Fej\'er representations for discrete quantum groups and applications

Jason Crann; Matthias Neufang; Soroush Kazemi

REVIEW 1 major objections 4 minor 38 references

Fej\'er representations for discrete quantum groups and applications

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Pith's one-line read A discrete quantum group has the approximation property exactly when its crossed products admit Fejér summability.

desk verdict A genuinely new AP/Fejér equivalence for discrete quantum groups with solid proofs and useful applications; the flagged concerns are mostly presentation issues, not real gaps. read the letter →

arxiv 2502.05125 v1 pith:I6ZQYLIQ submitted 2025-02-07 math.OA math.FA

Jason Crann , Soroush Kazemi , Matthias Neufang This is my paper · ORCID

classification math.OAmath.FA MSC 46L6746L5546L0746L8943A55

keywords discretequantumgroupscrossedproductsFejérrepresentationapproximationpropertycompletelyboundedmultipliersbimodulesFubiniproductslicemap

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that a discrete quantum group has the approximation property exactly when every element of any of its crossed products—von Neumann or C*-algebraic—can be recovered as the weak* limit (norm limit in the C*-case) of an explicit Cesàro-style Fejér average built from the dual group. The average is a finite sum over irreducible co-representations of the dual, with coefficients obtained by applying the canonical conditional expectation to the element twisted by the corresponding matrix coefficients. The equivalence is proved as Theorems 3.2 and 3.7, and it converts a stability condition about completely bounded multipliers into a concrete summability statement for Fourier series in crossed products. The paper then uses this representation to characterize the invariant $L^\infty(\hat{\mathbb G})$-bimodules of $\mathcal B(\ell^2(\mathbb G))$ and $C(\hat{\mathbb G})$-bimodules of $\mathcal K(\ell^2(\mathbb G))$, and to show that every action of such a quantum group has the slice map property. If the paper is right, the approximation property is exactly the summability condition that makes quantum crossed products behave like commutative ones.

What carries the argument

The load-bearing object is the operator-valued weight $E=E_{\hat\alpha}$ induced by the dual action $\hat\alpha$ of $\hat{\mathbb G}^{\mathrm{op}}$ on the crossed product, together with the completely bounded slice maps $\Theta^{\ell op}(f)$ implemented by the fundamental unitary $\widetilde W$ of $\hat{\mathbb G}^{\mathrm{op}}$. Lemma 3.1 shows that for $f\in\mathrm{Pol}(\hat{\mathbb G})\cdot\hat\phi$, the slice $(\Theta^{\ell op}(f)\otimes\mathrm{id})(T)$ can be rewritten as the displayed finite Fejér sum; the coefficients $d_\beta/\lambda^\beta_i\,\langle f,\hat u^\beta_{ji}\rangle$ come exactly from the Peter–Weyl orthogonality relations, so they match the Fourier coefficients of $f$ with respect to the dual co-representations. The approximation property supplies a net $f_\iota\to 1$ in the stable point-weak* topology of $M^\ell_{cb}(L^1(\hat{\mathbb G}))$, and the proof reduces to whether this stable convergence survives tensoring with an arbitrary von Neumann algebra $N$.

What would settle it

Find one discrete quantum group $\mathbb G$ without the approximation property and one action $\alpha$ on a von Neumann algebra $N$ for which the net in (18) fails to converge weak* to some $T\in\mathbb{G}\ltimes N$; equivalently, exhibit a Hilbert space $H$ and a von Neumann algebra $N$ such that the stable-convergence replacement used on page 10 (citing [21, Proposition 1.7]) breaks for a non-classical quantum group, because then the 'only if' direction of Theorem 3.2 collapses.

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Extended reading notes

Core claim

The paper's central claim is that a discrete quantum group $\mathbb G$ has the approximation property if and only if, for every action $\alpha$ on a von Neumann algebra $N$ and every $T$ in the crossed product $\mathbb G\ltimes N$, $$T=\mathrm{w}^*-\lim_\iota\sum_{\$\beta$\in F_\iota}\sum_{i,j,k=1}^{n_\$\beta$}\frac{d_\$\beta$}{\$\lambda$^\beta_i}\langle f_\iota,\hat u^\beta_{ji}\rangle E\bigl(T((\hat u^\beta_{ki})^*\otimes 1)\bigr)(\hat u^\beta_{kj}\otimes 1),$$ where $(f_\iota)$ is a net in $\mathrm{Pol}(\hat{\mathbb G})\cdot\hat\phi$, the $\hat u^\beta_{ij}$ are matrix coefficients of irreducible co-representations of $\hat{\mathbb G}$, $d_\beta$ is the quantum dimension, $\lambda^\beta_i$ are the weights from the Peter–Weyl orthogonality relations, and $E$ is the operator-valued weight induced by the dual action (a conditional expectation in the C*-case). The same statement holds in norm topology for C*-crossed products. Thus the AP is not merely a technical property of the dual Banach algebra of completely bounded multipliers; it is precisely the summability condition under which the Cesàro–Fourier series of every crossed-product element converges back to that element.

