Spaces of Bounded Measurable Functions Invariant under a Group Action

Round 1

Reviewer 1 Report

The author extends his previous work on the characterization of group invariant spaces of functions, to the space of bounded functions (L infinity).

The result seems quite incremental, which is also apparent in the length of the text about the new results (only lines 110 - 139 contain new results, all other text describes preliminaries).

The paper could be improved by adding at least one sentence of motivation at the beginning of the introduction. While reading I had wrote down a few comments that also may help:

* Remark 2 is not obvious to me at first glance, a brief statement why it is an "easy consequence" would help
* Def. 7 seems unnecessary: if alpha x = x, then f(alpha x)=f(x) for every f? I believe the author would like to express something else here.
* Line 67 is stated without explanation. Why is the dimension 1?
* The index set I in line 68 is not introduced. Can it be uncountable? (probably not, looking at later statements)
* line 89 "healthy amount" is too vague / colloquial.
* Line 91: the metric / norms are not stated.

 

Author Response

Thank you for the review and comments; your insights are appreciated.

Below are specific responses:

The introduction will be adjusted to give a motivation.

Remark 2 follows more specifically from the G-invariance of the spaces L^p(\mu). One precomposes the function f dot (g circ phi_alpha) with phi_{alpha inverse} to get the equality. Remark 2 is now Remark 5, and the phrase "easy consequence" has been changed.

Definition 7 is stated as intended. For a specific x, the space H(x) consists of those functions which are unchanged by all elements alpha of G that stabilize x. For example, each H(x) contains the constant functions.

Line 67 is part of Definition 8, which is why no explanation is given. The dimension must be 1 for the proof of Theorem 3.2, which has been left out of the paper for the sake of length.

The index set I is introduced to help with notation, so we can now talk about the projection pi_i onto H_i in the G-collection. This also helps with the notation of the direct sums E_Omega. I had not considered whether I can be uncountable, but this is an interesting question.

The phrase "healthy amount" has been changed.

The metric/norms are assumed to be the sup norm for C(X) and the usual L^p-norm for the L^p spaces. The particular definitions were omitted for brevity, but they can be stated in the introduction (around Lines 42-25) if necessary.

Reviewer 2 Report

This paper characterizes spaces of bounded functions on a compact Hausdorff space which are invariant under a transitive and continuous action by a compact group. The presented results and their potential impact seem to be relevance for wider audience in different areas including functional analysis and abstract harmonic analysis.

Suppose that G is a locally compact group acting on a locally compact space X continuously and transitively. In the framework of abstract/classical harmonic analysis, these spaces called as transitive G-spaces also known as homogeneous spaces, see Section 2.6 of [4]. Invoking Proposition 2.44 of [4], if G is a compact group then each transitive G-space has the form of a coset space G/H for some closed subgroup H of G. The mathematical theory of L^p-function spaces on these spaces studied at depth in the framework of abstract harmonic analysis on homogeneous space (or coset spaces) of compact subgroups. In this direction, the following references should be cited;

1. A. Ghaani Farashahi, Abstract measure algebras over homogeneous spaces of compact groups. Internat. J. Math. 29 (2018), no. 1, 1850005, 34 pp.

2. A. Ghaani Farashahi, Abstract harmonic analysis over spaces of complex measures on homogeneous spaces of compact groups. Bull. Korean Math. Soc. 54 (2017), no. 4, 1229–1240.

3. A. Ghaani Farashahi, Abstract convolution function algebras over homogeneous spaces of compact groups. Illinois J. Math. 59 (2015), no. 4, 1025–1042.

4. G.B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL 1995.
 

Author Response

Thank you for your review and comments. I especially appreciate the references provided, and I hope to incorporate them if possible. If not, they may still prove useful to answer some future questions I have related to this research.

Thank you again for your time in reviewing this manuscript.

Reviewer 3 Report

Spaces of Bounded Measurable Functions Invariant Under a Group Action

Samuel A. Hokamp

The author generalized the work in Hokamp, S.A. Certain invariant spaces of bounded measurable functions on a sphere. Positivity 2021, 25, 2081–2098. https: 162.

The paper can be accepted in its form.

The author used the class of complex continuous functions.

 Several inquiries for the author:

 

When one uses the space of analytical functions, is the output accurate? Is this class subject to any limitations?

Author Response

Thank you for the review; I greatly appreciate your time in reviewing the manuscript.

The question about analytic functions is interesting. It certainly seems that any closed G-invariant space of analytic functions can be decomposed into a closed sum of minimal G-invariant spaces that only contain analytic functions, since any space of non-analytic functions summed with a space of analytic functions would contain non-analytic functions. Thus, the subcollection of script G that sums to the space of all analytic functions would suffice to generate any space of analytic functions.

This is similar to a result of the 2021 paper which shows that for the unit sphere in C^n, script G consists of the spaces span{cz^p\bar{z}^q: p,q nonnegative integers}, and the space of functions that are radial limits ae of bounded holomorphic functions on the ball is the sum of all the above spaces with q=0.

An interesting question that follows is whether this subcollection of script G is somehow induced by a subgroup of G. In the previous example, it seems the subset of the (p,q), nonnegative integers, that determines the radial limits of bounded holomorphic functions has a monoidal structure, but it is not clear if a subgroup of the unitary operators is somehow related.

I am interested in exploring these questions further, and I thank you for raising them.

Reviewer 4 Report

The report is attached

 

Comments for author File: Comments.pdf

Author Response

Thank you for the review; I greatly appreciate your time in reviewing this manuscript.

The citations to [3] have been corrected. Thank you for catching these issues.

The results of Section 2 are indeed simple, but not having found them in another text, I was forced to include their proofs in the paper. If you have suggestions for how to treat these results, please let me know, but I certainly do not wish to claim them as particularly novel.

The proof of Lemma 5.2 of [3] was omitted for brevity, but it is not so long that it would encumber the paper. I have added the necessary proofs, and reworded them so as to make them clearer (hopefully).

Conditions under which (*) holds would make the paper much stronger. Unfortunately, aside from several specific, easy examples, I don't have further results in this direction. This remains a direction of research.

With regard to the specific corrections:

(1) I'll adjust the abstract to eliminate the citation.

(2) The abstract does have C^n in it, so I'm not sure what this correction is referencing.

(3) Corollary 16 follows from the continuity of the L_alpha map on the spaces of continuous and L^p functions. For brevity, this proof was omitted, but could be included.

(4) Line 130 does have L^infty(mu) to C, so again, I am unsure what this correction is referencing.

Again, thank you for your time in reviewing the manuscript.

Round 2

Reviewer 4 Report

The paper is improved according to the comments. So the revised version can be published in presented form.