A 'Fock'-type construction on a $C^*$-algebra

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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.

This post imported from StackExchange MathOverflow at 2018-01-20 17:45 (UTC), posted by SE-user Bedovlat
retagged Jan 20, 2018
I've never seen this in the literature.

This post imported from StackExchange MathOverflow at 2018-01-20 17:45 (UTC), posted by SE-user Nik Weaver
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 $A\to A\otimes A$.

This post imported from StackExchange MathOverflow at 2018-01-20 17:45 (UTC), posted by SE-user Bedovlat
Wow, comment disappeared? Now my comment looks weird :-/

This post imported from StackExchange MathOverflow at 2018-01-20 17:45 (UTC), posted by SE-user Bedovlat

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