Load-bearing premise

The key step assumes that stable point-weak* convergence of the net $\Theta^{\ell op}(f_\iota)$ — known when tensored with $\mathcal B(H)$ for Hilbert spaces $H$ — remains valid when $\mathcal B(H)$ is replaced by an arbitrary von Neumann algebra $N$; the paper cites a classical-group result for this replacement and supplies no quantum-group proof.

Editorial extensions

If this is right

For every discrete quantum group with the AP, each element of any von Neumann crossed product is a weak* limit of the displayed Fejér averages, and each element of any C*-crossed product is a norm limit of them (Theorems 3.2 and 3.7).
The AP is necessary for the representation: if a discrete quantum group lacks the AP, some crossed product element cannot be recovered by these Cesàro averages.
For an AP discrete quantum group, every weak*-closed $L^\infty(\hat{\mathbb G})$-bimodule in $\mathcal B(\ell^2(\mathbb G))$ that is invariant under completely bounded left multipliers is of the form $\mathrm{Bim}(J^\perp)=\mathrm{Ran}(J)^\perp$ for a closed left ideal $J$ of $\ell^1(\mathbb G)$, with the norm-closed analogue for $C(\hat{\mathbb G})$-bimodules in $\mathcal K(\ell^2(\mathbb G))$.
Every action of an AP discrete quantum group on a von Neumann algebra or C*-algebra has the slice map property: the Fubini crossed product of any invariant subspace coincides with its ordinary crossed-product subspace.
Compact groups, which always have the AP, obtain the new bimodule characterization in $\mathcal K(L^2(G))$ as a special case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the unproved replacement of $\mathcal B(H)$ by an arbitrary von Neumann algebra fails for a non-classical quantum group, the 'if' direction of Theorem 3.2 survives but the 'only if' may not; this hinge can be probed directly by testing the displayed net with $\mathbb G$ acting on $N=L^\infty(\mathbb G)$ by the co-multiplication, where the crossed product is concrete.
The closed-form coefficients suggest a truncation scheme: on finite-dimensional truncations of the dual (e.g., root-of-unity deformations), the Fejér average becomes a finite matrix expression, giving a direct numerical route to approximate crossed-product elements.
The bimodule characterization is a natural stepping stone to a Galois correspondence for intermediate subalgebras of discrete quantum group crossed products, which the authors explicitly postpone; in the classical setting the same representation supplies exactly that correspondence.
If the slice map property holds for all invariant subspaces, it should imply rigidity of crossed products under equivariant embeddings; conversely, known classical examples where slice maps fail should correspond to discrete quantum groups without the AP.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the main AP/Fejér equivalence for discrete quantum groups is real, and the applications are substantial. The paper deserves a serious referee; the issues I found are minor and fixable.

What's new: The equivalence between AP and Fejér representability in crossed products—Theorem 3.2 and 3.7—is genuinely new for quantum groups and doesn't reduce to the classical Crann–Neufang result. The proof via Peter–Weyl coefficients in Lemma 3.1 is clean and detailed. The bimodule applications, especially Corollary 4.7, extend known results and give something new even for discrete groups. The slice map property for AP actions is a natural and useful extension.

Soft spots: - The step in Theorem 3.2 replacing B(H) by an arbitrary von Neumann algebra N cites Haagerup–Kraus Prop 1.7 without stating it. The stress-test worried this is not justified for quantum groups. I don't think the concern lands: HK Prop 1.7 is a general statement about normal completely bounded maps on von Neumann algebras, not specific to groups. Once AP gives stable point-weak* convergence on L∞(Ĝop), the stable convergence passes to any N by that proposition. The paper should state the proposition and verify the maps are normal c.b., but that's a presentation issue, not a gap. - Proposition 4.8 states a Ditkin assumption for parts (ii) and (iii), but the proof never uses it. Either the assumption is extraneous or the proof is missing a step. This is worth asking about, but it's peripheral to the main theorem. - The pre-annihilator step in Prop 4.8(iii) (identifying (I1∩I2)^⊥ with I1^⊥+I2^⊥) is terse and needs a bit more justification, though it looks like a standard fact if the spaces are closed.

Overall: the central equivalence is well argued and the applications are meaningful. The paper is honest: no sign of circularity, no fitted parameters. It's a serious piece of work.

Bottom line: send it to a capable referee. I'd engage with it—cite it if I work in this area. For a reading group, it's a good choice for people interested in quantum groups and approximation properties.

