{"id":"6f2c2494-f086-4b20-a99f-f09fb031cbd7","arxiv_id":"2411.14624","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves surjectivity of restriction maps from Fourier and Fourier-Stieltjes algebras of locally compact groupoids to isotropy subgroups, and derives non-amenability conditions from them.","lead":"This paper studies what happens when you take functions on a groupoid and restrict them to a single isotropy group. It shows that under certain conditions this restriction map is onto, which lets the authors transfer results about group Fourier algebras to groupoids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.7's proof assumes Λ_{F,G}∈A(G) for bounded continuous sections F,G extended from a single fiber, but Definition 2.12 only gives A(G) as a closure of compactly supported coefficients; Remark 2.15 records that the equality is open.","rationale":"The single most load-bearing defect is exactly the one identified by the reader: the proof of Theorem 3.7, through Claim 3.8, relies on the membership of arbitrary left-regular coefficients Λ_{F,G} in A(G). Definition 2.12 defines A(G) as the closure of an algebra generated only by compactly supported functions on G, and the cited extension result [18, Proposition 10.1.10] does not provide such compact support. This is not a disagreement with an external consensus; it is an internal gap, and Remark 2.15 openly flags the relevant equality as unknown. The gap is serious because the surjectivity construction for A(G) is built entirely on these preimages. The concern does not by itself show Theorem 3.7 is false: in the model group bundle G=⊔_{x∈X}Z, one can approximate non-compactly-supported ℓ^2 coefficients by compactly supported sections with controlled norms, suggesting a repair may exist. For that reason, the reader's CONDITIONAL verdict is appropriate and I would not adjust it.","tokens_in":23844,"tokens_out":21927,"duration_ms":231742,"concrete_test":"Compute A(G) from Definition 2.12 for the étale groupoid G=⊔_{x∈X}Z, where X is a countably infinite discrete set and each fiber is a copy of Z, fixing u∈X. Take f,g∈ℓ^2(Z) with infinite support, and construct compactly supported sections F_N,G_N supported on the fiber over u that equal f·1_{[-N,N]} and g·1_{[-N,N]} on that fiber, with ‖F_N‖_Δ≤‖f·1_{[-N,N]}‖_2 and similarly for G_N. Then check whether Λ_{F_N,G_N} converges in the B(G)-norm to an element whose restriction to G^u_u is σ_{f,g}. If it converges, the unproved membership assertion in Claim 3.8 is repairable in this model; if not, Theorem 3.7 fails for this groupoid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Claim 3.8 in the proof of Theorem 3.7. For arbitrary f,g∈L^2(G^u), the authors use [18, Proposition 10.1.10] to obtain bounded continuous sections F,G of the left regular field with F^u=f and G^u=g, and then assert that R^u_{A(G)}(Λ_{F,G})=σ_{f,g}, concluding that σ_{f,g} belongs to the range of R^u. This inference is valid only if Λ_{F,G}∈A(G). By Definition 2.12, A(G) is the B(G)-closure of the algebra generated by coefficients Λ_{f,g} with f,g∈C_c(G); Proposition 10.1.10 produces bounded continuous sections, not compactly supported functions on G. Thus Λ_{F,G} need not lie in A(G). Remark 2.15 explicitly notes that it is unknown whether the full coefficient space of the left regular representation coincides with the closure of compactly supported coefficients, so the membership cannot be silently assumed. The same gap enters the subsequent absolute-summability argument: the inequality ∑‖Λ_{F_n,G_n}‖_{A(G)}≤∑‖f_n‖_2‖g_n‖_2 presupposes that every summand is in A(G). Without a proof that arbitrary bounded continuous regular coefficients are approximable in A(G)-norm by compactly supported regular coefficients, the preimage series is not known to be an element of A(G), and the surjectivity of Theorem 3.7 is left unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for locally compact Hausdorff groupoids G equipped with a Haar system, the restriction maps from the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) to the Fourier and Fourier-Stieltjes algebras of an isotropy subgroup G^u_u. The main results are: Proposition 3.4 (the restriction maps are continuous contractive algebra homomorphisms); Theorem 3.7 (surjectivity of A(G) → A(G^u_u) for étale G); Theorems 4.1 and Proposition 4.2 (surjectivity of B(G) → B(G^u_u) for transitive groupoids with a continuous section and for group bundles with discrete unit space); Theorem 4.6 and Corollary 4.7 (a necessary condition