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The name for the the following 2 mathematical objects:

  • $$H_2(G,\mathbb{Z})$$ and

  • $$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times G\longrightarrow\mathbb{C}\text{defined by} \\ S(g,h)=K(g,h)T(g,h)\text{also represents an element of}~B(l^2(G))\}$$ where $T$ and $S$ are seen as infinite matrices in the the canonical basis of $l^2(G)$

is the same: Schur multiplier of a group. Why? Is there a strong connection between them? I'd say it comes from the fact that originally $H_2(G,\mathbb{Z})$ was defined as $H^2(G,\mathbb{C}^*)$ for finite groups, which has to do with projective representations, and representations are related to the second object. But infinite groups interest me more.

I think Herz first defined and gave the name to the second object, but I don't know why he chose this name which already exited in the literature, unless there is a strong link between them. (the paper is in French and I can't read it).

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  • $\begingroup$ I have no idea why this second thing also deserves the name "Schur multiplier": there doesn't seem to be a cocycle condition on $K$, nor a quotienting by coboundaries. If $G$ is finite the condition is vacuous, right? So it definitely doesn't agree with the Schur multiplier in that case. Doesn't seem like a good name to me. (Also, you've mixed up indices: you mean $H_2(G, \mathbb{Z})$ and $H^2(G, \mathbb{C}^{\times})$.) $\endgroup$
    – Qiaochu Yuan
    Commented Jan 8, 2016 at 22:26
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    $\begingroup$ @QiaochuYuan the second thing is the space of multipliers for the operation of Schur product (aka entrywise product if you inreoduce co-ordinates and view operators as infinite matruces). So "Schur multiplier" really doesn't seem that bad a name to me. Cf. "Fourier multiplier", etc. Also I am not sure that Herz is responsible for the terminology Schur multiplier, see my answer $\endgroup$
    – Yemon Choi
    Commented Jan 8, 2016 at 22:30
  • $\begingroup$ @QiaochuYuan Edited the typo, thank you. $\endgroup$
    – Alin Galatan
    Commented Jan 8, 2016 at 22:38

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No. Well, I don't see any reason for any link between them. Note that in your second example, the group structure is irrelevant: what you are discussing there is the space of Schur multipliers on ${\mathcal B}(\ell^2(I))$ for an appropriate cardinal $I$. In particular, your speculation about representations and second homology seems misplaced.

I assume that the term "Schur product" (of matrices or operators) predates Herz's interest in what are now called Herz-Schur multipliers (basically, those Schur multipliers that respect the group action).

In other news, there is no connection between weakly amenable groups and weakly amenable Banach algebras; or, as far as I know, between Frobenius groups and Frobenius reciprocity.

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  • $\begingroup$ The reason I said that was this: Any representation gives rise to a Herz-Schur multiplier $\varphi(g)=<g\xi | \eta>$ which, as you said, gives the kernel $K(g, h)=\varphi(g^{-1}h)$. In my mind the question was rather about analogy between Schur multipliers as $H_2$ and Herz-Schur multiplier, but I decided to not put the Herz word, for a better impact on the reader. I really wanted both to really have the same name :) $\endgroup$
    – Alin Galatan
    Commented Jan 8, 2016 at 22:34
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    $\begingroup$ @AlinGatalan Well you may want them to have the same name but they don't. You are also missing some adjectives on what constitutes a "representation" (you need uniformly bounded, in general) and not all Herz-Schur multipliers arise this way; see the result of Gilbert/Jolissaint. $\endgroup$
    – Yemon Choi
    Commented Jan 8, 2016 at 22:36
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    $\begingroup$ I mean, is there some connection between normal subgroups and normal operators? No. $\endgroup$
    – Yemon Choi
    Commented Jan 8, 2016 at 22:37
  • $\begingroup$ I fully support Yemon's answer and comment! Sometimes it happens in mathematics that two completely unrelated objects have similar names... $\endgroup$
    – Alain Valette
    Commented Jan 8, 2016 at 22:39
  • $\begingroup$ @GeoffRobinson heh. perhaps I was hasty in my attempt to choose an example. That said, presumably the fact Frobenius's name is attached to both concepts does not reflect a mathematical link beyond the fact that F. worked on both? $\endgroup$
    – Yemon Choi
    Commented Jan 8, 2016 at 23:24

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