Referee Report

1 major / 4 minor

Summary. The paper proves an equivalence, for a discrete quantum group G, between the approximation property (AP) and the existence of Fejér-type representations for elements of arbitrary C*-algebraic or von Neumann algebraic crossed products by G (Theorems 3.2 and 3.7). The proof is built on Lemma 3.1, which expresses the action of the multiplier Θ̂ℓop(f) on crossed-product elements as an explicit finite sum involving the conditional expectation E and matrix coefficients of the dual compact quantum group. The authors then apply this Fejér theorem to obtain structural results for invariant L∞(Ĝ)-bimodules in B(ℓ2(G)) and C(Ĝ)-bimodules in K(ℓ2(G)), to characterize jointly invariant subspaces in terms of closed left ideals of ℓ1(G), and to prove a slice-map property for actions of discrete quantum groups with the AP.

Significance. If the main equivalence holds, it is a natural and substantial quantum-group analogue of the classical Fejér theorem for crossed products, and it identifies the AP precisely as the summability condition that recovers every crossed-product element from its Fourier coefficients. The paper is well organised, gives detailed proofs of the central statements, and contains several genuinely useful applications: the bimodule characterizations in Theorems 4.4, 4.5, 4.25 and 4.26, the harmonic-operator corollaries, and the Fubini crossed-product slice-map property. The main definitions are standard, and the formulas are explicit enough to be checked. The principal weakness is a load-bearing imported step in the proof of Theorem 3.2, which is discussed below; it is repairable but should be addressed before publication.

major comments (1)

[§3, proof of Theorem 3.2 (p. 10)] The forward direction uses the assertion, attributed to [21, Proposition 1.7], that stable point-weak* convergence of Θ̂ℓop(fι) on B(H) for Hilbert spaces H implies weak* convergence of Θ̂ℓop(fι)⊗idN on L∞(Ĝop)⊗N for every von Neumann algebra N. This assertion is the only step that upgrades the AP from its Hilbert-space formulation to arbitrary crossed products, and it is exactly what yields the Fejér representation (18) for arbitrary N. The manuscript neither states the proposition nor verifies its hypotheses (normality and complete boundedness of the maps, and a uniform cb-bound for the net). Please add this verification, or replace the appeal by the direct argument: represent N faithfully on a Hilbert space K, apply the stable convergence with H = L2(G)⊗K, and restrict to the subalgebra L∞(Ĝop)⊗(B(L2(G))⊗N).

minor comments (4)

[§4.1, Proposition 4.8(ii)-(iii)] The stated hypothesis that Ĝ has Ditkin's property is never used in the proof: (ii) follows from Theorem 4.4 and equation (27), and (iii) follows from (i) and Theorem 4.4. Moreover, Corollary 4.9 applies (ii) without the Ditkin assumption. Please either prove the stronger statements or explain the role of Ditkin.
[§3, AP definition] The definition of the AP says 'for any (separable) Hilbert space H' with the parentheses; this ambiguity matters for the passage to arbitrary von Neumann algebras. Please specify whether stable convergence is required for all Hilbert spaces or only separable ones, and ensure consistency with the definition used in [17].
[Lemma 3.3, displayed computation] In the displayed computation near equation (23), the sums written as 'β∑' and 'γ∑' should presumably be 'nβ∑' and 'nγ∑'; the notation is otherwise confusing.
[§3, proof of Theorem 3.7] The sentence 'Repeating the proof of Lemma 3.1' is not literally accurate, because Lemma 3.1 is stated for von Neumann actions, whereas Theorem 3.7 concerns C*-actions on reduced crossed products. Please add a sentence explaining that the C*-case follows by applying the lemma to the double-dual action α~ on A** and restricting to Gα⋉rA.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fejér representation is derived from the AP definition and prior independent structural theorems, not from the conclusion itself.

full rationale

The paper's central equivalence (Theorems 3.2 and 3.7) is not circular. Lemma 3.1 proves, by explicit Peter–Weyl orthogonality computations, the identity (17): for f in Pol(Ĝ)·φ, the displayed Fejér-type sum equals (Θ^ℓop(f)⊗id)(T). The approximation property is defined independently, as stable point-weak* convergence of Θ^ℓ(f_ι) to the identity on B(H) for Hilbert spaces H. The forward direction of Theorem 3.2 combines this definition with Lemma 3.1; the only step going beyond the definition is the replacement of B(H) by an arbitrary von Neumann algebra N, imported from Haagerup–Kraus [21, Proposition 1.7]. That is an external, previously proved result rather than a self-citation or a fitted input; whether it applies verbatim to quantum groups is a correctness/gap concern, not a circularity. The converse direction is also non-circular: it instantiates the assumed representation for the trivial action on B(H), where the representation reduces, by the same identity (17), to stable point-weak* convergence, which is exactly the AP. The applications—bimodule characterizations, harmonic operators, Fubini crossed products, and the slice map property—use the Fejér theorem rather than assuming it. No parameter is fitted, and no conclusion is identical to its hypothesis by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants or new postulates. Its results rest on the established Kustermans-Vaes framework, the AP characterization of Daws-Krajczok-Voigt, the representation theory for completely bounded multipliers of Junge-Neufang-Ruan, and standard spatial tensor product and slice map facts. These are prior results, not the target result, so the central equivalence is a genuine theorem rather than a repackaging.

