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I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are notes out there on this). When $A=\mathbb{C}$, one knows that if $(S,\mu)$ is a measure space and $f:S\rightarrow H$ is a Bochner-integrable function with values in $H$, then the identity

$$\langle x,\,\int_S f(s)\,d\mu(s)\rangle = \int_S\langle x,\, f(s)\rangle\,d\mu(s),$$

is satisfied for all $x\in H$.

Does there exist an integration theory for more general $A$ that still has this property? That is, if $(S,\mu)$ is a measure space and $f:S\rightarrow M$ is an integrable function in the sense of this theory, then is it true that

$$\langle x,\,\int_S f(s)\,d\mu(s)\rangle = \int_S\langle x,\, f(s)\rangle\,d\mu(s),$$

for all $x\in M$?

Thanks.

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  • $\begingroup$ What is the difference between the first and the second statement? Since the $A$-module structure does not appear in the statement, it seems obviously true. $\endgroup$
    – user1688
    Commented Dec 6, 2017 at 6:00
  • $\begingroup$ But the inner product in the second equality takes values in $A$, right? $\endgroup$
    – geometricK
    Commented Dec 6, 2017 at 6:33
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    $\begingroup$ The usual Bochner integral has the property that it commutes with every continuous linear map. $\endgroup$
    – user1688
    Commented Dec 6, 2017 at 8:25
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    $\begingroup$ Yes, but I don't know whether the direct analogue of the usual Bochner integral makes sense for Hilbert module-valued functions. I can see that uniqueness is guaranteed once existence is proved. Perhaps it would be more accurate to refer to this as a Gelfand-Pettis integral rather than a Bochner integral... $\endgroup$
    – geometricK
    Commented Dec 6, 2017 at 15:58
  • $\begingroup$ Bochner integrals exist for functions with values in any locally convex space, see arxiv.org/pdf/1403.3207.pdf $\endgroup$
    – user1688
    Commented Dec 7, 2017 at 7:24

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