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Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and right Haar measures on it coincide.

For locally compact groups, is there any relation or implication between these two notions? Abelian groups are trivially both, and compact groups are slightly less trivially both. Is there anything else that can be said there?

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A locally compact group whose left and right uniform structures coincide is usually called a SIN group (SIN= small invariant neighborhoods) — see the paper of Itzkowitz I linked to below. It is true that a SIN group is unimodular, see e.g. Theorem 12.1.9 on page 1273 of T.W. Palmer, Banach algebras and the general theory of $\ast$-algebras, Vol. 2, Cambridge University Press, 2001. The reason is that one can find an invariant compact neighborhood $K$ of the identity of $G$ and invariance means that $gKg^{-1} = K$ for all $g \in G$. Therefore

$$\lambda(K) = \lambda(gK) = \lambda(gKg^{-1}g) = \lambda(Kg) = \Delta(g) \lambda(K)$$

and since $0 \lt \lambda(K) \lt \infty$ we conclude that $\Delta \equiv 1$ and thus $G$ is unimodular.

The other inclusion is false: $\operatorname{SL}_{n}(\mathbb{R})$ is unimodular but its left and right uniform structures are distinct for $n \geq 2$, see Hewitt-Ross, Abstract Harmonic Analysis, I, Example 4.24 (a), p.28f.


Added: It is a result due to G. Itzkowitz, Uniform Structure in Topological Groups, Proc. Amer. Math. Soc. 57, vol. 2, (1976), pp. 363–366 that a locally compact group has a fundamental system of invariant neighborhoods of the identity if and only if the left and right uniform structures coincide.

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OK, thank you, I'll have to look this up! –  Harry Altman Jun 15 '11 at 4:46
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@Harry: I added a reference that may be useful and/or interesting. In any case, googling for "SIN group" and uniform structure should yield a number of papers pertinent to your question. The chapter on locally compact in the book of T.W. Palmer I linked to is quite an good source for finding out about implications between numerous properties a locally compact group may or may not have. –  t.b. Jun 15 '11 at 13:38
    
I'd second Theo's recommendation of Palmer's book. If you don't have access, then the following article contains similar information: projecteuclid.org/euclid.rmjm/1250129443 –  Matthew Daws Jun 15 '11 at 20:57

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