Fdtxjr

RSA: Danger of using p to create q

strTok function (thread safe, supports empty tokens, doesn't change string)

Can a Cauchy sequence converge for one metric while not converging for another?

How to determine what difficulty is right for the game?

What would happen to a modern skyscraper if it rains micro blackholes?

Paid for article while in US on F-1 visa?

NMaximize is not converging to a solution

Can I make popcorn with any corn?

"You are your self first supporter", a more proper way to say it

Does an object always see its latest internal state irrespective of thread?

Why do I get two different answers for this counting problem?

Can an x86 CPU running in real mode be considered to be basically an 8086 CPU?

Do infinite dimensional systems make sense?

If human space travel is limited by the G force vulnerability, is there a way to counter G forces?

Are the number of citations and number of published articles the most important criteria for a tenure promotion?

How old can references or sources in a thesis be?

Can a vampire attack twice with their claws using Multiattack?

How much RAM could one put in a typical 80386 setup?

Has there ever been an airliner design involving reducing generator load by installing solar panels?

Perform and show arithmetic with LuaLaTeX

Is it possible to run Internet Explorer on OS X El Capitan?

Intersection point of 2 lines defined by 2 points each

When a company launches a new product do they "come out" with a new product or do they "come up" with a new product?

Approximately how much travel time was saved by the opening of the Suez Canal in 1869?

a question on the definition of direct sum of $C^*$ algebras

A question about essential representation in C*-algebraWhat is the definition of hyperstonean space?Examples of $C^*$-algebras in Noncommutative Geometry from A. ConnesAre there infinite-dimensional, artinian C*-algebras?What does a homomorphism $phi: M_k to M_n$ look like?Irreducible representation of product of finite dimensional $C^*$-algebra(inner) direct sum of von Neumann algebras and pairwise orthogonal central projectionsProof Clarification type decomposition of von Neumann algebrasThe Hahn-Hellinger TheoremPure states of $M_n(Bbb C)$

0

$begingroup$

According to the definition in the Olsen's book,if $A=Bbigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there are many examples such as $Bbb CbigoplusBbb C$,$M_2(Bbb C)oplus M_2(Bbb C)$,the intersection is not zero.
What are the differences between two definitions?

operator-theory operator-algebras c-star-algebras

share|cite|improve this question

edited Mar 22 at 2:43

mathrookie

asked Mar 21 at 19:49

mathrookiemathrookie

936512

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you give a reference for when $M_2(mathbb C) oplus M_2(mathbb C)$ is defined in the way you described?
  $endgroup$
  – D_S
  Mar 21 at 19:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I mean if we define the direct sum,the intersection should be 0,but $Ccap C$ is not zero.
  $endgroup$
  – mathrookie
  Mar 22 at 2:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the title of the book, and where in the book is this stated?
  $endgroup$
  – Aweygan
  Mar 22 at 3:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You mean$A=Boplus C$if $A=B+C$ and $Bcap$C=0$?
  $endgroup$
  – mathrookie
  Mar 22 at 6:01
  

  
  
  
  

add a comment | 

0

$begingroup$

According to the definition in the Olsen's book,if $A=Bbigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there are many examples such as $Bbb CbigoplusBbb C$,$M_2(Bbb C)oplus M_2(Bbb C)$,the intersection is not zero.
What are the differences between two definitions?

operator-theory operator-algebras c-star-algebras

share|cite|improve this question

edited Mar 22 at 2:43

mathrookie

asked Mar 21 at 19:49

mathrookiemathrookie

936512

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Can you give a reference for when $M_2(mathbb C) oplus M_2(mathbb C)$ is defined in the way you described?
  $endgroup$
  – D_S
  Mar 21 at 19:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I mean if we define the direct sum,the intersection should be 0,but $Ccap C$ is not zero.
  $endgroup$
  – mathrookie
  Mar 22 at 2:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the title of the book, and where in the book is this stated?
  $endgroup$
  – Aweygan
  Mar 22 at 3:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You mean$A=Boplus C$if $A=B+C$ and $Bcap$C=0$?
  $endgroup$
  – mathrookie
  Mar 22 at 6:01
  

  
  
