Amenability and Inner Amenability of Transformation Groups

Ali Ebadian; Ali Jabbari

arxiv: 2005.08077 · v2 · submitted 2020-05-16 · 🧮 math.FA

Amenability and Inner Amenability of Transformation Groups

Ali Ebadian , Ali Jabbari This is my paper

Pith reviewed 2026-05-24 15:27 UTC · model grok-4.3

classification 🧮 math.FA

keywords amenabilitytransformation groupsFølner netinner amenabilitysemidirect productfunctional analysis

0 comments

The pith

Amenable transformation groups admit a net like the Følner net for groups, and inner amenability is introduced with a characterization.

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

The paper shows that amenable transformation groups possess a net that behaves like the Følner net for amenable groups. It studies how amenability carries over when a transformation group is built as a semidirect product of groups. The authors define inner amenability for transformation groups and give a characterization of the property. These steps adapt classical amenability tools to the transformation-group setting, which may support new classifications and examples.

Core claim

There is a net for amenable transformation groups like Følner net for amenable groups and amenability of a transformation group constructed by semidirect product of groups can be investigated. Inner amenability of transformation groups is introduced and characterized.

What carries the argument

Følner-type net for transformation groups that witnesses amenability, together with semidirect-product constructions and the new inner-amenability property.

If this is right

Amenability of semidirect-product transformation groups follows from amenability of the component groups under the given conditions.
Inner amenability supplies an additional invariant that distinguishes transformation groups.
Net-based criteria become available for verifying amenability of transformation groups.
Constructions from ordinary groups yield new families of amenable and inner-amenable transformation groups.

Where Pith is reading between the lines

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

The net criterion may transfer to the study of measure-preserving actions or dynamical systems.
Inner amenability could connect to fixed-point properties for the associated unitary representations.
Similar ideas might apply to C*-algebras or crossed products generated by the transformations.
The approach could extend to semigroups or other algebraic structures that act on spaces.

Load-bearing premise

Standard notions of amenability and transformation groups extend directly to admit Følner-type nets and allow semidirect product constructions to be analyzed for amenability without additional structural restrictions.

What would settle it

An explicit amenable transformation group that admits no net with the stated Følner-type properties, or a semidirect-product construction whose amenability does not follow the pattern predicted from the factors.

read the original abstract

In this paper, we show that there is a net for amenable transformation groups like F{\o}lner net for amenable groups and investigate amenability of a transformation group constructed by semidirect product of groups. We introduce inner amenability of transformation groups and characterize this property.

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 adds inner amenability for transformation groups plus a Følner-style net, but the abstract gives no proofs so the depth stays unclear.

read the letter

The paper shows that amenable transformation groups have a net resembling the Følner net, looks at amenability for semidirect products, and introduces inner amenability with a characterization. What stands out as new is the definition of inner amenability for these groups and the construction of the net. The semidirect product investigation applies existing ideas to this setting. The work is straightforward in extending classical notions, and the abstract lays out the program clearly without overclaiming. The main limitation is that the abstract gives no hint of the actual arguments or examples, so it's difficult to tell if the characterization of inner amenability is substantive or mostly definitional. The semidirect product results could turn out to be direct consequences without much new insight. Since the full text isn't detailed here, any assessment stays tentative. Readers working on amenability in dynamical systems or functional analysis might find the new property worth exploring further. It is not likely to change how most people think about the topic but adds a variant that could be studied. I think it should go to peer review. The claims are concrete enough for referees to evaluate the constructions and see if they hold up. The stress-test note finds no load-bearing gap in the stated claims, which matches what the abstract allows us to see. This is incremental work rather than a reorganization, but the constructions are specific enough to check.

Referee Report

0 major / 2 minor

Summary. The manuscript shows that amenable transformation groups admit a net analogous to the Følner net for amenable groups, investigates amenability criteria for transformation groups arising from semidirect-product constructions, and introduces the notion of inner amenability for transformation groups together with a characterization of the property.

Significance. If the stated existence and characterization results hold, the work supplies standard but useful extensions of amenability concepts from groups to transformation groups. The Følner-type net and the semidirect-product criteria are constructive moves that align with existing literature on amenable actions; the introduction of inner amenability supplies a new definitional tool whose utility will depend on subsequent applications.

minor comments (2)

The abstract states the three main results but does not indicate the precise hypotheses under which the semidirect-product amenability criterion holds; a sentence clarifying the standing assumptions on the groups or actions would improve readability.
Notation for the transformation group and the associated net is introduced without an explicit comparison table to the classical Følner net; adding such a comparison in the introduction would clarify the analogy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; claims are definitional extensions

full rationale

The paper introduces a Følner-type net for amenable transformation groups, examines amenability under semidirect product constructions, and defines plus characterizes inner amenability. These are presented as direct extensions of standard notions without any equations, fitted parameters, or self-citations that reduce the central results to their own inputs by construction. No load-bearing steps match the enumerated circularity patterns; the work is self-contained as constructive and definitional in the literature on amenable actions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work appears to rest on standard background notions of amenability and group actions.

