2
$\begingroup$

I am reading through the text "Complete Normed Algebras" by F.F. Bonsall and John Duncan and I'm having some troubles understanding proposition 2 in chapter 1, section 11 (page 58) regarding the existence of a bounded left approximate identity for a normed algebra.

The definition I am using for bounded left approximate identity is:

Definition: Let $ A $ be a normed algebra. A bounded left approximate identity is a net $ \{ e(\lambda) \}_{\lambda \in \Lambda} $ such that $ e(\lambda)x \to x $ for every $ x \in A $ and there exists a $ K > 0 $ such that $ \| e(\lambda) \| \leq K $ for all $ \lambda \in \Lambda $.**

Proposition 2 on page 58 states:

Proposition 2: Let $ A $ contain a bounded set $ U $ such that, given $ x \in A $, $ \epsilon > 0 $ there exists $ u \in U $ with $ \| x - ux \| < \epsilon $. Then $ A $ has a bounded left approximate identity.

For $ \epsilon > 0 $, it is shown in the proof that for any finite subset $ F $ of $ A $ there exists a $ w \in W = \{ v \circ u : v, u \in U \} $ such that:

$$ \| x - wx \| < \epsilon $$

Where $ v \circ u := v + u - vu $ is the quasi-product of $ v $ and $ u $. The authors prove this by induction (which is fairly straightforward to follow).

The authors then state that "the result is then clear by consideration of the product directed set $ \mathbb{N} \times F(A) $ where $ F(A) $ denotes the directed set of all finite subsets of $ A $."

I am not sure what this means, perhaps because I am not that familiar with nets. Explicitly, what net becomes the bounded left approximate identity for $ X $? Any help would be greatly appreciated!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.