 2013, Rivet et al., 2021).

Closed quantum subgroups are characterized via essential $*$-homomorphisms intertwining coproducts. Bornological quantum groups provide a dense algebraic model, generalizing algebraic quantum groups and facilitating the study of closed quantum subgroups in both analytic and algebraic settings (Rivet et al., 2021).

The quantum Bohr compactification attaches to any LCQG a compact quantum group capturing (almost) periodic elements, constructed as a compactification functor either at the universal or reduced level; the underlying Hopf $*$-algebra of admissible corepresentations is canonically the same (Daws, 2013). The compactification interpolates between classical, reduced, and universal completions, evidencing a rich spectrum even for cocommutative examples.

6. Quotients, Lattices, Extensions, and Structural Results

Quantum homogeneous spaces and quotients are encoded as fixed-point von Neumann subalgebras for actions of LCQGs. Invariant weights and states on these quotients relate directly to amenability and property (T) permanence. For a normal closed quantum subgroup $H\leq G$, the quotient $G/H$ admits a quantum group structure if and only if certain modular and scaling conditions are met. Lattices in LCQGs are discrete closed quantum subgroups with finite covolume, enabling the transference of rigidity properties such as property (T) from the lattice to the ambient quantum group (Brannan et al., 2019).

Quantum groups with projection generalize semidirect products, but are rarely extensions in the sense of Vainerman–Vaes unless coideals are normal. Braided quantum groups and coideals encode quotient structures beyond the group case, and normality under scaling and unitary antipode is fundamental for extension theory (Kasprzak et al., 2014).

The construction of LCQGs from quantization of the affine group (e.g., by cocycle twisting via oscillatory functions over p-adic groups) provides tangible, nontrivial compact and noncompact examples, illustrating the operator-algebraic toolkit (Galois objects, dual cocycles) in construction and classification (Jondreville, 2018).

7. Further Directions and Open Problems

Foundational open questions include the fine structure of Poisson boundaries for noncompact quantum groups of compact type, connections with exactness and the Haagerup property, and the development of quantum groupoid analogues. Structural properties of LCQGs are constrained by their intrinsic group (quantum point-masses), as this invariant preserves compactness/discreteness and transmits properties such as amenability; moreover, sufficiently small intrinsic group and centralizer data imply that the quantum group is essentially Kac and nearly classical (Kalantar et al., 2011). Classification by cohomological properties of convolution algebras, module projectivity, and biprojectivity seamlessly mediate between analytic approximation and topological quantum symmetry (Kalantar et al., 2011).

Research continues into the precise interplay between approximation properties, representation-theoretic rigidity, and nontrivial quantum group extensions. The framework of LCQGs thus remains central in the ongoing synthesis of functional-analytic, operator-algebraic, and quantum group-theoretic phenomena.