Exactness and Fell bundles with the approximation property over inverse semigroups

Changyuan Gao; Julian Kranz

arxiv: 2601.07572 · v2 · pith:ELQMWORUnew · submitted 2026-01-12 · 🧮 math.OA

Exactness and Fell bundles with the approximation property over inverse semigroups

Changyuan Gao , Julian Kranz This is my paper

Pith reviewed 2026-05-21 16:07 UTC · model grok-4.3

classification 🧮 math.OA

keywords Fell bundlesinverse semigroupsexact C*-algebrasapproximation propertycross-sectional algebrasreduced algebras

0 comments

The pith

The reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber is exact.

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

The paper establishes an if-and-only-if criterion for exactness in certain C*-algebras built from Fell bundles. For any such bundle that satisfies the approximation property over an inverse semigroup, the reduced cross-sectional algebra is exact precisely when the fiber over the unit element is exact. A reader would care because exactness is a fundamental regularity condition in operator algebras, and the result reduces the check to a simpler object while extending similar statements from groupoid actions.

Core claim

The central claim is that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. The proof proceeds by reducing exactness questions via the approximation property and by reproving selected results on Fell bundle ideals.

What carries the argument

The approximation property of the Fell bundle, which reduces exactness of the reduced cross-sectional algebra to exactness of the unit fiber.

If this is right

Exactness of the reduced cross-sectional algebra is completely determined by the unit fiber alone.
The result extends earlier exactness criteria from second-countable locally compact Hausdorff groupoid actions to inverse semigroups.
The same methods yield new proofs of selected facts about ideals in Fell bundles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The criterion may simplify classification of exact C*-algebras arising from semigroup dynamical systems.
Cases without the approximation property remain open and may require additional conditions to recover a similar reduction.
Concrete examples from specific inverse semigroups could be used to test exactness predictions in practice.

Load-bearing premise

The Fell bundle must possess the approximation property.

What would settle it

An explicit Fell bundle over an inverse semigroup that has the approximation property, together with a direct computation showing the reduced cross-sectional algebra is exact while the unit fiber is not (or vice versa), would contradict the claimed equivalence.

read the original abstract

We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable $C^*$-algebras. Along the way, we reprove some results of Kwa\'sniewski--Meyer on Fell bundle ideals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes a clean if-and-only-if for exactness of the reduced cross-sectional algebra of an approximation-property Fell bundle over an inverse semigroup, precisely when the unit fiber is exact.

read the letter

The main takeaway is that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber is exact. This is framed as a direct generalization of the first author's earlier result on second-countable groupoid actions, and the paper also includes a reproof of some ideal results from Kwaśsniewski-Meyer. Both pieces add some value for people already working in this corner of C*-algebras from semigroups and bundles. The equivalence itself looks like a reasonable extension of the groupoid case, and the if-and-only-if structure suggests the two directions are built separately rather than leaning on each other. The reproof of the ideal results is a secondary but honest contribution that could save readers from chasing the original reference. One soft spot worth checking is the handling of countability or separability. The groupoid version explicitly used second-countable groupoids and separable algebras, yet the new statement does not list parallel restrictions. The approximation property is often used to get countable approximate units or to preserve exact sequences under limits, so if those steps are still present in the proofs, the claim as written might need those hypotheses to hold in full generality. If the full text makes the reduction work without them, that would be worth noting explicitly. This is a standard incremental result inside operator algebras. It is aimed at specialists who already care about exactness, Fell bundles, and inverse semigroups. A reader in that subfield can extract the generalization and the reproof without much trouble. It is solid enough to deserve a serious referee who can verify the technical steps and any implicit assumptions on countability.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber is exact. This generalizes a recent result for actions of second-countable locally compact Hausdorff groupoids on separable C*-algebras, and includes reproofs of some results of Kwaśniewski-Meyer on Fell bundle ideals.

