functional calculus for non-unital $C^*$ algebras
up vote
1
down vote
favorite
If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(sigma_{A}(a))$ to $C^*(a)$.
I have a question:what is the set of $C_0(sigma_{A}(a))$?Is it the set of all continuous functions $f$ defined on $sigma_{A}(a) $ which vanish at infinity? or besides, we need $f(0)=0$? see continuous functional calculus for nonunital $c^*$-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 18 at 11:20
mathrookie
722512
add a comment |
up vote
1
down vote
favorite
If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(sigma_{A}(a))$ to $C^*(a)$.
I have a question:what is the set of $C_0(sigma_{A}(a))$?Is it the set of all continuous functions $f$ defined on $sigma_{A}(a) $ which vanish at infinity? or besides, we need $f(0)=0$? see continuous functional calculus for nonunital $c^*$-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 18 at 11:20
mathrookie
722512
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(sigma_{A}(a))$ to $C^*(a)$.
I have a question:what is the set of $C_0(sigma_{A}(a))$?Is it the set of all continuous functions $f$ defined on $sigma_{A}(a) $ which vanish at infinity? or besides, we need $f(0)=0$? see continuous functional calculus for nonunital $c^*$-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 18 at 11:20
mathrookie
722512
If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(sigma_{A}(a))$ to $C^*(a)$.
I have a question:what is the set of $C_0(sigma_{A}(a))$?Is it the set of all continuous functions $f$ defined on $sigma_{A}(a) $ which vanish at infinity? or besides, we need $f(0)=0$? see continuous functional calculus for nonunital $c^*$-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked Nov 18 at 11:20
mathrookie
722512
asked Nov 18 at 11:20
mathrookie
722512
asked Nov 18 at 11:20
mathrookie
722512
asked Nov 18 at 11:20
mathrookie
722512
asked Nov 18 at 11:20
mathrookie
722512
722512
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
No, it's just bad notation. When $A$ is not unital and you consider the C$^*$-algebra generated by $a$ in $A$, it cannot contain the unit of $A$. So when $a$ is normal C$^*(a)$ is the closure of the span of the words in $a$ and $a^*$, without constants.
The spectrum $sigma(A)$ is compact, so $C(sigma(A)$ is unital. For the identification $C^*(a)simeq C_0(sigma(a))$, one is taking the continuous functions where $f(0)=0$. That's what the subscript means. It's bad notation because it is not the set of functions that vanish at infinity.
answered Nov 18 at 12:09
Martin Argerami
122k1174172
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
No, it's just bad notation. When $A$ is not unital and you consider the C$^*$-algebra generated by $a$ in $A$, it cannot contain the unit of $A$. So when $a$ is normal C$^*(a)$ is the closure of the span of the words in $a$ and $a^*$, without constants.
The spectrum $sigma(A)$ is compact, so $C(sigma(A)$ is unital. For the identification $C^*(a)simeq C_0(sigma(a))$, one is taking the continuous functions where $f(0)=0$. That's what the subscript means. It's bad notation because it is not the set of functions that vanish at infinity.
answered Nov 18 at 12:09
Martin Argerami
122k1174172
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
add a comment |
up vote
1
down vote
accepted
No, it's just bad notation. When $A$ is not unital and you consider the C$^*$-algebra generated by $a$ in $A$, it cannot contain the unit of $A$. So when $a$ is normal C$^*(a)$ is the closure of the span of the words in $a$ and $a^*$, without constants.
The spectrum $sigma(A)$ is compact, so $C(sigma(A)$ is unital. For the identification $C^*(a)simeq C_0(sigma(a))$, one is taking the continuous functions where $f(0)=0$. That's what the subscript means. It's bad notation because it is not the set of functions that vanish at infinity.
answered Nov 18 at 12:09
Martin Argerami
122k1174172
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
No, it's just bad notation. When $A$ is not unital and you consider the C$^*$-algebra generated by $a$ in $A$, it cannot contain the unit of $A$. So when $a$ is normal C$^*(a)$ is the closure of the span of the words in $a$ and $a^*$, without constants.
The spectrum $sigma(A)$ is compact, so $C(sigma(A)$ is unital. For the identification $C^*(a)simeq C_0(sigma(a))$, one is taking the continuous functions where $f(0)=0$. That's what the subscript means. It's bad notation because it is not the set of functions that vanish at infinity.
answered Nov 18 at 12:09
Martin Argerami
122k1174172
No, it's just bad notation. When $A$ is not unital and you consider the C$^*$-algebra generated by $a$ in $A$, it cannot contain the unit of $A$. So when $a$ is normal C$^*(a)$ is the closure of the span of the words in $a$ and $a^*$, without constants.
The spectrum $sigma(A)$ is compact, so $C(sigma(A)$ is unital. For the identification $C^*(a)simeq C_0(sigma(a))$, one is taking the continuous functions where $f(0)=0$. That's what the subscript means. It's bad notation because it is not the set of functions that vanish at infinity.
answered Nov 18 at 12:09
Martin Argerami
122k1174172
answered Nov 18 at 12:09
Martin Argerami
122k1174172
answered Nov 18 at 12:09
Martin Argerami
122k1174172
answered Nov 18 at 12:09
Martin Argerami
122k1174172
122k1174172
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
add a comment |
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
So $C_0(sigma(a))$ is the set of continuous functions $f$ defined on $sigma(a)$ such that $f(0)=0$?
– mathrookie
Nov 18 at 12:26
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
Not usually, but it is in what you wrote. It's a really bad choice of notation.
– Martin Argerami
Nov 18 at 12:54
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
what is the standard notaion when we talk about the functional calculus for the non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 13:05
Standard notation for what ?
– André S.
Nov 18 at 15:19
Standard notation for what ?
– André S.
Nov 18 at 15:19
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003416%2ffunctional-calculus-for-non-unital-c-algebras%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
