elements in $C^*$ algebra
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0
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If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$?
My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
operator-theory operator-algebras c-star-algebras
asked Nov 17 at 17:11
mathrookie
724512
add a comment |
up vote
0
down vote
favorite
If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$?
My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
operator-theory operator-algebras c-star-algebras
asked Nov 17 at 17:11
mathrookie
724512
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$?
My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
operator-theory operator-algebras c-star-algebras
asked Nov 17 at 17:11
mathrookie
724512
If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$?
My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
operator-theory operator-algebras c-star-algebras
operator-theory operator-algebras c-star-algebras
asked Nov 17 at 17:11
mathrookie
724512
asked Nov 17 at 17:11
mathrookie
724512
asked Nov 17 at 17:11
mathrookie
724512
asked Nov 17 at 17:11
mathrookie
724512
asked Nov 17 at 17:11
mathrookie
724512
724512
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Let's first assume that $A$ is unital. Then let's define, for $x in A$
begin{equation}
exp(x) = lim_{N rightarrow infty} sum_{n=0}^{N} x^{n}/n!
end{equation}
The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence
begin{equation}
a_{N} = sum_{n=0}^{N} x^{n}/n!
end{equation}
is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)
answered Nov 17 at 17:29
Peter
1,269321
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
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
Let's first assume that $A$ is unital. Then let's define, for $x in A$
begin{equation}
exp(x) = lim_{N rightarrow infty} sum_{n=0}^{N} x^{n}/n!
end{equation}
The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence
begin{equation}
a_{N} = sum_{n=0}^{N} x^{n}/n!
end{equation}
is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)
answered Nov 17 at 17:29
Peter
1,269321
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
add a comment |
up vote
1
down vote
accepted
Let's first assume that $A$ is unital. Then let's define, for $x in A$
begin{equation}
exp(x) = lim_{N rightarrow infty} sum_{n=0}^{N} x^{n}/n!
end{equation}
The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence
begin{equation}
a_{N} = sum_{n=0}^{N} x^{n}/n!
end{equation}
is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)
answered Nov 17 at 17:29
Peter
1,269321
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let's first assume that $A$ is unital. Then let's define, for $x in A$
begin{equation}
exp(x) = lim_{N rightarrow infty} sum_{n=0}^{N} x^{n}/n!
end{equation}
The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence
begin{equation}
a_{N} = sum_{n=0}^{N} x^{n}/n!
end{equation}
is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)
answered Nov 17 at 17:29
Peter
1,269321
Let's first assume that $A$ is unital. Then let's define, for $x in A$
begin{equation}
exp(x) = lim_{N rightarrow infty} sum_{n=0}^{N} x^{n}/n!
end{equation}
The question is now if this limit converges to anything in our C$^{*}$-algebra. The answer to this question is yes, this can be shown by proving that the sequence
begin{equation}
a_{N} = sum_{n=0}^{N} x^{n}/n!
end{equation}
is Cauchy, which I'll leave as an exercise. (Hint: the norm is submultiplicative.)
answered Nov 17 at 17:29
Peter
1,269321
answered Nov 17 at 17:29
Peter
1,269321
answered Nov 17 at 17:29
Peter
1,269321
answered Nov 17 at 17:29
Peter
1,269321
1,269321
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
add a comment |
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
Thanks,Imade a mistake.I thought $exp^x$ is also in A even if $A$ is non-unital.Besides the polynomial of $x,x^*$,do there exist special elements which lie in $A$ when $A$ is non-unital?If $A$ is non-unital,can we take some function $f$ act on x such $f(x) in A$?
– mathrookie
Nov 17 at 18:19
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I'm not sure how te define $exp(x)$ if $A$ is non-unital. For example, what would you propose $exp(0)$ to be? For more general functions, you might want to look into the functional calculus. Any book about C$^{*}$-algebras should treat this at some point.
– Peter
Nov 17 at 19:02
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
I've found the functional calculus for the unital $C^*$ algebras,would you mind recommending some reference books about functional calculus. for non-unital $C^*$ algebras?
– mathrookie
Nov 18 at 6:06
add a comment |
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