Significance. If the central equivalence holds in the stated generality, the result supplies a useful exactness criterion for reduced cross-sectional algebras arising from Fell bundles over inverse semigroups. The generalization from the groupoid setting broadens applicability within operator algebra theory, and the reproof of ideal results may simplify subsequent arguments involving Fell bundle ideals.

major comments (1)

[§1, Theorem 1.1] §1 (Introduction) and Theorem 1.1: The main if-and-only-if statement is formulated without any countability, separability, or second-countability hypotheses on the inverse semigroup or on the fibers of the Fell bundle. The generalized result for groupoids explicitly requires second-countable groupoids and separable C*-algebras; the approximation property is typically invoked to produce countable approximate units or to pass exactness through inductive limits and reduced crossed products. If the proof in §§3–4 relies on such restrictions implicitly, the claimed equivalence does not hold in the stated generality. A concrete test is whether the constructions remain well-defined and exactness-preserving when the semigroup is uncountable.

minor comments (2)

[§2.2] §2.2: The definition of the reduced cross-sectional algebra could include an explicit pointer to the precise formula used for the reduced norm, to aid readers comparing with the groupoid case.
[Abstract] Abstract: The phrase 'along the way, we reprove some results' would be clearer if it named the specific Kwaśniewski–Meyer theorems being reproved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for raising this important point about the generality of the main result. We maintain that the stated theorem holds without countability or separability assumptions, because the approximation property of the Fell bundle supplies the necessary approximations independently of any countability on the inverse semigroup. We address the comment in detail below and will incorporate a clarifying remark.

read point-by-point responses

Referee: [§1, Theorem 1.1] §1 (Introduction) and Theorem 1.1: The main if-and-only-if statement is formulated without any countability, separability, or second-countability hypotheses on the inverse semigroup or on the fibers of the Fell bundle. The generalized result for groupoids explicitly requires second-countable groupoids and separable C*-algebras; the approximation property is typically invoked to produce countable approximate units or to pass exactness through inductive limits and reduced crossed products. If the proof in §§3–4 relies on such restrictions implicitly, the claimed equivalence does not hold in the stated generality. A concrete test is whether the constructions remain well-defined and exactness-preserving when the semigroup is uncountable.

Authors: The proof does not rely on countability. The approximation property furnishes a net of finite-rank completely positive contractive maps on the fibers that approximate the identity uniformly on compact sets in the appropriate sense; this net is used directly to show that exactness of the unit fiber implies exactness of the reduced cross-sectional algebra via a diagram-chasing argument that works for arbitrary (possibly uncountable) inverse semigroups. The reduced cross-sectional algebra itself is defined as the completion of the algebraic cross-section with respect to the reduced norm coming from the regular representation on the Hilbert module over the unit fiber; this construction is purely algebraic and does not require a countable dense subset. In contrast to the groupoid setting, where second-countability is used to guarantee a countable basis for the topology and hence a countable approximate unit in the groupoid C*-algebra, the discrete nature of an inverse semigroup together with the bundle AP suffices here. We have verified that the same arguments apply verbatim when the semigroup is uncountable. We will add a short paragraph after Theorem 1.1 explaining this distinction and confirming that the result is stated in full generality. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper proves an if-and-only-if equivalence between exactness of the reduced cross-sectional algebra and exactness of the unit fiber for Fell bundles possessing the approximation property over inverse semigroups. This rests on independent constructions for the reduced algebra and the unit fiber rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. The reference to generalizing the first-named author's prior groupoid result provides context for the extension but does not substitute for the new proof; the approximation property is stated explicitly as a hypothesis. No equations or steps in the abstract or described theorem reduce by construction to the inputs, and the result is presented as holding under the given technical condition without hidden self-referential assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of C*-algebras, Fell bundles, and inverse semigroups. No free parameters or invented entities are introduced in the abstract. Axioms are background facts from the literature on exact C*-algebras and bundle ideals.

