Approximate identity for induced algebra
$begingroup$
Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $alpha$. Then we can define an induced algebra $mathrm{Ind}A$ with a $G$-action by taking all bounded continuous functions $f:Grightarrow A$ such that $f(sh)=alpha_{h^{-1}}(f(s))$ for all $sin G$ and $hin H$ and such that $sHmapsto||f(s)||$ is in $C_0(G/H)$. It is a Banach algebra under the supremum norm.
If $A$ has a bounded approximate identity, does $mathrm{Ind}A$ have a bounded approximate identity?
Given any $ain A$, I can produce some $finmathrm{Ind}A$ such that the value of $f$ at the identity is close to $a$. But I cannot see how to construct appropriate functions using the approximate identity in $A$ to get one for $mathrm{Ind}A$.
More generally, if $A$ is nondegenerate in the sense that the linear span of products of pairs of elements in $A$ is dense in $A$, is $mathrm{Ind}A$ nondegenerate?
banach-algebras
asked Jan 6 at 16:11
cyccyc
1,445713
$endgroup$
add a comment |
$begingroup$
Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $alpha$. Then we can define an induced algebra $mathrm{Ind}A$ with a $G$-action by taking all bounded continuous functions $f:Grightarrow A$ such that $f(sh)=alpha_{h^{-1}}(f(s))$ for all $sin G$ and $hin H$ and such that $sHmapsto||f(s)||$ is in $C_0(G/H)$. It is a Banach algebra under the supremum norm.
If $A$ has a bounded approximate identity, does $mathrm{Ind}A$ have a bounded approximate identity?
Given any $ain A$, I can produce some $finmathrm{Ind}A$ such that the value of $f$ at the identity is close to $a$. But I cannot see how to construct appropriate functions using the approximate identity in $A$ to get one for $mathrm{Ind}A$.
More generally, if $A$ is nondegenerate in the sense that the linear span of products of pairs of elements in $A$ is dense in $A$, is $mathrm{Ind}A$ nondegenerate?
banach-algebras
asked Jan 6 at 16:11
cyccyc
1,445713
$endgroup$
add a comment |
$begingroup$
Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $alpha$. Then we can define an induced algebra $mathrm{Ind}A$ with a $G$-action by taking all bounded continuous functions $f:Grightarrow A$ such that $f(sh)=alpha_{h^{-1}}(f(s))$ for all $sin G$ and $hin H$ and such that $sHmapsto||f(s)||$ is in $C_0(G/H)$. It is a Banach algebra under the supremum norm.
If $A$ has a bounded approximate identity, does $mathrm{Ind}A$ have a bounded approximate identity?
Given any $ain A$, I can produce some $finmathrm{Ind}A$ such that the value of $f$ at the identity is close to $a$. But I cannot see how to construct appropriate functions using the approximate identity in $A$ to get one for $mathrm{Ind}A$.
More generally, if $A$ is nondegenerate in the sense that the linear span of products of pairs of elements in $A$ is dense in $A$, is $mathrm{Ind}A$ nondegenerate?
banach-algebras
asked Jan 6 at 16:11
cyccyc
1,445713
$endgroup$
Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $alpha$. Then we can define an induced algebra $mathrm{Ind}A$ with a $G$-action by taking all bounded continuous functions $f:Grightarrow A$ such that $f(sh)=alpha_{h^{-1}}(f(s))$ for all $sin G$ and $hin H$ and such that $sHmapsto||f(s)||$ is in $C_0(G/H)$. It is a Banach algebra under the supremum norm.
If $A$ has a bounded approximate identity, does $mathrm{Ind}A$ have a bounded approximate identity?
Given any $ain A$, I can produce some $finmathrm{Ind}A$ such that the value of $f$ at the identity is close to $a$. But I cannot see how to construct appropriate functions using the approximate identity in $A$ to get one for $mathrm{Ind}A$.
More generally, if $A$ is nondegenerate in the sense that the linear span of products of pairs of elements in $A$ is dense in $A$, is $mathrm{Ind}A$ nondegenerate?
banach-algebras
banach-algebras
asked Jan 6 at 16:11
cyccyc
1,445713
asked Jan 6 at 16:11
cyccyc
1,445713
edited Jan 6 at 16:19
cyc
asked Jan 6 at 16:11
cyccyc
1,445713
asked Jan 6 at 16:11
cyccyc
1,445713
asked Jan 6 at 16:11
cyccyc
1,445713
1,445713
add a comment |
add a comment |
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