Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the Stone-Cech compactification $\beta(X)$, can be viewed as a topological left semigroup whose structure is compatible with that if $X$. My question is: what about $ \text{Spec}(C_b(X))$? Can it be viewed as a (topological/algebraic) group/semigroup whose structure is compatible with that of $X$?
My attempt is as follows: Let $\mu \colon X \times X \to X$ be the multiplication map. Then we have an induced map $\tilde{\mu} \colon \text{Spec}(C_b(X \times X)) \to \text{Spec}(C_b(X))$. But from [1] we know that $C_b(X \times X) \cong C_b(X) \hat{\otimes}_\epsilon C_b(X)$ (where $\hat{\otimes}_\epsilon$ is the injective tensor product), and from [2] its spectrum need not equal to $\text{Spec}(C_b(X)) \times \text{Spec}(C_b(X))$. Yet we have an injective map $\iota \colon C_b(X) \otimes C_b(X) \to C_b(X) \hat{\otimes}_\epsilon C_b(X)$, which gives us a map $\tilde{\iota} \colon \text{Spec}(C_b(X \times X) \to \text{Spec}(C_b(X)) \times \text{Spec}(C_b(X))$, whose image is dense. But can we extend $\tilde{\mu}$ along $\tilde{\iota}$ to get a map $\text{Spec}(C_b(X)) \times \text{Spec}(C_b(X)) \to \text{Spec}(C_b(X))$?
Is there an alternative way to give $\text{Spec}(C_b(X))$ a structure of a group/semigroup?
Sources:
[1] https://math.stackexchange.com/questions/4392116/ring-of-continuous-functions-of-product-space
[2] https://math.stackexchange.com/questions/4733187/spectrum-of-an-injective-tensor-product