axioms (2)

standard math Standard properties of reduced cross-sectional algebras and exactness for C*-algebras hold as in the literature.
Invoked implicitly when defining the reduced algebra and exactness.
domain assumption The approximation property for the Fell bundle is a well-defined technical condition from prior work.
Used as the hypothesis that enables the equivalence.

pith-pipeline@v0.9.0 · 5589 in / 1322 out tokens · 35709 ms · 2026-05-21T16:07:21.793286+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A. Let A be a Fell bundle over a unital inverse semigroup S with the approximation property. Then the reduced cross-sectional algebra C*_r(A) is exact if and only if its unit fiber A_1 is exact.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Amenability and approximation properties for partial actions and Fell bundles.Bull

[ABF22] Fernando Abadie, Alcides Buss, and Dami´ an Ferraro. Amenability and approximation properties for partial actions and Fell bundles.Bull. Braz. Math. Soc. (N.S.), 53(1):173– 227, 2022.doi:10.1007/s00574-021-00255-8. [AD87] Claire Anantharaman-Delaroche. Syst` emes dynamiques non commutatifs et moyennabilit´ e.Math. Ann., 279(2):297–315, 1987.doi:10...

work page doi:10.1007/s00574-021-00255-8 2022
[2]

[ADR00] Claire Anantharaman-Delaroche and Jean Renault.Amenable groupoids

URL:https://arxiv.org/ abs/1605.05117,arXiv:1605.05117. [ADR00] Claire Anantharaman-Delaroche and Jean Renault.Amenable groupoids. L’Enseignement Math´ ematique, Geneva,

work page arXiv
[3]

Fourier series and twisted C ∗-crossed products.J

[BC15] Erik B´ edos and Roberto Conti. Fourier series and twisted C ∗-crossed products.J. Fourier Anal. Appl., 21(1):32–75, 2015.doi:10.1007/s00041-014-9360-3. [BE12] Alcides Buss and Ruy Exel. Fell bundles over inverse semigroups and twisted ´ etale groupoids.J. Oper. Theory, pages 153–205,

work page doi:10.1007/s00041-014-9360-3 2015
[4]

[BEM17] Alcides Buss, Ruy Exel, and Ralf Meyer

URL:https://jot.theta.ro/jot/ archive/2012-067-001/2012-067-001-007.html. [BEM17] Alcides Buss, Ruy Exel, and Ralf Meyer. ReducedC ∗-algebras of Fell bun- dles over inverse semigroups.Israel. J. Math., 220:225–274, 2017.doi:10.1007/ s11856-017-1516-9. [BEW24] Alcides Buss, Siegfried Echterhoff, and Rufus Willett. Amenability and weak contain- ment for act...

work page doi:10.1090/memo/1513 2012
[5]

[BM17] Alcides Buss and Ralf Meyer

Springer Science & Business Media, 2006.doi:10.1007/3-540-28517-2. [BM17] Alcides Buss and Ralf Meyer. Inverse semigroup actions on groupoids.Rocky Mountain J. Math., 47(1):53–159, 2017.doi:10.1216/RMJ-2017-47-1-53. [BM23] Alcides Buss and Diego Mart´ ınez. Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids.Adv. M...

work page doi:10.1007/3-540-28517-2 2006
[6]

URL:https://arxiv.org/abs/2501.01775,arXiv:2501. 01775. [CE78] Man Duen Choi and Edward G. Effros. NuclearC ∗-algebras and the approximation property.Amer. J. Math., 100(1):61–79, 1978.doi:10.2307/2373876. [EN02] Ruy Exel and Chi-Keung Ng. Approximation property ofC ∗-algebraic bundles.Math. Proc. Cambridge Philos. Soc., 132(3):509–522, 2002.doi:10.1017/S...

work page doi:10.2307/2373876 1978
[7]