assumptions (6)

standard math Kustermans-Vaes locally compact quantum group axioms and Woronowicz Peter-Weyl orthogonality relations for compact quantum groups.
Used throughout Section 2 and in Lemma 3.1 to expand elements of Pol(Ĝ) and compute the Fejér formula; the paper cites [29, 37, 38].
standard math Daws-Krajczok-Voigt characterization: G has the AP iff a net in L1(Ĝ) converges weak* to 1 in M^l_cb(L1(Ĝ)), and AP is equivalent for G and G'.
Invoked in Section 3 to move between AP, stable point-weak* convergence of Θ^ℓop, and the existence of multiplier nets; if this equivalence failed, Theorems 3.2 and 3.7 would not follow.
standard math Junge-Neufang-Ruan representation theorem: for discrete quantum groups, Θ^ℓ and Θ^r are completely isometric isomorphisms onto the indicated CBσ bimodule maps (Theorem 2.1 and Remark 2.2), and the identity T·f = (Θ^ℓop(f)⊗id)(T) for the dual action (equation (13)).
Central computational tool in Lemmas 3.1 and 3.3 and in the converse directions of Theorems 3.2 and 3.7.
domain assumption The identification B(ℓ²(G)) with the crossed product Gα⋉ℓ∞(G) via the extended right co-multiplication Γr, and K(ℓ²(G)) with Gα⋉r c0(G) via [23, Corollary 3.6].
Used in Theorems 4.4 and 4.25 to apply the Fejér representation to bimodule characterizations; relies on discreteness of G and the density equation (5).
standard math B(ℓ²(G)) has Property S_σ, giving the slice map identity B(ℓ²(G))⊗X = B(ℓ²(G))⊗_F X for weak*-closed subspaces X.
Used in Proposition 4.32 to identify intersections of tensor products when proving the Fubini crossed product description.
standard math Stable point-weak* convergence of Θ^ℓop(f_ι) to the identity on L∞(Ĝop) can be tensored with an arbitrary von Neumann algebra N, not only with B(H).
Invoked in Theorem 3.2 to go from the AP definition, which uses B(H), to crossed products over arbitrary N; the paper cites Haagerup-Kraus [21, Proposition 1.7] for the group case, and the quantum-group version is not separately proved.

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Pith. "Pith review of Fej\'er representations for discrete quantum groups and applications." pith.science (2026). https://pith.science/paper/I6ZQYLIQ

@misc{pith2026250205125,
  author       = {Pith},
  title        = {Pith review of: Fej\'er representations for discrete quantum groups and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6ZQYLIQ}},
  note         = {Machine review of arXiv:2502.05125}
}

abstract

We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fej\'{e}r-type representation holds for its $C^*$-algebraic or von Neumann algebraic crossed products. As applications, we extend several results from the literature to the context of discrete quantum groups with the approximation property. Additionally, we provide new characterizations of invariant $L^\infty(\widehat{\mathbb{G}})$-bimodules of $\mathcal{B}(\ell^2(\mathbb{G}))$ and invariant $C(\widehat{\mathbb{G}})$-bimodules of $\mathcal{K}(\ell^2(\mathbb{G}))$, some of which are new in the group setting. Finally, we study Fubini crossed products of discrete quantum group actions.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Massoud Amini, Mehrdad Kalantar, Alireza Medghalchi, A hmad Mollakhalili, and Matthias Neu- fang, Compact elements and operators of quantum groups , Glasg. Math. J. 59 (2017), no. 2, 445–462. MR3628940

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Benjamin Anderson-Sackaney, On ideals of L1-algebras of compact quantum groups , Internat. J. Math. 33 (2022), no. 12, Paper No. 2250074, 38. MR4514300

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Anoussis, A

M. Anoussis, A. Katavolos, and I. G. Todorov, Ideals of A(G) and bimodules over maximal abelian selfadjoint algebras, J. Funct. Anal. 266 (2014), no. 11, 6473–6500. MR3192459

work page 2014

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, Ideals of the Fourier algebra, supports and harmonic operators , Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 2, 223–235. MR3530505

work page 2016

[5] [5]

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, Bimodules over VN(G), harmonic operators and the non-commutative Poisson boundary , Studia Math. 249 (2019), no. 2, 193–213. MR3990189

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Erik Bédos and Roberto Conti, Fourier series and twisted C∗ -crossed products, J. Fourier Anal. Appl. 21 (2015), no. 1, 32–75. MR3302101

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