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This answer is concerned with the first question. The $C^\ast$-subalgebra $B$ you are considering is in fact a closed $^\ast$-ideal and hence given by $$\{f\in C[0,1]\mid f_C=0\}$$ for some closed subset $C\subseteq [0,1]$. Clearly all functions in $B$ vanish at $x$ when $g(x)=0$. Conversely, if $g(x)\neq 0$, then we have already found a function in $B$ ...

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Let $B$ be your algebra, which has the set $\{f_i:0\leq i<n\}$ as a basis. The identity element is $f_0$, which we can write simply $1$, and the element $f_1$ generates $B$ as an algebra, as $f_i=f_1^i$ for each $0\leq i<n$. Moreover, $f_1^n=1$, and one can easily see that $B$ is generated as an algebra by $f_1$, which I will writ more simply just $x$, ...

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Regarding your question on notation in Taylor's paper $C$ means the field of complex numbers and $\epsilon$ stands for the natural embedding of $C$ into unital algebra $A$. As for the deinition of relative homology you must understand that you can't transfer Mac Lane's definition word by word to the realm of topological modules for one simple reason - their ...

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It is perhaps easier to extend the linear functional $\tau$ to a positive linear functional $\tilde{\tau}$ on the unitization $\tilde{A}$. You know get a cyclic representation $(\pi, H, v)$ of $\tilde{A}$. Now restrict $\pi$ to $A$, and we get a representation of $A$. Now we claim that $v$ is a cyclic vector for $\pi\lvert_A$: Note that $\|\tau\| = ... 1 The range of the Volterra operator consists of all absolutely continuous functions vanishing at$0$. These are$L^2$-dense. This also implies the (seemingly) stronger statement you gave. 1 You should provide more information. If you mean that we view$C(X)$as a$C^*$-algebra and isomorphism means unital$^*$-isomorphism, then the answer is yes. In fact, any unital$^*$-homomorphism between two unital$C^*$-algebras is continuous. Even weaker, every unital algebra automorphism of$C(X)$is continuous. 1 When you suppose that$G$is amenable we have$L^1(G)$as an amenable Banach algebra. Now if any ideal of it has left or right bounded approximate identity then it has to be weakly complemented in$A\$ and vice versa. Also if ideal has two sided bounded approximate identity then it has to be amenable and vice versa.

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