for surjectivity for HLS groupoids in terms of property FD, with SL_n(Z), n ≥ 3, giving non-surjective examples); Corollary 3.11 (hereditary Banach-algebra obstructions); and Proposition 5.1/Corollary 5.5 (box decompositions B(G) ≅ ℓ∞-⊕ B(G_t) and A(G) ≅ c_0-⊕ A(G_t)). The paper is clearly organized and systematically follows the known group-theoretic analogues. The chief concern, detailed below, is that the proof of the central Theorem 3.7 relies on a membership assertion for A(G) that is not established and is in fact false for a simple étale groupoid.","tokens_in":24163,"tokens_out":48667,"duration_ms":439817,"significance":"If Theorem 3.7 were established, it would be a valuable groupoid analogue of Herz's restriction theorem, and the hereditary-property criteria of Corollary 3.11 would give practical obstructions to amenability-type properties of A(G). The Fourier-Stieltjes portion of the paper is largely independent of this gap and appears sound: Theorems 4.1 and 4.2 are proved by explicit preimage constructions, and Theorem 4.6 yields falsifiable predictions (failure of surjectivity for HLS groupoids built from SL_n(Z), n ≥ 3) in the spirit of the Lubotzky-Shalom theory. The box-decomposition results of Section 5 are also natural and useful. However, the gap in Theorem 3.7 is load-bearing: the paper's own Remark 2.15 records that the identification of A(G) with the full coefficient space of the left regular representation is open, and the proof of Claim 3.8 silently assumes exactly that identification. As a consequence the main theorem and the statements that depend on it (Corollary 3.11, and the citation in Corollary 5.5) are not established as written.","major_comments":[{"comment":"The proof of Claim 3.8 assumes that the coefficient Λ_{F,G} of the left regular representation, with F and G the bounded continuous sections produced by [18, Proposition 10.1.10], belongs to A(G). This does not follow from Definition 2.12, where A(G) is the B(G)-closure of the algebra generated by coefficients Λ_{f,g} with f,g ∈ C_c(G); Proposition 10.1.10 provides no compact support and no approximability by compactly supported coefficients, and Remark 2.15 explicitly records that the equality of the closure-based and full coefficient-space definitions is unknown for general groupoids. The membership assertion is not merely unproved: it is false for a simple étale groupoid. Take G = ⊔_{n∈ℕ} ℤ, the group bundle over the discrete unit space ℕ (every point of G is open, so G is étale), and fix u ∈ ℕ. The function F(n,k) = δ_{k0} is a bounded continuous section of the left regular field through δ_0 ∈ L²(G^u), and one computes Λ_{F,F}(n,k) = δ_{k0} for every n. Under the isometric isomorphism Φ: B(G) → ℓ∞-⊕_{n}B(ℤ) of Proposition 5.1, Φ(Λ_{F,F}) = (δ_0, δ_0, ...), which is a constant nonzero sequence and so does not lie in c_0-⊕_{n}A(ℤ) ≅ A(G). Hence Λ_{F,F} ∉ A(G), and the preimage construction in Claim 3.8 does not produce elements of A(G). Consequently the summability estimate Σ‖Λ_{F_n,G_n}‖_{A(G)} ≤ Σ‖f_n‖₂‖g_n‖₂ is unjustified, and the surjectivity of R^u: A(G) → A(G^u_u) in Theorem 3.7, and everything that depends on it (Corollary 3.11), is left unproved.","section":"§3, Claim 3.8 (proof of Theorem 3.7)"},{"comment":"Section 1.1 claims that 'the result of Theorem 3.7 holds more generally for any locally compact groupoid admitting a Haar system that satisfies Condition (*)', and Remark 3.10 repeats this. The proof of Theorem 3.7 does not establish this generality. Claim 3.9 uses the étale condition in two ways that Condition (*) does not imply: the fibers G^u must be discrete so that L²(G^u) is the orthogonal direct sum ⊕_v L²(G^u_v) over the units, and the Haar system must consist of counting measures so that the operators U_v f(z) = f(zγ_v) are isometries. The unimodularity relaxation suggested in Remark 3.10 still leaves the direct-sum decomposition unaddressed for groupoids with non-discrete fibers. Either the missing argument for the general case should be supplied, or the introduction and Remark 3.10 should be revised to state the étale hypothesis (or a hypothesis making the fibers discrete and the Haar system counting) as the actual scope of the theorem.","section":"Introduction, Section 1.1; Remark 3.10"}],"minor_comments":[{"comment":"The equality ‖F‖_Δ = ‖f‖₂ (and likewise for G) is attributed to [18, Proposition 10.1.10], but that proposition guarantees a continuous section through a given vector and does not by itself prescribe the section norm; the norm