[FD88] J

URL:https://nyjm.albany.edu/j/2011/17-17p.pdf. [FD88] J. M. G. Fell and R. S. Doran.Representations of ∗-algebras, locally compact groups, and Banach ∗-algebraic bundles. Vol. 1, volume 126 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA,

work page 2011
[8]

[Gao25] Changyuan Gao

Banach ∗-algebraic bundles, induced represen- tations, and the generalized Mackey analysis.doi:10.1016/S0079-8169(09)60018-0. [Gao25] Changyuan Gao. On the exactness of groupoid crossed products.Ann. Funct. Anal., 16(2):Paper No. 17, 17, 2025.doi:10.1007/s43034-025-00409-5. [GK02] Erik Guentner and Jerome Kaminker. Exactness and the Novikov conjecture.Top...

work page doi:10.1016/s0079-8169(09)60018-0 2025
[9]

[Kir94] Eberhard Kirchberg

doi:10.1002/mana.19770760115. [Kir94] Eberhard Kirchberg. Commutants of unitaries in UHF algebras and functorial properties of exactness.J. Reine Angew. Math., 452:39–77, 1994.doi:10.1515/crll.1994.452.39. [Kir95a] Eberhard Kirchberg. ExactC ∗-algebras, tensor products, and the classification of purely infinite algebras. InProceedings of the International...

work page doi:10.1002/mana.19770760115 1994
[10]

[Law98] Mark V

A toolkit for operator algebraists.doi:10.1017/CBO9780511526206. [Law98] Mark V. Lawson.Inverse semigroups. World Scientific Publishing Co., Inc., River Edge, NJ,

work page doi:10.1017/cbo9780511526206
[11]

[MT21] Andrew McKee and Lyudmila Turowska

The theory of partial symmetries.doi:10.1142/9789812816689. [MT21] Andrew McKee and Lyudmila Turowska. Exactness and SOAP of crossed products via Herz-Schur multipliers.J. Math. Anal. Appl., 496(2):Paper No. 124812, 16,

work page doi:10.1142/9789812816689
[12]

[Mur90] Gerald J

doi:10.1016/j.jmaa.2020.124812. [Mur90] Gerald J. Murphy.C ∗-algebras and operator theory. Academic press, 1990.doi:10. 1016/C2009-0-22289-6. [MW08] Paul S. Muhly and Dana P. Williams. Equivalence and disintegration theorems for Fell bundles and theirC ∗-algebras.Dissertationes Math. (Rozprawy Mat.), 456:1–57,

work page doi:10.1016/j.jmaa.2020.124812 2020
[13]

[OS21] Narutaka Ozawa and Yuhei Suzuki

doi:10.4064/dm456-0-1. [OS21] Narutaka Ozawa and Yuhei Suzuki. On characterizations of amenableC ∗- dynamical systems and new examples.Selecta Math. (N.S.), 27, 2021.doi:10.1007/ s00029-021-00699-2. [Ren87] Jean Renault. Repr´ esentation des produits crois´ es d’alg` ebres de groupo¨ ıdes.J. Oper. Theory, 18(1):67–97,

work page doi:10.4064/dm456-0-1 2021
[14]

[RW98] Iain Raeburn and Dana P

URL:https://jot.theta.ro/jot/archive/1987-018-001/ 1987-018-001-005.pdf. [RW98] Iain Raeburn and Dana P. Williams.Morita equivalence and continuous-traceC ∗- algebras, volume 60 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.doi:10.1090/surv/060. [Sie98] N´ andor Sieben. Fell bundles over r-discrete groupoids an...

work page doi:10.1090/surv/060 1987
[15]

On the cross-norm of the direct product ofC ∗-algebras.Tohoku Math

[Tak64] Masamichi Takesaki. On the cross-norm of the direct product ofC ∗-algebras.Tohoku Math. J. (2), 16:111–122, 1964.doi:10.2748/tmj/1178243737. [Tak14] Takuya Takeishi. On nuclearity ofC ∗-algebras of Fell bundles over ´ etale groupoids. Publ. Res. Inst. Math. Sci., 50(2):251–268, 2014.doi:10.4171/PRIMS/132. (Changyuan Gao)Chern Institute of Mathemat...

