Mixing
Banach algebra $l^p$ is not isomorphic to $C^{*}$ algebra
$begingroup$
Consider commutative Banach algebra $l^p$, $p in [1,infty)$ with multiplication by coordinates. I know, that $Delta (l^{p})={e_n : n in mathbb{N}}$ - set of canonical functionals. We know that $widehat{x}(e_n)=x_n$ and $widehat{x}: l^{p} to C_0(Delta (l^{p}))$. Because there exists $xin l^{p}$ such that $x_ineq x_j$ for $ineq j$, and Gelfand transformation is continuous, GT must be discrete. I would like to show, that GT is or is not surjective, but I do not have any idea. Many thanks
abstract-algebra lp-spaces group-isomorphism banach-algebras gelfand-representation
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
$endgroup$
add a comment |
$begingroup$
Consider commutative Banach algebra $l^p$, $p in [1,infty)$ with multiplication by coordinates. I know, that $Delta (l^{p})={e_n : n in mathbb{N}}$ - set of canonical functionals. We know that $widehat{x}(e_n)=x_n$ and $widehat{x}: l^{p} to C_0(Delta (l^{p}))$. Because there exists $xin l^{p}$ such that $x_ineq x_j$ for $ineq j$, and Gelfand transformation is continuous, GT must be discrete. I would like to show, that GT is or is not surjective, but I do not have any idea. Many thanks
abstract-algebra lp-spaces group-isomorphism banach-algebras gelfand-representation
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
$endgroup$
add a comment |
$begingroup$
Consider commutative Banach algebra $l^p$, $p in [1,infty)$ with multiplication by coordinates. I know, that $Delta (l^{p})={e_n : n in mathbb{N}}$ - set of canonical functionals. We know that $widehat{x}(e_n)=x_n$ and $widehat{x}: l^{p} to C_0(Delta (l^{p}))$. Because there exists $xin l^{p}$ such that $x_ineq x_j$ for $ineq j$, and Gelfand transformation is continuous, GT must be discrete. I would like to show, that GT is or is not surjective, but I do not have any idea. Many thanks
abstract-algebra lp-spaces group-isomorphism banach-algebras gelfand-representation
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
$endgroup$
Consider commutative Banach algebra $l^p$, $p in [1,infty)$ with multiplication by coordinates. I know, that $Delta (l^{p})={e_n : n in mathbb{N}}$ - set of canonical functionals. We know that $widehat{x}(e_n)=x_n$ and $widehat{x}: l^{p} to C_0(Delta (l^{p}))$. Because there exists $xin l^{p}$ such that $x_ineq x_j$ for $ineq j$, and Gelfand transformation is continuous, GT must be discrete. I would like to show, that GT is or is not surjective, but I do not have any idea. Many thanks
abstract-algebra lp-spaces group-isomorphism banach-algebras gelfand-representation
abstract-algebra lp-spaces group-isomorphism banach-algebras gelfand-representation
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
edited Jan 7 at 23:16
Martin Sleziak
44.7k9117272
44.7k9117272
asked Jan 7 at 16:25
LeoLeeLeoLee
112
asked Jan 7 at 16:25
LeoLeeLeoLee
112
asked Jan 7 at 16:25
LeoLeeLeoLee
112
112
add a comment |
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%2f3065186%2fbanach-algebra-lp-is-not-isomorphic-to-c-algebra%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%2f3065186%2fbanach-algebra-lp-is-not-isomorphic-to-c-algebra%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