control used later in the summability estimate needs a separate (and easy) argument, so this should be proved or cited precisely.","section":"§3, Claim 3.8"},{"comment":"The proof invokes Theorem 3.7 to assert surjectivity of R^t: A(G) → A(G_t), but a group bundle with discrete unit space need not be étale (its fibers are arbitrary locally compact groups), so the hypotheses of Theorem 3.7 need not be satisfied. The surjectivity citation is in fact unnecessary for the conclusion: the chain Φ(A(G)) = c_0-⊕A(G_t) follows from Definition 2.12, Proposition 5.1, and the standard fact that the sup-norm closure of the algebraic direct sum ⊕A_c(G_t) in ℓ∞-⊕B(G_t) is c_0-⊕A(G_t). The citation should be removed and the argument rewritten accordingly.","section":"§5, Corollary 5.5"},{"comment":"The approximating functions ψ_m are defined for x ∈ G_m but then evaluated at z ∈ G_∞; the notation should be clarified, and the text should state explicitly that each ψ_n is the function z ↦ (π_n ∘ q_n)_{ξ(n),ξ(n)}(z) on G_∞.","section":"§4.3, Proposition 4.4"},{"comment":"The standard fact that every finitely generated residually finite group admits an approximating sequence is cited to the authors' own in-preparation paper [17]; a published reference should be used.","section":"§4.3, Remark 4.5"},{"comment":"The inclusion C_c(G^u_u) ∩ B(G^u_u) ⊆ A(G^u_u) is asserted without reference; in the group case this is a classical theorem of Eymard [20] (see also [31]), and a citation should be provided.","section":"§3, Proposition 3.4"},{"comment":"There are several presentation issues: in Definition 2.4 the range map is printed as 'r : G → G' and should read 'r : G → G^(0)'; in Proposition 4.2 the word 'restriciton' appears; and in the proof of Corollary 5.5 the same notation Φ(A_c(G)) with an overline is used ambiguously for the image of the closure and the closure of the image.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The strongest result (Theorem 3.7) is the one most in doubt; the referee's major comment documents a concrete étale counterexample to the key membership assertion of Claim 3.8. The authors are advised to either prove the needed approximation result for bounded continuous sections of the left regular field of an étale groupoid, or to reformulate Theorem 3.7 and the dependent statements (Corollary 3.11 and the citation in Corollary 5.5) under hypotheses for which the preimage can be constructed explicitly. The reference [17] is to the authors' own in-preparation paper for a standard fact; a published citation should be requested. Attention should also be drawn to the tension between Remark 2.15, which records that the relevant identification is open, and the unproved use of that identification in Claim 3.8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a useful, mostly correct extension of group restriction-map theory to groupoids, with one real but patchable gap in the proof of the headline A(G) surjectivity theorem. I agree with the reader's conditional verdict, though I think the paper is in better shape than the stress test alone suggests.\n\nWhat's actually new: the framework for restriction to isotropy subgroups, the surjectivity results for B(G) for transitive groupoids with a continuous section and for group bundles with discrete unit space, the necessary property-FD condition for HLS groupoids, and the box decomposition of Section 5. The HLS examples giving non-surjectivity are a concrete payoff. These parts look sound; the proofs use standard weak containment criteria and the computations check out.\n\nThe soft spot is in Claim 3.8. The authors take a coefficient sigma_{f,g} of the restricted left regular representation, extend f,g to bounded continuous sections F,G via [18, Proposition 10.1.10], and then treat Lambda_{F,G} as an element of A(G) with the usual norm bound. But by their own Definition 2.12, A(G) is the B(G)-closure of compactly supported coefficients, and Remark 2.15 notes that it is open whether arbitrary bounded continuous regular coefficients lie in that closure. Without that, the preimage series need not land in A(G). That is a genuine gap in the proof as written.\n\nThe saving grace is that this is likely fixable: in an etale groupoid, G^u is discrete and each point sits in a bisection, so finitely supported coefficient functions on the isotropy group can be realized exactly by compactly supported coefficients on G. Approximating f and g by finite-support sequences should let you rebuild the series argument with sections that are genuinely in A(G). I would expect Theorem 3.7 and Corollary 3.11 to survive, but the authors need to supply this argument.