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.