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Albert harold
Albert harold
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Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)_{\cal U}$ we have $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0,$$ so$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$

When do we have that $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ?$$

In general, whenWhen can we "displace" an ultrafilter limit with another limit?

Thank you so much!

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)_{\cal U}$ we have $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0,$$ so$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$

When do we have that $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ?$$

In general, when can we "displace" an ultrafilter limit with another limit?

Thank you so much!

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)_{\cal U}$ we have $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0,$$ so$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$

When do we have that $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ?$$

When can we "displace" an ultrafilter limit with another limit?

Thank you so much!

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Andrés E. Caicedo
Andrés E. Caicedo
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when displace When can we "displace" an ultrafilter limit with otheranother limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$,and and suppose that for every $(a_i)\in (\cal A)_{\cal U}$ we have: $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$$$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0,$$ Soso$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$ WHEN WE HAVE $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ??????$$

When do we have that $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ?$$

In general, when we can displacewe "displace" an ultrafilter limit with another limit? THANK YOU SO MUUUUUUCH

Thank you so much!

when displace ultrafilter limit with other limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$,and for every $(a_i)\in (\cal A)_{\cal U}$ we have: $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$ So$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$ WHEN WE HAVE $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ??????$$

In general, when we can displace ultrafilter limit with another limit? THANK YOU SO MUUUUUUCH

When can we "displace" an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)_{\cal U}$ we have $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0,$$ so$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$

When do we have that $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ?$$

In general, when can we "displace" an ultrafilter limit with another limit?

Thank you so much!

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Albert harold
Albert harold
  • 565
  • 2
  • 9

when displace ultrafilter limit with other limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$,and for every $(a_i)\in (\cal A)_{\cal U}$ we have: $$\lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$ So$$\lim_{\cal U} \lim_{\alpha}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0.$$ WHEN WE HAVE $$\lim_{\alpha}\lim_{\cal U}\|a_i w_{\alpha}-\phi(a_i)w_{\alpha}\|=0 ??????$$

In general, when we can displace ultrafilter limit with another limit? THANK YOU SO MUUUUUUCH