\n\nOne minor annoyance: the introduction says Theorem 3.7 holds more generally under Condition (*), but the proof and Remark 3.10 make clear they only know this under extra hypotheses. That should be reworded. Also, an in-preparation citation for a standard residual finiteness fact is unnecessary, but that is cosmetic.\n\nWho this is for: harmonic analysts working on groupoid Fourier algebras. It gives them a usable toolkit and a partial answer to a Paterson question. It deserves a serious referee. I would send it out and ask for the Claim 3.8 fix before acceptance.","headline":"Useful groupoid restriction-map paper with a real but likely patchable gap in the proof of the main A(G) surjectivity theorem.","tokens_in":24681,"tokens_out":9168,"would_cite":true,"duration_ms":92568,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["43A30","46J99","22A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's main theorem: on étale groupoids, every Fourier function on an isotropy subgroup extends to the whole groupoid.","keywords":["locally compact groupoids","Fourier algebra","Fourier-Stieltjes algebra","restriction map","isotropy subgroup","étale groupoid","HLS groupoid","property FD"],"falsifier":"Concrete observation: for the HLS groupoid built from $SL_n(\\mathbb{Z})$ with $n\\ge 3$, Theorem 4.6 predicts that the restriction map $B(G)\\to B(SL_n(\\mathbb{Z}))$ is not surjective, because $SL_n(\\mathbb{Z})$ is known to lack property FD. If one could exhibit any coefficient function in $B(SL_n(\\mathbb{Z}))$ that nevertheless extends to an element of $B(G)$, the theorem would be false. On the Fourier-algebra side, the decisive check is whether some coefficient $\\Lambda_{F,G}$ of the left regular representation, with bounded continuous sections $F,G$, fails to lie in $A(G)$; such a coefficient would invalidate the preimage step, so the paper's proof of Theorem 3.7 would not establish the claimed surjectivity.","tokens_in":23624,"feed_emoji":"","tokens_out":17748,"duration_ms":145765,"temperature":0.7,"pith_summary":"This paper studies restriction maps that take functions in the Fourier algebra $A(G)$ (the norm closure of compactly supported coefficients of the left regular representation) or the Fourier-Stieltjes algebra $B(G)$ of a locally compact groupoid $G$ and restrict them to an isotropy subgroup $G^u_u$. Its central claim is that for étale groupoids this restriction map on $A(G)$ is surjective: every Fourier function on an isotropy subgroup has a Fourier extension to the whole groupoid. For $B(G)$, surjectivity is proved for transitive groupoids with a continuous section and for group bundles with discrete unit space, while an HLS-groupoid analysis shows that surjectivity of the restriction map forces property FD on the fiber at infinity, yielding examples where the map is not surjective. These results matter because surjective restriction maps pass hereditary Banach-algebra properties upward, so non-amenability or absence of a bounded approximate identity in an isotropy subgroup obstructs the corresponding property in $A(G)$.","feed_headline":"Every isotropy Fourier function extends to the whole étale groupoid","feed_subtitle":"Non-amenable isotropy subgroups block bounded approximate identities in the whole groupoid Fourier algebra.","key_machinery":"The central object is the isotropy restriction map $R^u\\colon \\phi\\mapsto \\phi|_{G^u_u}$, with $G^u_u$ the isotropy subgroup at the unit $u$. The load-bearing mechanism for Theorem 3.7 is the restricted left regular representation $\\sigma$ acting on $L^2(G^u)$, a fiber of the left regular Hilbert field: because $G$ is étale, $G^u$ is discrete, so $L^2(G^u)$ splits as $\\bigoplus_{v:G^u_v\\neq\\emptyset} L^2(G^u_v)$, and each summand is unitarily equivalent to $L^2(G^u_u)$ via the shift $z\\mapsto z\\gamma_v$. Hence $\\sigma\\cong \\bigoplus L_{G^u_u}$, where $L_{G^u_u}$ is the left regular representation of the isotropy group, and this identifies the coefficient space of $\\sigma$ with $A(G^u_u)$; a standard extension result for continuous sections of Hilbert fields then supplies preimages in $A(G)$. For the Fourier-Stieltjes surjectivity results, the machinery is a representation built on a constant Hilbert field by conjugating the given isotropy representation with a continuous section of the range map, and in the HLS case it is weak containment of the fiber-at-infinity