Let $\mu$ and $\nu$ be two Radon measures on a locally compact group $G$. For every irreducible unitary representation $\pi$ of $G$ and vectors $u$ and $v$ from the corresponding Hilbert space $H_\pi$ denote by $\pi_{u,v}$ the respective matrix coefficient, i.e. the function from $G$ to $\mathbb{C}$ that maps $g$ to $\langle u,\pi(g)v\rangle$.

Assume that $\int_G\pi_{u,v}(g)\mu(dg)=\int_G\pi_{u,v}(g)\nu(dg)$, i.e. the Fourier transforms of the meausures coincide. Can we conclude that $\mu=\nu$?

My idea to proove this is the following: The matrix coefficients of irreducible representations are closed under multiplication (since the tensor product of two irreducible representations is again irreducible) and complex conjugates (consider adjoint operators). Also the trivial representation always exists and by the Gelfand-Raikov theorem, the matrix coefficients separate points. Thus by a (more or less standard) application of the Stone-Weierstraß theorem, we can conclude that the measures coincide on compact sets and hence on all Borel sets (because of inner regularity).

My questions are:
(i) Is the reasoning I have sketched correct?
(ii) Does this result have a name?
(iii) Where do I find an explicit formulation of this result in the standard literature? E.g. it seems not to be mentioned in Folland's book on abstract harmonic analysis.

I am thankful for any kind of advice.

  • 1
    $\begingroup$ It is certainly not true that the tensor product of irreducible $G$-reps is irreducible as a $G$-rep! (This can be seen in the smallest non-abelian group, for instance.) $\endgroup$ – Yemon Choi Jun 8 '14 at 23:38
  • $\begingroup$ Two further questions. (1) When you say "Radon measure" do you assume that the measure is finite? (2) Do you only want this result for Type I groups, or for more general locally compact groups? $\endgroup$ – Yemon Choi Jun 8 '14 at 23:46
  • $\begingroup$ It surprises me, that you say that the tensor product is not necessary irreducible, if the original representations are. This precise statement is given in Folland's book (Thm. 7.20 on page 217). Furthermore, I do not assume that the Radon measure is finite (even though I am actually only interested in probability measures on Polish groups). Unfortunately I have not heard the term "Type I" before, but I have looked it up and I think the result should hold for arbitrary locally compact groups. $\endgroup$ – bla Jun 10 '14 at 18:40
  • $\begingroup$ I do not have a copy of that book, but hopefully it does not literally assert that tensor products of irreducibles are irreducible. As @YemonChoi noted, for non-abelian groups this is rarely the case. $\endgroup$ – paul garrett Jun 10 '14 at 19:08
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    $\begingroup$ @paulgarrett Given the general good standard of Folland's book, I assume he means that the tensor product of a G-irrep with an H-irrep is a $(G\times H)$-irrep. $\endgroup$ – Yemon Choi Jun 10 '14 at 20:50

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