Lemma 5.1.5 from Garth Dales, Introduction to Banach algebra
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The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5
Lemma: Let $(A, |.|)$ be a unital Banach algebra, let $ain A$ and let $epsilon>0.$ then there is a norm $newcommand{vertiii}[1]{{leftvertkern-0.25exleftvertkern-0.25exleftvert #1 rightvertkern-0.25exrightvertkern-0.25exrightvert}}vertiii{.}$ on $A$ such that $vertiii.$ is equivalent to $|.|,$ $vertiii{e}=1$ and $vertiii{a}leq nu(a)+epsilon $ where $$nu(a)=lim_{ntoinfty}|a^n|^{frac{1}{n}}$$
Proof: If we let $b=frac {a}{nu(a)+epsilon}$ the $S={b^n: nin mathbb{Z^+}}$ is bounded. For $cin A$ let $$p(c)=sup{|sc|: sin S}, vertiii{c}=sup{p(cd): din A, p(d)leq 1}$$
It is easy to prove that $S$ is bounded but I couldn't check $vertiii{a}leq nu(a)+epsilon $
Any piece of advice would be much appreciated
normed-spaces banach-algebras
edited Dec 1 at 13:14
user10354138
6,570623
asked Dec 1 at 12:55
user62498
1,889613
add a comment |
up vote
0
down vote
favorite
The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5
Lemma: Let $(A, |.|)$ be a unital Banach algebra, let $ain A$ and let $epsilon>0.$ then there is a norm $newcommand{vertiii}[1]{{leftvertkern-0.25exleftvertkern-0.25exleftvert #1 rightvertkern-0.25exrightvertkern-0.25exrightvert}}vertiii{.}$ on $A$ such that $vertiii.$ is equivalent to $|.|,$ $vertiii{e}=1$ and $vertiii{a}leq nu(a)+epsilon $ where $$nu(a)=lim_{ntoinfty}|a^n|^{frac{1}{n}}$$
Proof: If we let $b=frac {a}{nu(a)+epsilon}$ the $S={b^n: nin mathbb{Z^+}}$ is bounded. For $cin A$ let $$p(c)=sup{|sc|: sin S}, vertiii{c}=sup{p(cd): din A, p(d)leq 1}$$
It is easy to prove that $S$ is bounded but I couldn't check $vertiii{a}leq nu(a)+epsilon $
Any piece of advice would be much appreciated
normed-spaces banach-algebras
edited Dec 1 at 13:14
user10354138
6,570623
asked Dec 1 at 12:55
user62498
1,889613
Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5
Lemma: Let $(A, |.|)$ be a unital Banach algebra, let $ain A$ and let $epsilon>0.$ then there is a norm $newcommand{vertiii}[1]{{leftvertkern-0.25exleftvertkern-0.25exleftvert #1 rightvertkern-0.25exrightvertkern-0.25exrightvert}}vertiii{.}$ on $A$ such that $vertiii.$ is equivalent to $|.|,$ $vertiii{e}=1$ and $vertiii{a}leq nu(a)+epsilon $ where $$nu(a)=lim_{ntoinfty}|a^n|^{frac{1}{n}}$$
Proof: If we let $b=frac {a}{nu(a)+epsilon}$ the $S={b^n: nin mathbb{Z^+}}$ is bounded. For $cin A$ let $$p(c)=sup{|sc|: sin S}, vertiii{c}=sup{p(cd): din A, p(d)leq 1}$$
It is easy to prove that $S$ is bounded but I couldn't check $vertiii{a}leq nu(a)+epsilon $
Any piece of advice would be much appreciated
normed-spaces banach-algebras
edited Dec 1 at 13:14
user10354138
6,570623
asked Dec 1 at 12:55
user62498
1,889613
The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5
Lemma: Let $(A, |.|)$ be a unital Banach algebra, let $ain A$ and let $epsilon>0.$ then there is a norm $newcommand{vertiii}[1]{{leftvertkern-0.25exleftvertkern-0.25exleftvert #1 rightvertkern-0.25exrightvertkern-0.25exrightvert}}vertiii{.}$ on $A$ such that $vertiii.$ is equivalent to $|.|,$ $vertiii{e}=1$ and $vertiii{a}leq nu(a)+epsilon $ where $$nu(a)=lim_{ntoinfty}|a^n|^{frac{1}{n}}$$
Proof: If we let $b=frac {a}{nu(a)+epsilon}$ the $S={b^n: nin mathbb{Z^+}}$ is bounded. For $cin A$ let $$p(c)=sup{|sc|: sin S}, vertiii{c}=sup{p(cd): din A, p(d)leq 1}$$
It is easy to prove that $S$ is bounded but I couldn't check $vertiii{a}leq nu(a)+epsilon $
Any piece of advice would be much appreciated
normed-spaces banach-algebras
normed-spaces banach-algebras
edited Dec 1 at 13:14
user10354138
6,570623
asked Dec 1 at 12:55
user62498
1,889613
edited Dec 1 at 13:14
user10354138
6,570623
asked Dec 1 at 12:55
user62498
1,889613
edited Dec 1 at 13:14
user10354138
6,570623
edited Dec 1 at 13:14
user10354138
6,570623
edited Dec 1 at 13:14
user10354138
6,570623
6,570623
asked Dec 1 at 12:55
user62498
1,889613
asked Dec 1 at 12:55
user62498
1,889613
asked Dec 1 at 12:55
user62498
1,889613
1,889613
Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24
add a comment |
Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24
Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24
add a comment |
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Hint: show that the effect of multiplying by $a$ is at most multiplying by $nu(a)+epsilon$ on $p$ directly from definition of $p(a)$ (or more appropriately, $p(b)$).
– user10354138
Dec 1 at 13:10
@user10354138, please explain it a little more
– user62498
Dec 1 at 13:24