representation by the finite-quotient representations.","core_discovery":"The paper's central claim is Theorem 3.7: if $G$ is a locally compact Hausdorff étale groupoid and $u\\in G^{(0)}$, then $R^u_{A(G)}\\colon A(G)\\to A(G^u_u)$ is surjective. The proof restricts the left regular representation $\\Lambda$ of $G$ to the isotropy group $G^u_u$, proves that the restricted representation is unitarily equivalent to a direct sum of copies of the left regular representation of $G^u_u$, and then lifts each coefficient of the latter to a coefficient $\\Lambda_{F,G}$ of $\\Lambda$ using bounded continuous sections of the left regular Hilbert field. On the Fourier-Stieltjes side, the paper proves surjectivity for transitive groupoids admitting a continuous section (Theorem 4.1) and for group bundles with discrete unit space (Proposition 4.2). It also proves a necessary condition for HLS groupoids, the group bundles built from a discrete group and a nested sequence of finite-index normal subgroups with a ray topology: if $R_{B(G)}\\colon B(G)\\to B(G_\\infty)$ is surjective, then $G_\\infty$ has property FD, meaning every unitary representation is weakly contained in the family of representations factoring through finite quotients (Theorem 4.6). Since $SL_n(\\mathbb{Z})$ for $n\\ge 3$ lacks property FD, this produces HLS groupoids for which the Fourier-Stieltjes restriction map is not surjective. The final application decomposes $A(G)$ and $B(G)$ as $c_0$- and $\\ell^\\infty$-direct sums of fiber algebras for group bundles with discrete unit space.","pith_inferences":["If the unproved inclusion of all left-regular coefficients into $A(G)$ is filled in, Theorem 3.7 would give a groupoid-level route from non-amenability of any single isotropy group to absence of a bounded approximate identity, extending the classical group theorem for Fourier algebras to étale groupoids.","The same coefficient-lifting strategy may transfer to the measurable groupoid Fourier-algebra setting, where measurability rather than continuity could make the extension step easier; the paper does not pursue that setting.","For group bundles with discrete unit space, the box decomposition suggests that questions about multipliers or derivations on $A(G)$ reduce fiberwise; one could test whether the isometric isomorphism preserves the multiplier algebra, which the paper does not discuss."],"forward_implications":["For every étale groupoid, each Fourier function on an isotropy subgroup admits an extension to a Fourier function on the whole groupoid.","If an isotropy subgroup of an étale groupoid is non-amenable, then $A(G)$ has no bounded approximate identity; and failures of amenability, weak amenability, or contractibility in $A(G^u_u)$ force the same failure in $A(G)$.","For transitive groupoids with a continuous section, every Fourier-Stieltjes function on an isotropy subgroup extends to the whole groupoid; the same holds for group bundles with discrete unit space.","For group bundles with discrete unit space, $B(G)$ is isometrically isomorphic to the $\\ell^\\infty$-direct sum of the fiber algebras and $A(G)$ to their $c_0$-direct sum.","For HLS groupoids, surjectivity of the Fourier-Stieltjes restriction map forces property FD on the fiber at infinity, so HLS groupoids built from $SL_n(\\mathbb{Z})$, $n\\ge3$, have non-surjective restriction maps."],"supporting_citations":[{"why":"It supplies the group-level Fourier and Fourier-Stieltjes algebras whose restriction behaviour the paper extends.","marker":"[20]"},{"why":"It provides the continuous-field definitions of $B(G)$ and $A(G)$ for groupoids and the left regular representation.","marker":"[45]"},{"why":"It gives the extension result for continuous sections of Hilbert fields used to lift vectors on one fiber to bounded sections.","marker":"[18]"},{"why":"It supplies the coefficient-space representation theorem and weak-containment facts used in the surjectivity arguments.","marker":"[31]"},{"why":"It proves surjectivity of restriction for Fourier algebras of groups, the result the paper generalizes to étale groupoids.","marker":"[26]"},{"why":"It gives the theorem connecting bounded approximate identities in Fourier algebras to amenability, used in Corollary 3.11.","marker":"[34]"},{"why":"It introduces property FD, the necessary condition for surjectivity of the Fourier-Stieltjes restriction map in the HLS case.","marker":"[37]"},{"why":"It originates the HLS groupoid construction used as the test class for non-surjective restriction