pith-pipeline@v0.9.0 · 5553 in / 1000 out tokens · 33675 ms · 2026-05-24T15:27:16.881652+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C∗-algebras, Trans

C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C∗-algebras, Trans. Amer. Math. Soc., 364(10)(2002), 4153-4178

work page 2002
[2]

Anantharaman-Delaroche, Amenability and exactness for groups, group actions and ope rator algebras, ´Ecole th´ ematique

C. Anantharaman-Delaroche, Amenability and exactness for groups, group actions and ope rator algebras, ´Ecole th´ ematique. Erwin Schrdinger Institute, Mars 2007, 2007

work page 2007
[3]

Bodaghi, A

A. Bodaghi, A. Jabbari, M. Amini, Character inner amenability for semigroups , Semigroup Forum, (2019), https://doi.org/10.1007/s00233-019-10004-5

work page doi:10.1007/s00233-019-10004-5 2019
[4]

Choda, The factors of inner amenable groups , Math

M. Choda, The factors of inner amenable groups , Math. Japon., 24(1979), 145-152

work page 1979
[5]

Choda, Inner amenability and fullness , Proc

M. Choda, Inner amenability and fullness , Proc. Amer. Math. Soc., 86(1982), 663-666

work page 1982
[6]

E. G. Eﬀros, Property T and inner amenability , Proc. Amer. Math. Soc., 47(1975), 483-486

work page 1975
[7]

Hewitt and K

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I , second ed., Springer-Verlag, Berlin, 1979

work page 1979
[8]

Janzen, Følner nets for semidirect products of amenable groups , Canad

D. Janzen, Følner nets for semidirect products of amenable groups , Canad. Math. Bull., 51(1)(2008), 60-66

work page 2008
[9]

Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Følner conditions , Paciﬁc J

M. Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Følner conditions , Paciﬁc J. Math., 73(1)(1977), 91-106

work page 1977
[10]

A. T-M. Lau, Topological semigroups with invariant means in the convex h ull of multiplicative means , Trans. Amer. Math. Soc., 148(1970), 69-84. AMENABILITY AND INNER AMENABILITY OF TRANSFORMATION GROUP S 13

work page 1970
[11]

A. T-M. Lau and L. T. Paterson, Inner amenable locally compact groups , Trans. Amer. Math. Soc., 325(1)(1991), 155-169

work page 1991
[12]

Losert and H

V. Losert and H. Rindler, Conjugate invariant means , Colloq. Math., 15(1987), 221-225

work page 1987
[13]

J. P. Pier, Amenable locally compact groups , Wiley-Interscience, 1984

work page 1984
[14]

Picioroaga, The inner amenability of the generalized Thompson group , Proc

G. Picioroaga, The inner amenability of the generalized Thompson group , Proc. Amer. Math. Soc., 134(7), 1995-2002

work page 1995
[15]

Sakai, Amenable transformation Groups

K. Sakai, Amenable transformation Groups. II , Proc. Japan Acad., 49(1973), 428-431

work page 1973
[16]

Sakai, Amenable transformation groups , Sci

K. Sakai, Amenable transformation groups , Sci. Report of Kagoshima Univ., 22(1973), 1-7

work page 1973
[17]

Stokke, Quasi-central bounded approximate identities in group alg ebras of locally compact groups , Illinois J

R. Stokke, Quasi-central bounded approximate identities in group alg ebras of locally compact groups , Illinois J. Math., 48 (2004), 151-170

work page 2004
[18]

Stokke, Approximate diagonals and Følner conditions for amenable g roup and semigroup algebras , Studia Math., 164(2)(2004), 139-159

R. Stokke, Approximate diagonals and Følner conditions for amenable g roup and semigroup algebras , Studia Math., 164(2)(2004), 139-159

work page 2004
[19]

Wilison, Reiter nets for semidirect products of amenable groups and s emigroups, Proc

B. Wilison, Reiter nets for semidirect products of amenable groups and s emigroups, Proc. Amer. Math., 137(11)(2009), 3823-3832

work page 2009
[20]

C. K. Yuan, The existence of inner invariant means on L∞(G), J. Math. Anal. Appl., 130(1988), 514-524

work page 1988
[21]

C. K. Yuan, Inner amenable groups and the induced C∗-algebra Properties, Acta Mathematica Sinica, 5(2)(1989), 165-169. Young Researchers and Elite Club, Islamic Azad University, Arda bil Branch, Ardabil, Iran E-mail address : jabbari al@yahoo.com Department of Mathematics, F aculty of Science, Urmia Univers ity, Iran E-mail address : ebadian.ali@gmail.com

work page 1989

[1] [1]

Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C∗-algebras, Trans

C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C∗-algebras, Trans. Amer. Math. Soc., 364(10)(2002), 4153-4178

work page 2002

[2] [2]

Anantharaman-Delaroche, Amenability and exactness for groups, group actions and ope rator algebras, ´Ecole th´ ematique

C. Anantharaman-Delaroche, Amenability and exactness for groups, group actions and ope rator algebras, ´Ecole th´ ematique. Erwin Schrdinger Institute, Mars 2007, 2007

work page 2007

[3] [3]

Bodaghi, A

A. Bodaghi, A. Jabbari, M. Amini, Character inner amenability for semigroups , Semigroup Forum, (2019), https://doi.org/10.1007/s00233-019-10004-5

work page doi:10.1007/s00233-019-10004-5 2019

[4] [4]

Choda, The factors of inner amenable groups , Math

M. Choda, The factors of inner amenable groups , Math. Japon., 24(1979), 145-152

work page 1979

[5] [5]

Choda, Inner amenability and fullness , Proc

M. Choda, Inner amenability and fullness , Proc. Amer. Math. Soc., 86(1982), 663-666

work page 1982

[6] [6]

E. G. Eﬀros, Property T and inner amenability , Proc. Amer. Math. Soc., 47(1975), 483-486

work page 1975

[7] [7]

Hewitt and K

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I , second ed., Springer-Verlag, Berlin, 1979

work page 1979

[8] [8]

Janzen, Følner nets for semidirect products of amenable groups , Canad

D. Janzen, Følner nets for semidirect products of amenable groups , Canad. Math. Bull., 51(1)(2008), 60-66

work page 2008

[9] [9]

Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Følner conditions , Paciﬁc J

M. Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Følner conditions , Paciﬁc J. Math., 73(1)(1977), 91-106

work page 1977

[10] [10]

A. T-M. Lau, Topological semigroups with invariant means in the convex h ull of multiplicative means , Trans. Amer. Math. Soc., 148(1970), 69-84. AMENABILITY AND INNER AMENABILITY OF TRANSFORMATION GROUP S 13

work page 1970

[11] [11]

A. T-M. Lau and L. T. Paterson, Inner amenable locally compact groups , Trans. Amer. Math. Soc., 325(1)(1991), 155-169

work page 1991

[12] [12]

Losert and H

V. Losert and H. Rindler, Conjugate invariant means , Colloq. Math., 15(1987), 221-225

work page 1987

[13] [13]

J. P. Pier, Amenable locally compact groups , Wiley-Interscience, 1984

work page 1984

[14] [14]

Picioroaga, The inner amenability of the generalized Thompson group , Proc

G. Picioroaga, The inner amenability of the generalized Thompson group , Proc. Amer. Math. Soc., 134(7), 1995-2002

work page 1995

[15] [15]

Sakai, Amenable transformation Groups

K. Sakai, Amenable transformation Groups. II , Proc. Japan Acad., 49(1973), 428-431

work page 1973

[16] [16]

Sakai, Amenable transformation groups , Sci

K. Sakai, Amenable transformation groups , Sci. Report of Kagoshima Univ., 22(1973), 1-7

work page 1973

[17] [17]

Stokke, Quasi-central bounded approximate identities in group alg ebras of locally compact groups , Illinois J

R. Stokke, Quasi-central bounded approximate identities in group alg ebras of locally compact groups , Illinois J. Math., 48 (2004), 151-170

work page 2004

[18] [18]

Stokke, Approximate diagonals and Følner conditions for amenable g roup and semigroup algebras , Studia Math., 164(2)(2004), 139-159

R. Stokke, Approximate diagonals and Følner conditions for amenable g roup and semigroup algebras , Studia Math., 164(2)(2004), 139-159

work page 2004

[19] [19]

Wilison, Reiter nets for semidirect products of amenable groups and s emigroups, Proc

B. Wilison, Reiter nets for semidirect products of amenable groups and s emigroups, Proc. Amer. Math., 137(11)(2009), 3823-3832

work page 2009

[20] [20]

C. K. Yuan, The existence of inner invariant means on L∞(G), J. Math. Anal. Appl., 130(1988), 514-524

work page 1988

[21] [21]

C. K. Yuan, Inner amenable groups and the induced C∗-algebra Properties, Acta Mathematica Sinica, 5(2)(1989), 165-169. Young Researchers and Elite Club, Islamic Azad University, Arda bil Branch, Ardabil, Iran E-mail address : jabbari al@yahoo.com Department of Mathematics, F aculty of Science, Urmia Univers ity, Iran E-mail address : ebadian.ali@gmail.com

work page 1989