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In an article I'm revising, I spend some time giving a self-contained proof of the following result

Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary representations $G \to {\mathcal U}(H_i)$, each of which are continuous with respect to the WOT on ${\mathcal B}(H_i)$. Let $H$ be the Hilbert space direct sum of the family $(H_i)$ and let $\pi$ be the corresponding direct product of the family $\pi_i$. Then $\pi: G \to {\mathcal U}(H)$ is continuous with respect to the WOT on ${\mathcal B}(H)$.

The referee has indicated that this is standard knowledge. I agree that the proof is routine, but I would feel happier if I could provide a reference from the literature.

I don't need the earliest reference, even if such were possible; ideally this would be in a book on harmonic analysis or ${\rm C}^*$-algebras. In fact, for the intended applications we can assume $G$ is locally compact Hausdorff.

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Perhaps 3.4.13 in Kowalsky, An introduction to the representation theory of groups

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    $\begingroup$ Why perhaps? What does it say? $\endgroup$
    – Will Sawin
    Commented Jul 19, 2015 at 18:57
  • $\begingroup$ If you mean this book of Emmanuel Kowalski books.google.co.uk/books?id=sBdnBAAAQBAJ then I cannot see 3.4.13 in preview, and in the preliminary version which Kowalski has on his webpage, 3.4.13 appears to be about something different $\endgroup$
    – Yemon Choi
    Commented Jul 19, 2015 at 21:00
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    $\begingroup$ It's 3.4.12 (3) in Kowalski preliminary version $\endgroup$
    – user75485
    Commented Jul 19, 2015 at 21:31
  • $\begingroup$ Aha! this is the kind of thing I wanted, thanks. If you don't mind, I'll wait a day or two in case someone finds a reference in an older, "more established" book $\endgroup$
    – Yemon Choi
    Commented Jul 19, 2015 at 21:53

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