maps.","marker":"[28]"},{"why":"It provides the specific HLS groupoid topology and approximating-sequence setup used in Section 4.3.","marker":"[58]"},{"why":"It provides the example of $SL_n(\\mathbb{Z})$, $n\\ge3$, lacking property FD, which yields non-surjective restriction maps.","marker":"[6]"}],"fun_headline_variants":["Étale groupoids: every isotropy Fourier function extends","Fourier algebra restriction is surjective for étale groupoids","Fourier-Stieltjes restriction surjectivity hinges on property FD","Groupoid restriction maps: surjective for Fourier, not Stieltjes","Isotropy subgroups decide Fourier and Fourier-Stieltjes surjectivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that every coefficient function of the groupoid's left regular representation obtained from bounded continuous sections of the Hilbert field belongs to the Fourier algebra $A(G)$, even though the paper notes it is unknown whether $A(G)$ contains the full coefficient space; if some such coefficient lies outside $A(G)$, the preimage construction in the proof of Theorem 3.7 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Étale groupoids: every isotropy Fourier function extends","Fourier algebra restriction is surjective for étale groupoids","Fourier-Stieltjes restriction surjectivity hinges on property FD","Groupoid restriction maps: surjective for Fourier, not Stieltjes","Isotropy subgroups decide Fourier and Fourier-Stieltjes surjectivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1979,"prompt_tokens":1172,"completion_tokens":807,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":788,"completion_tokens_details":{"reasoning_tokens":728}},"tokens_in":788,"tokens_out":807,"duration_ms":8468,"temperature":1.0,"reasoning_tokens":728,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:07:58.748833+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Concrete observation: for the HLS groupoid built from $SL_n(\\mathbb{Z})$ with $n\\ge 3$, Theorem 4.6 predicts that the restriction map $B(G)\\to B(SL_n(\\mathbb{Z}))$ is not surjective, because $SL_n(\\mathbb{Z})$ is known to lack property FD. If one could exhibit any coefficient function in $B(SL_n(\\mathbb{Z}))$ that nevertheless extends to an element of $B(G)$, the theorem would be false. On the Fourier-algebra side, the decisive check is whether some coefficient $\\Lambda_{F,G}$ of the left regular representation, with bounded continuous sections $F,G$, fails to lie in $A(G)$; such a coefficient would invalidate the preimage step, so the paper's proof of Theorem 3.7 would not establish the claimed surjectivity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the group-level Fourier and Fourier-Stieltjes algebras whose restriction behaviour the paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the continuous-field definitions of $B(G)$ and $A(G)$ for groupoids and the left regular representation."},{"cited_title":"15, North-Holl and Pub- lishing Co., Amsterdam-New York-Oxford, 1977, Translated from the French by Francis Jellett","cited_arxiv_id":null,"evidence_quote":"It gives the extension result for continuous sections of Hilbert fields used to lift vectors on one fiber to bounded sections."},{"cited_title":"Lau, Fourier and Fourier-Stieltjes algebras on locally com- pact groups, Mathematical Surveys and Monographs, vol","cited_arxiv_id":null,"evidence_quote":"It supplies the coefficient-space representation theorem and weak-containment facts used in the surjectivity arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It proves surjectivity of restriction for Fourier algebras of groups, the result the paper generalizes to étale groupoids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the theorem connecting bounded approximate identities in Fourier algebras to amenability, used in Corollary 3.11."},{"cited_title":"Math., vol","cited_arxiv_id":null,"evidence_quote":"It introduces property FD, the necessary condition for surjectivity of the Fourier-Stieltjes restriction map in the HLS case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It originates the HLS groupoid construction used as the test class for non-surjective restriction maps."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the specific HLS groupoid topology and approximating-sequence setup used in Section 4.3."},{"cited_title":"11 (1999), no","cited_arxiv_id":null,"evidence_quote":"It provides the example of $SL_n(\\mathbb{Z})$, $n\\ge3$, lacking property FD, which yields non-surjective restriction maps."}],"review_version":1}