  
  

add a comment | 

$begingroup$

According to the definition in the Olsen's book,if $A=Bbigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there are many examples such as $Bbb CbigoplusBbb C$,$M_2(Bbb C)oplus M_2(Bbb C)$,the intersection is not zero.
What are the differences between two definitions?

operator-theory operator-algebras c-star-algebras

share|cite|improve this question

edited Mar 22 at 2:43

mathrookie

asked Mar 21 at 19:49

mathrookiemathrookie

936512

$endgroup$

share|cite|improve this question

edited Mar 22 at 2:43

mathrookie

asked Mar 21 at 19:49

mathrookiemathrookie

936512

share|cite|improve this question

edited Mar 22 at 2:43

mathrookie

asked Mar 21 at 19:49

mathrookiemathrookie

936512

asked Mar 21 at 19:49

mathrookiemathrookie

936512

asked Mar 21 at 19:49

mathrookiemathrookie

936512

2 Answers
2

active

oldest

votes

1

$begingroup$

The same term "direct sum" is used for two different things (that, in the end, are basically the same in spirit).

If you have two subspaces $B,Csubset A$, you say that $A$ is the (internal) direct sum of $B$ and $C$ if $A=B+C$ and $Bcap C=0$.

If, on the other hand, you have two spaces (not necessarily living in the same environment) you may construct the (external) direct sum $A=Boplus C=(b,c): bin B, cin C$. Now $A$ is an internal direct sum $A=(Aoplus 0) + (0oplus B)$.

share|cite|improve this answer

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In the definition of external direct sum.B and C can be any $C^*$-algebras,the inersection of $B$ and $C$ may not be 0.Is it correct?
  $endgroup$
  – mathrookie
  Mar 24 at 14:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  They don't even have to be the same kind of objects. Say $C[0,1]oplus M_2(mathbb C)$.
  $endgroup$
  – Martin Argerami
  Mar 24 at 14:24
  

  
  
  
  

add a comment | 

1

$begingroup$

By definition, $mathbb C oplus mathbb C$ is the set of ordered pairs $(x,y)$ with $x$ and $y$ both in $mathbb C$.

Inside $mathbb C oplus mathbb C$, there are two copies of $mathbb C$. One of them is $ (x,0) : x in mathbb C $, the other one is $ (0,x) : x in mathbb C $. These have intersection $(0,0)$ as required.

share|cite|improve this answer

edited Mar 22 at 4:08

answered Mar 22 at 4:02

D_SD_S

14.2k61653

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You mean $Coplus 0$ is the subspace of $Coplus C$,$C$ is not the subspace of $Coplus C$?
  $endgroup$
  – mathrookie
  Mar 22 at 6:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's right. They are each isomorphic to $mathbb C$, so I call them a "copy" of $mathbb C$.
  $endgroup$
  – D_S
  Mar 22 at 12:41
  

  
  
  
  

add a comment | 

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3157310%2fa-question-on-the-definition-of-direct-sum-of-c-algebras%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

The same term "direct sum" is used for two different things (that, in the end, are basically the same in spirit).

If you have two subspaces $B,Csubset A$, you say that $A$ is the (internal) direct sum of $B$ and $C$ if $A=B+C$ and $Bcap C=0$.

If, on the other hand, you have two spaces (not necessarily living in the same environment) you may construct the (external) direct sum $A=Boplus C=(b,c): bin B, cin C$. Now $A$ is an internal direct sum $A=(Aoplus 0) + (0oplus B)$.

share|cite|improve this answer

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In the definition of external direct sum.B and C can be any $C^*$-algebras,the inersection of $B$ and $C$ may not be 0.Is it correct?
  $endgroup$
  – mathrookie
  Mar 24 at 14:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  They don't even have to be the same kind of objects. Say $C[0,1]oplus M_2(mathbb C)$.
  $endgroup$
  – Martin Argerami
  Mar 24 at 14:24
  

  
  
  
  

add a comment | 

1

$begingroup$

The same term "direct sum" is used for two different things (that, in the end, are basically the same in spirit).

If you have two subspaces $B,Csubset A$, you say that $A$ is the (internal) direct sum of $B$ and $C$ if $A=B+C$ and $Bcap C=0$.