work page doi:10.2748/tmj/1178243737 1964

[1] [1]

Amenability and approximation properties for partial actions and Fell bundles.Bull

[ABF22] Fernando Abadie, Alcides Buss, and Dami´ an Ferraro. Amenability and approximation properties for partial actions and Fell bundles.Bull. Braz. Math. Soc. (N.S.), 53(1):173– 227, 2022.doi:10.1007/s00574-021-00255-8. [AD87] Claire Anantharaman-Delaroche. Syst` emes dynamiques non commutatifs et moyennabilit´ e.Math. Ann., 279(2):297–315, 1987.doi:10...

work page doi:10.1007/s00574-021-00255-8 2022

[2] [2]

[ADR00] Claire Anantharaman-Delaroche and Jean Renault.Amenable groupoids

URL:https://arxiv.org/ abs/1605.05117,arXiv:1605.05117. [ADR00] Claire Anantharaman-Delaroche and Jean Renault.Amenable groupoids. L’Enseignement Math´ ematique, Geneva,

work page arXiv

[3] [3]

Fourier series and twisted C ∗-crossed products.J

[BC15] Erik B´ edos and Roberto Conti. Fourier series and twisted C ∗-crossed products.J. Fourier Anal. Appl., 21(1):32–75, 2015.doi:10.1007/s00041-014-9360-3. [BE12] Alcides Buss and Ruy Exel. Fell bundles over inverse semigroups and twisted ´ etale groupoids.J. Oper. Theory, pages 153–205,

work page doi:10.1007/s00041-014-9360-3 2015

[4] [4]

[BEM17] Alcides Buss, Ruy Exel, and Ralf Meyer

URL:https://jot.theta.ro/jot/ archive/2012-067-001/2012-067-001-007.html. [BEM17] Alcides Buss, Ruy Exel, and Ralf Meyer. ReducedC ∗-algebras of Fell bun- dles over inverse semigroups.Israel. J. Math., 220:225–274, 2017.doi:10.1007/ s11856-017-1516-9. [BEW24] Alcides Buss, Siegfried Echterhoff, and Rufus Willett. Amenability and weak contain- ment for act...

work page doi:10.1090/memo/1513 2012

[5] [5]

[BM17] Alcides Buss and Ralf Meyer

Springer Science & Business Media, 2006.doi:10.1007/3-540-28517-2. [BM17] Alcides Buss and Ralf Meyer. Inverse semigroup actions on groupoids.Rocky Mountain J. Math., 47(1):53–159, 2017.doi:10.1216/RMJ-2017-47-1-53. [BM23] Alcides Buss and Diego Mart´ ınez. Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids.Adv. M...

work page doi:10.1007/3-540-28517-2 2006

[6] [6]

URL:https://arxiv.org/abs/2501.01775,arXiv:2501. 01775. [CE78] Man Duen Choi and Edward G. Effros. NuclearC ∗-algebras and the approximation property.Amer. J. Math., 100(1):61–79, 1978.doi:10.2307/2373876. [EN02] Ruy Exel and Chi-Keung Ng. Approximation property ofC ∗-algebraic bundles.Math. Proc. Cambridge Philos. Soc., 132(3):509–522, 2002.doi:10.1017/S...

work page doi:10.2307/2373876 1978

[7] [7]

[FD88] J

URL:https://nyjm.albany.edu/j/2011/17-17p.pdf. [FD88] J. M. G. Fell and R. S. Doran.Representations of ∗-algebras, locally compact groups, and Banach ∗-algebraic bundles. Vol. 1, volume 126 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA,

work page 2011

[8] [8]

