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In very rough terms, let $A$ be a complex unital $C^*$-algebra. Assume it nuclear for convenience, but it doesn't matter much. Consider the 'Fock'-type $C^*$-algebra (don't know a better name for it) $$ \mathbb{C}\bigoplus A\bigoplus A\otimes A\bigoplus\ldots $$ This can be thought of as the $C^*$-algebra of continuous sections in the 'power bundle' $n\mapsto A^{\otimes n}$ vanishing at infinity. It can be made unital by considering the 1-point compactification etc. etc. If $A$ is simple, this should be the Dauns-Hofmann representation of the resulting algebra, I think. This is a graded $C^*$-algebra, but I am not sure what that gives.

The corresponding construction for von Neumann algebras is a bit easier and is related to the Fock representations in QFT.

Question: Has this thing been studied in the literature and does it have a proper name?

Thank you.

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    $\begingroup$ I've never seen this in the literature. $\endgroup$
    – Nik Weaver
    Jan 6, 2018 at 14:14
  • $\begingroup$ Well, right, I would rather reformulate it as follows; the infinite convex linear combination of powers of a fixed faithful state on $A$ is a faithful state on the above algebra. But I don't see this as a subrepresentation of the universal representation, since, for instance, $A\otimes A$ is not the image of a representation of $A$. There is in general no surjective homomorphism $A\to A\otimes A$. $\endgroup$
    – Bedovlat
    Jan 6, 2018 at 16:30
  • $\begingroup$ Wow, comment disappeared? Now my comment looks weird :-/ $\endgroup$
    – Bedovlat
    Jan 6, 2018 at 16:31

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