If, on the other hand, you have two spaces (not necessarily living in the same environment) you may construct the (external) direct sum $A=Boplus C=(b,c): bin B, cin C$. Now $A$ is an internal direct sum $A=(Aoplus 0) + (0oplus B)$.

share|cite|improve this answer

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In the definition of external direct sum.B and C can be any $C^*$-algebras,the inersection of $B$ and $C$ may not be 0.Is it correct?
  $endgroup$
  – mathrookie
  Mar 24 at 14:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  They don't even have to be the same kind of objects. Say $C[0,1]oplus M_2(mathbb C)$.
  $endgroup$
  – Martin Argerami
  Mar 24 at 14:24
  

  
  
  
  

add a comment | 

$begingroup$

The same term "direct sum" is used for two different things (that, in the end, are basically the same in spirit).

If you have two subspaces $B,Csubset A$, you say that $A$ is the (internal) direct sum of $B$ and $C$ if $A=B+C$ and $Bcap C=0$.

If, on the other hand, you have two spaces (not necessarily living in the same environment) you may construct the (external) direct sum $A=Boplus C=(b,c): bin B, cin C$. Now $A$ is an internal direct sum $A=(Aoplus 0) + (0oplus B)$.

share|cite|improve this answer

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

$endgroup$

share|cite|improve this answer

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

answered Mar 22 at 18:15

Martin ArgeramiMartin Argerami

129k1184185

1

$begingroup$

By definition, $mathbb C oplus mathbb C$ is the set of ordered pairs $(x,y)$ with $x$ and $y$ both in $mathbb C$.

Inside $mathbb C oplus mathbb C$, there are two copies of $mathbb C$. One of them is $ (x,0) : x in mathbb C $, the other one is $ (0,x) : x in mathbb C $. These have intersection $(0,0)$ as required.

share|cite|improve this answer

edited Mar 22 at 4:08

answered Mar 22 at 4:02

D_SD_S

14.2k61653

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You mean $Coplus 0$ is the subspace of $Coplus C$,$C$ is not the subspace of $Coplus C$?
  $endgroup$
  – mathrookie
  Mar 22 at 6:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's right. They are each isomorphic to $mathbb C$, so I call them a "copy" of $mathbb C$.
  $endgroup$
  – D_S
  Mar 22 at 12:41
  

  
  
  
  

add a comment | 

1

$begingroup$

By definition, $mathbb C oplus mathbb C$ is the set of ordered pairs $(x,y)$ with $x$ and $y$ both in $mathbb C$.

Inside $mathbb C oplus mathbb C$, there are two copies of $mathbb C$. One of them is $ (x,0) : x in mathbb C $, the other one is $ (0,x) : x in mathbb C $. These have intersection $(0,0)$ as required.

share|cite|improve this answer

edited Mar 22 at 4:08

answered Mar 22 at 4:02

D_SD_S

14.2k61653

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You mean $Coplus 0$ is the subspace of $Coplus C$,$C$ is not the subspace of $Coplus C$?
  $endgroup$
  – mathrookie
  Mar 22 at 6:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's right. They are each isomorphic to $mathbb C$, so I call them a "copy" of $mathbb C$.
  $endgroup$
  – D_S
  Mar 22 at 12:41
  

  
  
  
  

add a comment | 

$begingroup$

By definition, $mathbb C oplus mathbb C$ is the set of ordered pairs $(x,y)$ with $x$ and $y$ both in $mathbb C$.

Inside $mathbb C oplus mathbb C$, there are two copies of $mathbb C$. One of them is $ (x,0) : x in mathbb C $, the other one is $ (0,x) : x in mathbb C $. These have intersection $(0,0)$ as required.

share|cite|improve this answer

edited Mar 22 at 4:08

answered Mar 22 at 4:02

D_SD_S

14.2k61653

$endgroup$

share|cite|improve this answer

edited Mar 22 at 4:08

answered Mar 22 at 4:02

D_SD_S

14.2k61653

answered Mar 22 at 4:02

D_SD_S

14.2k61653

answered Mar 22 at 4:02

D_SD_S

14.2k61653