the spectrum of self-adjoint element
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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 at 16:37
mathrookie
787512
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up vote
0
down vote
favorite
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 at 16:37
mathrookie
787512
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 16:37
mathrookie
787512
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 at 16:37
mathrookie
787512
asked Dec 4 at 16:37
mathrookie
787512
asked Dec 4 at 16:37
mathrookie
787512
asked Dec 4 at 16:37
mathrookie
787512
asked Dec 4 at 16:37
mathrookie
787512
787512
add a comment |
add a comment |
1 Answer
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up vote
3
down vote
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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 at 16:47
Robert Israel
316k23206457
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
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 at 16:47
Robert Israel
316k23206457
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
add a comment |
up vote
3
down vote
accepted
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 at 16:47
Robert Israel
316k23206457
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
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 at 16:47
Robert Israel
316k23206457
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 at 16:47
Robert Israel
316k23206457
answered Dec 4 at 16:47
Robert Israel
316k23206457
answered Dec 4 at 16:47
Robert Israel
316k23206457
answered Dec 4 at 16:47
Robert Israel
316k23206457
316k23206457
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
add a comment |
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
2
2
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
I gotta learn to type faster...
– David C. Ullrich
Dec 4 at 16:53
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
haha.....。。。。。。
– mathrookie
Dec 4 at 16:58
add a comment |
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