the spectrum of self-adjoint element
$begingroup$
If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $sigma_A(x)subset mathbb{R}$,my question is :Is the following form possible for $sigma_A(x)$?1.$sigma_A(x)$ be unions of intervals in $mathbb{R}$.
2.$sigma_A(x)$ is the set of isolated points.
Does there exist other possibilites?Can anyone show me some examples?Thanks
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
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add a comment |
$begingroup$
If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $sigma_A(x)subset mathbb{R}$,my question is :Is the following form possible for $sigma_A(x)$?1.$sigma_A(x)$ be unions of intervals in $mathbb{R}$.
2.$sigma_A(x)$ is the set of isolated points.
Does there exist other possibilites?Can anyone show me some examples?Thanks
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
$endgroup$
add a comment |
$begingroup$
If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $sigma_A(x)subset mathbb{R}$,my question is :Is the following form possible for $sigma_A(x)$?1.$sigma_A(x)$ be unions of intervals in $mathbb{R}$.
2.$sigma_A(x)$ is the set of isolated points.
Does there exist other possibilites?Can anyone show me some examples?Thanks
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
$endgroup$
If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $sigma_A(x)subset mathbb{R}$,my question is :Is the following form possible for $sigma_A(x)$?1.$sigma_A(x)$ be unions of intervals in $mathbb{R}$.
2.$sigma_A(x)$ is the set of isolated points.
Does there exist other possibilites?Can anyone show me some examples?Thanks
operator-theory operator-algebras c-star-algebras
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
asked Dec 4 '18 at 16:37
mathrookiemathrookie
826512
826512
add a comment |
add a comment |
1 Answer
1
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The spectrum can be any compact subset of $mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
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2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
The spectrum can be any compact subset of $mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
$endgroup$
2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
add a comment |
$begingroup$
The spectrum can be any compact subset of $mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
$endgroup$
2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
add a comment |
$begingroup$
The spectrum can be any compact subset of $mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
$endgroup$
The spectrum can be any compact subset of $mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
answered Dec 4 '18 at 16:47
Robert IsraelRobert Israel
319k23208458
319k23208458
2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
add a comment |
2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
2
2
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
I gotta learn to type faster...
$endgroup$
– David C. Ullrich
Dec 4 '18 at 16:53
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
$begingroup$
haha.....。。。。。。
$endgroup$
– mathrookie
Dec 4 '18 at 16:58
add a comment |
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