Mixing
Is strong topology on a W*-algebra a dual topology?
$begingroup$
I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8.
Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and $V$ a norm-dense subspace of the predual $A_{*}subseteq A^{*}$ of $A$. $V$ is assumed to be invariant in the sense that for each $ain A$ and $varphiin V$, all of $(xmapstovarphi(ax))$, $(xmapstovarphi(xa))$, and $(xmapstooverline{varphi(x^{*})})$ are still in $V$.
For example, if $A=mathcal{B}(H)$ for some Hilbert space $H$, then we might choose $V$ to be the set of finite-rank operators on $H$, which is a dense subspace of $mathcal{B}(H)_{*}$, the set of trace-class operators on $H$.
Or, $A_{*}$ itself is an example of invariant subspace.
We can define the $V$-strong topology on $A$ as the locally convex topology generated by the set of seminorms given by $$xmapstovarphi(x^{*}x)^{1/2}$$ for each positive linear functional $varphiin V$. Let us denote this topology as $s(A,V)$.
Question: Is $s(A,V)$ a dual topology for the pairing $(A,V)$? That is, is the following true? $$sigma(A,V)subseteq s(A,V)subseteqtau(A,V)$$
I believe I could show the followings:
For $A=mathcal{B}(H)$ and $V=H^{*}otimes H$ the set of finite rank operators, then $s(A,V)$ is the usual strong operator topology. In this case, the above claim is true.
$sigma(A,V)subseteq s(A,V)subseteqtau(A,A_{*})$. Hence, the dual of $(A,s(A,V))$ is a subspace of $A_{*}$ containing $V$.
If $B$ is the unit ball in $A$, then $mathbf{1}_{B}:(B,tau(A,V))rightarrow(B,s(A,V))$ is continuous.
Edit
Here is the precise definition of $sigma(A,V)$ and $tau(A,V)$.
$sigma(A,V)$ is the coarsest topology making each element in $V$ a continuous linear functional on $A$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:|varphi(a)|<epsilonright}$$
for $varphiin V$ and $epsilon>0$.
$tau(A,V)$ is the Mackey topology of the pairing $(A,V)$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:sup_{varphiin K}|varphi(a)|<epsilonright}$$
for $sigma(V,A)$-compact absolutely convex subset $Ksubseteq V$ and $epsilon>0$. In other words, $K$ is a weakly compact absolutely convex subset of the Banach space $A_{*}$, and at the same time $K$ is contained in $V$.
functional-analysis operator-theory von-neumann-algebras
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
$endgroup$
add a comment |
$begingroup$
I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8.
Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and $V$ a norm-dense subspace of the predual $A_{*}subseteq A^{*}$ of $A$. $V$ is assumed to be invariant in the sense that for each $ain A$ and $varphiin V$, all of $(xmapstovarphi(ax))$, $(xmapstovarphi(xa))$, and $(xmapstooverline{varphi(x^{*})})$ are still in $V$.
For example, if $A=mathcal{B}(H)$ for some Hilbert space $H$, then we might choose $V$ to be the set of finite-rank operators on $H$, which is a dense subspace of $mathcal{B}(H)_{*}$, the set of trace-class operators on $H$.
Or, $A_{*}$ itself is an example of invariant subspace.
We can define the $V$-strong topology on $A$ as the locally convex topology generated by the set of seminorms given by $$xmapstovarphi(x^{*}x)^{1/2}$$ for each positive linear functional $varphiin V$. Let us denote this topology as $s(A,V)$.
Question: Is $s(A,V)$ a dual topology for the pairing $(A,V)$? That is, is the following true? $$sigma(A,V)subseteq s(A,V)subseteqtau(A,V)$$
I believe I could show the followings:
For $A=mathcal{B}(H)$ and $V=H^{*}otimes H$ the set of finite rank operators, then $s(A,V)$ is the usual strong operator topology. In this case, the above claim is true.
$sigma(A,V)subseteq s(A,V)subseteqtau(A,A_{*})$. Hence, the dual of $(A,s(A,V))$ is a subspace of $A_{*}$ containing $V$.
If $B$ is the unit ball in $A$, then $mathbf{1}_{B}:(B,tau(A,V))rightarrow(B,s(A,V))$ is continuous.
Edit
Here is the precise definition of $sigma(A,V)$ and $tau(A,V)$.
$sigma(A,V)$ is the coarsest topology making each element in $V$ a continuous linear functional on $A$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:|varphi(a)|<epsilonright}$$
for $varphiin V$ and $epsilon>0$.
$tau(A,V)$ is the Mackey topology of the pairing $(A,V)$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:sup_{varphiin K}|varphi(a)|<epsilonright}$$
for $sigma(V,A)$-compact absolutely convex subset $Ksubseteq V$ and $epsilon>0$. In other words, $K$ is a weakly compact absolutely convex subset of the Banach space $A_{*}$, and at the same time $K$ is contained in $V$.
functional-analysis operator-theory von-neumann-algebras
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
$endgroup$
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38
add a comment |
$begingroup$
I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8.
Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and $V$ a norm-dense subspace of the predual $A_{*}subseteq A^{*}$ of $A$. $V$ is assumed to be invariant in the sense that for each $ain A$ and $varphiin V$, all of $(xmapstovarphi(ax))$, $(xmapstovarphi(xa))$, and $(xmapstooverline{varphi(x^{*})})$ are still in $V$.