[Gao25] Changyuan Gao

Banach ∗-algebraic bundles, induced represen- tations, and the generalized Mackey analysis.doi:10.1016/S0079-8169(09)60018-0. [Gao25] Changyuan Gao. On the exactness of groupoid crossed products.Ann. Funct. Anal., 16(2):Paper No. 17, 17, 2025.doi:10.1007/s43034-025-00409-5. [GK02] Erik Guentner and Jerome Kaminker. Exactness and the Novikov conjecture.Top...

work page doi:10.1016/s0079-8169(09)60018-0 2025

[9] [9]

[Kir94] Eberhard Kirchberg

doi:10.1002/mana.19770760115. [Kir94] Eberhard Kirchberg. Commutants of unitaries in UHF algebras and functorial properties of exactness.J. Reine Angew. Math., 452:39–77, 1994.doi:10.1515/crll.1994.452.39. [Kir95a] Eberhard Kirchberg. ExactC ∗-algebras, tensor products, and the classification of purely infinite algebras. InProceedings of the International...

work page doi:10.1002/mana.19770760115 1994

[10] [10]

[Law98] Mark V

A toolkit for operator algebraists.doi:10.1017/CBO9780511526206. [Law98] Mark V. Lawson.Inverse semigroups. World Scientific Publishing Co., Inc., River Edge, NJ,

work page doi:10.1017/cbo9780511526206

[11] [11]

[MT21] Andrew McKee and Lyudmila Turowska

The theory of partial symmetries.doi:10.1142/9789812816689. [MT21] Andrew McKee and Lyudmila Turowska. Exactness and SOAP of crossed products via Herz-Schur multipliers.J. Math. Anal. Appl., 496(2):Paper No. 124812, 16,

work page doi:10.1142/9789812816689

[12] [12]

[Mur90] Gerald J

doi:10.1016/j.jmaa.2020.124812. [Mur90] Gerald J. Murphy.C ∗-algebras and operator theory. Academic press, 1990.doi:10. 1016/C2009-0-22289-6. [MW08] Paul S. Muhly and Dana P. Williams. Equivalence and disintegration theorems for Fell bundles and theirC ∗-algebras.Dissertationes Math. (Rozprawy Mat.), 456:1–57,

work page doi:10.1016/j.jmaa.2020.124812 2020

[13] [13]

[OS21] Narutaka Ozawa and Yuhei Suzuki

doi:10.4064/dm456-0-1. [OS21] Narutaka Ozawa and Yuhei Suzuki. On characterizations of amenableC ∗- dynamical systems and new examples.Selecta Math. (N.S.), 27, 2021.doi:10.1007/ s00029-021-00699-2. [Ren87] Jean Renault. Repr´ esentation des produits crois´ es d’alg` ebres de groupo¨ ıdes.J. Oper. Theory, 18(1):67–97,

work page doi:10.4064/dm456-0-1 2021

[14] [14]

[RW98] Iain Raeburn and Dana P

URL:https://jot.theta.ro/jot/archive/1987-018-001/ 1987-018-001-005.pdf. [RW98] Iain Raeburn and Dana P. Williams.Morita equivalence and continuous-traceC ∗- algebras, volume 60 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.doi:10.1090/surv/060. [Sie98] N´ andor Sieben. Fell bundles over r-discrete groupoids an...

work page doi:10.1090/surv/060 1987

[15] [15]

On the cross-norm of the direct product ofC ∗-algebras.Tohoku Math

[Tak64] Masamichi Takesaki. On the cross-norm of the direct product ofC ∗-algebras.Tohoku Math. J. (2), 16:111–122, 1964.doi:10.2748/tmj/1178243737. [Tak14] Takuya Takeishi. On nuclearity ofC ∗-algebras of Fell bundles over ´ etale groupoids. Publ. Res. Inst. Math. Sci., 50(2):251–268, 2014.doi:10.4171/PRIMS/132. (Changyuan Gao)Chern Institute of Mathemat...

work page doi:10.2748/tmj/1178243737 1964