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I am interested in the general question of

When is a map between algebra of "Clifford-valued continuous functions" homomorphism?

As a starter, I would like to first understand the case for finite-dimensional vector space. Let $V$ be a finite-dimensional real vector space. $Cliff(V\oplus V, Q)$ be the Clifford algebra on $V\oplus V$ equipped with the Quadratic Form Q. (For definition of Clifford algebra, one may refer to this MO post below: José Figueroa-O'Farrill (, Clifford algebra as an adjunction?, Clifford algebra as an adjunction? (version: 2009-12-03))

Let $B$ be $C_0(V\oplus V, Cliff(V\oplus V,Q))$, the algebra of all continuous functions from $V \oplus V$ to the Clifford algebra of $V \oplus V$ which vanish at infinity. (this definition would have to be changed when $V$ is infinite-dimensional.)

I am wondering when would a linear map $\phi \colon B \to B$ be a homomorphism. I understand the basic requirement for a linear map to be algebra homomorphism is that $\phi(a)\phi(b)=\phi(ab)$ for all elements $a, b \in B$. However, the involvement of $Clifford(V \oplus V, Q)$ here makes me feel like we may need more condition, because there is a universal property requirement for homomorphism of Clifford algebras.

I am confused about the following: 1. Is this Clifford condition needed for determination homomorphism of $B$?
2. If so, how do we check this in general? Can we check it on a "canonical set" of functions and claim that it is homomorphism on the whole algebra?
3. As an example, I am wondering why the Remark on p.12 of this paper is true:,%20Kasparov,%20Trout%20-%201998%20-%20A%20Bott%20periodicity%20theorem%20for%20infinite%20dimensional%20Euclidean%20space.pdf

Thanks a lot!!

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