Non-invertible elements in an algebra which do not belong to the kernel of any character
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Is there a commutative unital algebra $A$ over $mathbb C$ such that for some $fin A$, $chi(f)not=0$ for every (non-zero)
homomorphism $chi:Atomathbb C$, but for which $f$ is not invertible? If $A$ is the algebra of rational functions in $mathbb C$,
then this holds, but in that case there isn't any such character at all. So I ask this question for the case where
$X(A)not=emptyset$ (the set of all non-zero homomorphisms into $mathbb C$).
The algebra of all holomorphic functions in $mathbb C$ is not such an example, and of course no Banach-algebra has such a `weird' property.
complex-analysis functions commutative-algebra
asked Nov 8 at 10:12
ray
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up vote
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down vote
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Is there a commutative unital algebra $A$ over $mathbb C$ such that for some $fin A$, $chi(f)not=0$ for every (non-zero)
homomorphism $chi:Atomathbb C$, but for which $f$ is not invertible? If $A$ is the algebra of rational functions in $mathbb C$,
then this holds, but in that case there isn't any such character at all. So I ask this question for the case where
$X(A)not=emptyset$ (the set of all non-zero homomorphisms into $mathbb C$).
The algebra of all holomorphic functions in $mathbb C$ is not such an example, and of course no Banach-algebra has such a `weird' property.
complex-analysis functions commutative-algebra
asked Nov 8 at 10:12
ray
629
This question has an open bounty worth +50
reputation from ray ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
1
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Is there a commutative unital algebra $A$ over $mathbb C$ such that for some $fin A$, $chi(f)not=0$ for every (non-zero)
homomorphism $chi:Atomathbb C$, but for which $f$ is not invertible? If $A$ is the algebra of rational functions in $mathbb C$,
then this holds, but in that case there isn't any such character at all. So I ask this question for the case where
$X(A)not=emptyset$ (the set of all non-zero homomorphisms into $mathbb C$).
The algebra of all holomorphic functions in $mathbb C$ is not such an example, and of course no Banach-algebra has such a `weird' property.
complex-analysis functions commutative-algebra
asked Nov 8 at 10:12
ray
629
Is there a commutative unital algebra $A$ over $mathbb C$ such that for some $fin A$, $chi(f)not=0$ for every (non-zero)
homomorphism $chi:Atomathbb C$, but for which $f$ is not invertible? If $A$ is the algebra of rational functions in $mathbb C$,
then this holds, but in that case there isn't any such character at all. So I ask this question for the case where
$X(A)not=emptyset$ (the set of all non-zero homomorphisms into $mathbb C$).
The algebra of all holomorphic functions in $mathbb C$ is not such an example, and of course no Banach-algebra has such a `weird' property.
complex-analysis functions commutative-algebra
complex-analysis functions commutative-algebra
asked Nov 8 at 10:12
ray
629
asked Nov 8 at 10:12
ray
629
asked Nov 8 at 10:12
ray
629
asked Nov 8 at 10:12
ray
629
asked Nov 8 at 10:12
ray
629
629
This question has an open bounty worth +50
reputation from ray ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from ray ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
1
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago
add a comment |
why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
1
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago
why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
1
1
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago
add a comment |
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why does no Banach algebra have such a weird property
– mathworker21
17 hours ago
1
In a commutative unital complex Banach algebra $A$ an element $ain A$ is invertible if and only if its Gelfand transform $hat a$ does not vanish on $X(A)$. This is Gelfand's theory and can be found e.g. in Rudin's book ``Real and Complex Analysis".
– ray
17 hours ago