Operator K-theory and Topological K-theory
$begingroup$
I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_infty (C(X))$ and the Vector bundles on $X$ by $p$ goes to $xi_p=(E_p,pi,X)$ where $E_p ={(x,v) in Xtimes mathbb{C}^n: vin p(x)(mathbb{C}^n)}$ and $pi$ is the natural projection. I wonder why this is well defined in the equivalence classes of $P_infty (C(X))$. i.e why $psim q$ $iff$ there exists an isomorphism of vector bundles between $xi_p$ and $xi_q$.
operator-theory operator-algebras vector-bundles k-theory topological-k-theory
asked Jan 25 at 16:29
sirjoesirjoe
726
$endgroup$
add a comment |
$begingroup$
I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_infty (C(X))$ and the Vector bundles on $X$ by $p$ goes to $xi_p=(E_p,pi,X)$ where $E_p ={(x,v) in Xtimes mathbb{C}^n: vin p(x)(mathbb{C}^n)}$ and $pi$ is the natural projection. I wonder why this is well defined in the equivalence classes of $P_infty (C(X))$. i.e why $psim q$ $iff$ there exists an isomorphism of vector bundles between $xi_p$ and $xi_q$.
operator-theory operator-algebras vector-bundles k-theory topological-k-theory
asked Jan 25 at 16:29
sirjoesirjoe
726
$endgroup$
add a comment |
$begingroup$
I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_infty (C(X))$ and the Vector bundles on $X$ by $p$ goes to $xi_p=(E_p,pi,X)$ where $E_p ={(x,v) in Xtimes mathbb{C}^n: vin p(x)(mathbb{C}^n)}$ and $pi$ is the natural projection. I wonder why this is well defined in the equivalence classes of $P_infty (C(X))$. i.e why $psim q$ $iff$ there exists an isomorphism of vector bundles between $xi_p$ and $xi_q$.
operator-theory operator-algebras vector-bundles k-theory topological-k-theory
asked Jan 25 at 16:29
sirjoesirjoe
726
$endgroup$
I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_infty (C(X))$ and the Vector bundles on $X$ by $p$ goes to $xi_p=(E_p,pi,X)$ where $E_p ={(x,v) in Xtimes mathbb{C}^n: vin p(x)(mathbb{C}^n)}$ and $pi$ is the natural projection. I wonder why this is well defined in the equivalence classes of $P_infty (C(X))$. i.e why $psim q$ $iff$ there exists an isomorphism of vector bundles between $xi_p$ and $xi_q$.
operator-theory operator-algebras vector-bundles k-theory topological-k-theory
operator-theory operator-algebras vector-bundles k-theory topological-k-theory
asked Jan 25 at 16:29
sirjoesirjoe
726
asked Jan 25 at 16:29
sirjoesirjoe
726
edited Jan 26 at 11:12
sirjoe
asked Jan 25 at 16:29
sirjoesirjoe
726
asked Jan 25 at 16:29
sirjoesirjoe
726
asked Jan 25 at 16:29
sirjoesirjoe
726
726
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%2f3087294%2foperator-k-theory-and-topological-k-theory%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%2f3087294%2foperator-k-theory-and-topological-k-theory%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