For example, if $A=mathcal{B}(H)$ for some Hilbert space $H$, then we might choose $V$ to be the set of finite-rank operators on $H$, which is a dense subspace of $mathcal{B}(H)_{*}$, the set of trace-class operators on $H$.
Or, $A_{*}$ itself is an example of invariant subspace.
We can define the $V$-strong topology on $A$ as the locally convex topology generated by the set of seminorms given by $$xmapstovarphi(x^{*}x)^{1/2}$$ for each positive linear functional $varphiin V$. Let us denote this topology as $s(A,V)$.
Question: Is $s(A,V)$ a dual topology for the pairing $(A,V)$? That is, is the following true? $$sigma(A,V)subseteq s(A,V)subseteqtau(A,V)$$
I believe I could show the followings:
For $A=mathcal{B}(H)$ and $V=H^{*}otimes H$ the set of finite rank operators, then $s(A,V)$ is the usual strong operator topology. In this case, the above claim is true.
$sigma(A,V)subseteq s(A,V)subseteqtau(A,A_{*})$. Hence, the dual of $(A,s(A,V))$ is a subspace of $A_{*}$ containing $V$.
If $B$ is the unit ball in $A$, then $mathbf{1}_{B}:(B,tau(A,V))rightarrow(B,s(A,V))$ is continuous.
Edit
Here is the precise definition of $sigma(A,V)$ and $tau(A,V)$.
$sigma(A,V)$ is the coarsest topology making each element in $V$ a continuous linear functional on $A$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:|varphi(a)|<epsilonright}$$
for $varphiin V$ and $epsilon>0$.
$tau(A,V)$ is the Mackey topology of the pairing $(A,V)$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:sup_{varphiin K}|varphi(a)|<epsilonright}$$
for $sigma(V,A)$-compact absolutely convex subset $Ksubseteq V$ and $epsilon>0$. In other words, $K$ is a weakly compact absolutely convex subset of the Banach space $A_{*}$, and at the same time $K$ is contained in $V$.
functional-analysis operator-theory von-neumann-algebras
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
$endgroup$
I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8.
Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and $V$ a norm-dense subspace of the predual $A_{*}subseteq A^{*}$ of $A$. $V$ is assumed to be invariant in the sense that for each $ain A$ and $varphiin V$, all of $(xmapstovarphi(ax))$, $(xmapstovarphi(xa))$, and $(xmapstooverline{varphi(x^{*})})$ are still in $V$.
For example, if $A=mathcal{B}(H)$ for some Hilbert space $H$, then we might choose $V$ to be the set of finite-rank operators on $H$, which is a dense subspace of $mathcal{B}(H)_{*}$, the set of trace-class operators on $H$.
Or, $A_{*}$ itself is an example of invariant subspace.
We can define the $V$-strong topology on $A$ as the locally convex topology generated by the set of seminorms given by $$xmapstovarphi(x^{*}x)^{1/2}$$ for each positive linear functional $varphiin V$. Let us denote this topology as $s(A,V)$.
Question: Is $s(A,V)$ a dual topology for the pairing $(A,V)$? That is, is the following true? $$sigma(A,V)subseteq s(A,V)subseteqtau(A,V)$$
I believe I could show the followings:
For $A=mathcal{B}(H)$ and $V=H^{*}otimes H$ the set of finite rank operators, then $s(A,V)$ is the usual strong operator topology. In this case, the above claim is true.
$sigma(A,V)subseteq s(A,V)subseteqtau(A,A_{*})$. Hence, the dual of $(A,s(A,V))$ is a subspace of $A_{*}$ containing $V$.
If $B$ is the unit ball in $A$, then $mathbf{1}_{B}:(B,tau(A,V))rightarrow(B,s(A,V))$ is continuous.
Edit
Here is the precise definition of $sigma(A,V)$ and $tau(A,V)$.
$sigma(A,V)$ is the coarsest topology making each element in $V$ a continuous linear functional on $A$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:|varphi(a)|<epsilonright}$$
for $varphiin V$ and $epsilon>0$.
$tau(A,V)$ is the Mackey topology of the pairing $(A,V)$. More concretely, a neighborhood base at $0$ is given as
$$left{ain A:sup_{varphiin K}|varphi(a)|<epsilonright}$$
for $sigma(V,A)$-compact absolutely convex subset $Ksubseteq V$ and $epsilon>0$. In other words, $K$ is a weakly compact absolutely convex subset of the Banach space $A_{*}$, and at the same time $K$ is contained in $V$.
functional-analysis operator-theory von-neumann-algebras
functional-analysis operator-theory von-neumann-algebras
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
edited Jan 1 at 8:38
Junekey Jeon
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
asked Dec 27 '18 at 12:35
Junekey JeonJunekey Jeon
21216
21216
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38
add a comment |
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3053882%2fis-strong-topology-on-a-w-algebra-a-dual-topology%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3053882%2fis-strong-topology-on-a-w-algebra-a-dual-topology%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
$begingroup$
What are $sigma(A,V)$ and $tau(A,V)$? I've seen $sigma(X,X_*)$ for the weak-star topology.
$endgroup$
– Ashwin Trisal
Dec 31 '18 at 23:04
$begingroup$
@AshwinTrisal I've edited the question to answer your comment. Thanks.
$endgroup$
– Junekey Jeon
Jan 1 at 8:38