Results 1 to 2 of 2

Math Help - Banach algebra example

  1. #1
    Member Mauritzvdworm's Avatar
    Joined
    Aug 2009
    From
    Pretoria
    Posts
    122

    Banach algebra example

    Construct a banach algebra \mathcal{B} and a unital subalgebra \mathcal{A} such that \sigma_{\mathcal{A}}(x)\neq\sigma_{\mathcal{B}}(x) for some x\in\mathcal{A}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Nov 2009
    Posts
    3

    different spectra in diff alg.

    Let $A = A(D)$, the disk algebra. That is $D = \{z \in C : |z|\le 1\}$ and $A$ be the algebra of all functions $f: D \to C$ which are continuous on $D$ and analytic on the interior of $D$. Let $\Gamma = \{z \in C : |z|= 1\}$ and $B = C(\Gamma)$, the algebra of all continuous functions on $\Gamma$. Then with the sup norm, both of them are Banach Algebras. Also the elements of $A(D)$ can be realized as elements of $C(\Gamma)$ by restricting them to $\Gamma$. Thus, in this way, we can say that $A(D)$ is a Banach subalgebra of $C(\Gamma)$. Now let $f : D \to C$ be defined by $f(z) = z$, then the spectrum of $f$ computed in $A$ is $D$ and computed in $B$ is $\Gamma$. I hope this answers your question.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Spectral radius (Banach algebra)
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 4th 2012, 10:44 PM
  2. product rule in banach algebra
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: June 23rd 2011, 12:14 AM
  3. Banach Algebra
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 4th 2010, 09:53 AM
  4. Banach Algebra
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: August 3rd 2010, 09:09 PM
  5. Banach Algebra question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 4th 2010, 03:16 AM

Search Tags


/mathhelpforum @mathhelpforum