Matrix with a bounded mapping as an entry is bounded

In summary, for the given exercise, it is shown that the map $L_x: \mathcal{A} \rightarrow \mathcal{A}$ is bounded, with a norm of $||L_x||_{\infty} \leq ||x||_1$. The Banach space $X = \mathcal{A} \oplus \mathbb{C}$ is introduced, with a norm defined as $||(a, \lambda)||_{max} = max \{ ||a||, |\lambda| \}$. The map $\rho(x): X \rightarrow X$ is defined as a matrix, and it is proven that $\rho(x)$ is a bounded operator with a norm of $||
  • #1
HeinzBor
39
1
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..
 
Last edited:
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  • #2
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..
 
  • #3
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
 

Related to Matrix with a bounded mapping as an entry is bounded

1. What is a matrix with a bounded mapping as an entry?

A matrix with a bounded mapping as an entry is a matrix where each element is associated with a function that maps a set of numbers to another set of numbers, and the range of this function is limited or bounded.

2. How is a matrix with a bounded mapping as an entry different from a regular matrix?

A regular matrix contains only numerical values as elements, while a matrix with a bounded mapping as an entry has functions as elements. This allows for more flexibility and complexity in representing data and solving mathematical problems.

3. What are the benefits of using a matrix with a bounded mapping as an entry?

Using a matrix with a bounded mapping as an entry allows for more accurate and precise representation of data, as well as more efficient solutions to mathematical problems. It also allows for the incorporation of non-numerical data, such as functions and equations, in a matrix format.

4. Can a matrix with a bounded mapping as an entry be used in any field of science?

Yes, a matrix with a bounded mapping as an entry can be used in various fields of science, including physics, engineering, economics, and computer science. It is a versatile tool that can be applied to a wide range of problems and data sets.

5. How can one determine if a matrix with a bounded mapping as an entry is bounded?

A matrix with a bounded mapping as an entry is considered bounded if the range of each function associated with its elements is limited or finite. This can be determined by analyzing the functions and their corresponding domains and ranges. If the ranges are finite, the matrix is considered bounded.

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