I've discussed a problem with good mathematicians here. We consider again a non-unital $C^\ast$-algebra.
Let's consider the following $C^\ast$-algebra. That is, the $C^\ast$-algebra $\mathcal{A}$ with norm $\|\cdot\|$. Let $\tilde{\mathcal{A}}=\mathcal{A}\oplus \mathbb{C}$ as a vector space. We endow it with multiplication and involution,
$$(a,\lambda)\cdot (b,\mu):=(ab+\lambda b+\mu a,\lambda \mu)$$ and $$(a,\lambda)^\ast:=(a^\ast,\bar{\lambda})$$
I already did some proofs including proof of homomorphism and norm etc. so we will take that for granted.
Now take $x\in\tilde{\mathcal{A}}$ then we consider the linear map $\tilde{L}_x:\tilde{\mathcal{A}}\rightarrow \tilde{\mathcal{A}}$ by $y\mapsto xy$ restricts to a map, $L_x:\mathcal{A}\rightarrow \mathcal{A}$ which is bounded with $\|L_x\|_\infty \leq \|x\|_1$. This is already proven. Notice that we know the one-norm is given by $x=(a,\lambda)\in \tilde{\mathcal{A}}$ by $\|x\|_1=\|a\|+|\lambda|$ and of course $\|\cdot\|_\infty$ denote the operator norm.
Let's consider the following Banach space, call it $\mathscr{X}:=\mathcal{A}\oplus \mathbb{C}$ with the following norm. $$\|(a,\lambda)\|_{\max}=\max\{\|a\|,|\lambda|\}.$$ Set $x=(a,\lambda)$ and let $x\in \tilde{\mathcal{A}}$, then I define the following: $p(x):\mathscr{X}\rightarrow \mathscr{X}$ by using a matrix given below, i.e. $$p(x)=\begin{pmatrix} L_x & 0 \\ 0 & \lambda \\ \end{pmatrix}$$
My questions:
- Can we verify that $p(x)\in\mathbb{B}(\mathscr{X})$ and further show $\|p(x)\|_\infty=\max\{\|L_x\|_\infty,|\lambda|\}$?
- Is $p$ a unital, injective homomorphism of algebras?
Here $\mathbb{\mathscr{X}}$ is bounded operators on a Banach space $\mathscr{X}$. For my second question I know the idea of homomorphism but now injective homomorphism? Any suggestion would be helpful for me! If there are more conditions to show homomorphism, I would like to see how one of them can be done then I will do the other conditions.
Further let's define a new norm on $\tilde{\mathcal{A}}$ by setting $\|(a,\lambda)\|_\sim=\|p(a,\lambda)\|_\infty$.
My third and last question.
- Is it possible to show the norm $\|\cdot \|_\infty$ restricts to the original $C^\ast$-norm on $\mathcal{A}\subset\tilde{\mathcal{A}}$?
I know that there is a lot of questions and I hope someone could help me to understand this way more better than I already do. I hope someone can give me an detailed answers/sketches so I can understand this and fill details. Thanks in advance. If I get a great answer I will mark it accepted.