Matrix with a bounded mapping as an entry is bounded

[HeinzBor]

Homework Statement

C* algebras, showing that a matrix is bounded, and a homomorphism

Relevant Equations

Definitions of boundedness and homomorphisms.

In a previous exercise I have shown that for a $$C^{*} algebra \ \mathcal{A}$$ which may or may not have a unit the map $$L_{x} : \mathcal{A} \rightarrow \mathcal{A}, \ L_{x}(y)=xy$$ is bounded. I.e. $$||L_{x}||_{\infty} \leq ||x||_{1}$$, $$x=(a, \lambda) \in \mathcal{\hat{A}} = \mathcal{A} \oplus \mathbb{C} \ with \ ||x||_{1}:= ||a|| + |\lambda| \ and \ ||.||_{\infty}$$ is the usual operator norm.For this exercise, consider $$X = \mathcal{A} \oplus \mathbb{C}$$ with $X$ being a Banach space given by the following norm $$||(a, \lambda)||_{max}:= max \{ ||a||, |\lambda| \}$$.

For $$x = (a,\lambda) \in \mathcal{\hat{A}} \ define \ \rho(x): X \rightarrow X, \ \rho(x) :=
\begin{pmatrix}
L_{x} & 0 \\
0 & \lambda
\end{pmatrix}
$$.

Alright... Then I must show that $$\rho(x) \in B(X)$$ and that $$||\rho(x)||_{\infty} = max \{ ||L_{x}||_{\infty}, |\lambda| \}$$. also show that $$\rho$$ is a homomorphism of algebras.

I think this exercise should be a straightforward calculation honestly, but it has been bothering me for a while since I am not really sure on how I should work with a matrix in this regard as opposed to the usual way of working with functions between two spaces. So I think the reason why I can't get started is first of all how do I take the norm of this matrix and if I know that I can at least write out $$||\rho(x)||_{\infty}$$ and then probably try to get some calculation started, but until then I am stuck..

 

[HeinzBor]

So According to the definition which I found $$||\rho(x)||_{\infty} $$ should be the maximum value of $$1: L_{x} + |0|$$ vs $$2: |0| + |\lambda|$$.

So
$$||\rho(x)||_{\infty} = |\lambda| \ or ||\rho(x)||_{\infty} = |L_{x}| $$ depending on whether lambda or L_x is the largest in terms of absolute value..

 

[HeinzBor]

I realized one can show boundedness of $$\rho$$ by showing that $$||\rho(x)||_{\infty} = max \{ ||L_{x}||_{\infty}, \lambda \},$$ since both $$|\lambda| \ and \ ||L_{x}||_{\infty}$$ are finite