 to back up your system databases.
(I'm assuming that you're on SQL 2008, based on your tags; please always mention the version of SQL Server that you have.)

 Q:

K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book:
Rordam, Larsen and Laustsen.
I am having a problem with the the Exercise 3.4, which is:
Let $X$ be any compact Housdorff space.
In the part (i) of the exercise, I have shown that there is a surjective group homomorphism
$$\text{dim}: K_0(C(X)) \to C(X, \mathbb{Z})$$
which satisfies $\text{dim}([p]_0)(x)=\text{Tr}(p(x))$.
In the part (ii) of the exercise, I have shown that $\text{dim}([p]_0)=\text{dim}([q]_0)$ iff for each $x \in X$ there exists $v_x \in M_{m,n}(\mathbb{C})$ such that $v_xv_x^*=p(x)$ and $v_x^*v_x=q(x)$, for $p$ is a projection in $M_m(C(X))$ and $q$ is a projection in $M_m(C(X))$.
My problem is the part (iii) of this exercise, that is, I can not show that the $\text{dim}$ map in (i) is injective if $X$ is totally disconected.
Does anyone have have an ideia to help me?
Thank you!

A:

By continuity of $p, q :X \rightarrow M_n \mathbb(C)$, and (ii), and the total disconnectedess of $X$, find a partition of X into clopen sets $X_1,\cdots, X_k$ and complex matrix $v_1,\cdots,v_k$ such that $\|v_i^*v_i - p(x)\|<1$ and $\|v_iv_i^* - q(x)\|<1$ for all $x \in X_i$. 
Now, define the map $f: X \rightarrow M_n (C)$, $f(x) = v
to back up your system databases. 
 (I'm assuming that you're on SQL 2008, based on your rag; please constantly mention the version of SQL Server that you have .) 

  Q: 

 K - hypothesis of $ C(X)$ for $ X$ totally disconnected 

 I am studying K - Theory for C*-algebras by the following bible: 
 Rordam, Larsen and Laustsen. 
 I am having a problem with the the Exercise 3.4, which is: 
 permit $ X$ be any compact Housdorff space. 
 In the share (i) of the exercise, I have testify that there is a surjective group homomorphism 
 $ $ \text{dim }: K_0(C(X) ) \to C(X, \mathbb{Z})$$ 
 which satisfies $ \text{dim}([p]_0)(x)=\text{Tr}(p(x))$. 
 In the part (ii) of the exercise, I have shown that $ \text{dim}([p]_0)=\text{dim}([q]_0)$ iff for each $ x \in X$ there exists $ v_x \in M_{m, n}(\mathbb{C})$ such that $ v_xv_x^*=p(x)$ and $ v_x^*v_x = q(x)$, for $ p$ is a project in $ M_m(C(X))$ and $ q$ is a projection in $ M_m(C(X))$. 
 My problem is the part (iii) of this use, that is, I cannot show that the $ \text{dim}$ map in (iodine) is injective if $ X$ is totally disconected. 
 Does anyone have experience an ideia to help me? 
 Thank you! 

 A: 

 By continuity of $ p, q :X \rightarrow M_n \mathbb(C)$, and (ii), and the total disconnectedess of $ X$, find a division of X into clopen sets $ X_1,\cdots, X_k$ and complex matrix $ v_1,\cdots, v_k$ such that $ \|v_i^*v_i - p(x)\|<1 $ and $ \|v_iv_i^ * - q(x)\|<1 $ for all $ x \in X_i$. 
 Now, define the map $ f: ten \rightarrow M_n (C)$, $ f(x) = v
 to back up your system datxbases.
(I'm assumiut that you're on SQL 2008, based on your tags; please apwqys mtution the version of SQL Servvr that yiu heve.)

 Q:

K-Theory of $R(S)$ for $X$ totalmn diseoinected

I am stucying K-Themry for C*-algebsar yy the following book:
Rordam, Larsen agd Lausysfn.
I am having w prpflem with the the Exercise 3.4, which is:
Lef $X$ be eny compact Houxdorff space.
In the part (i) lf tje exercise, I have shown that theww is a surjeztive grouk komomorphisj
$$\text{dim}: K_0(C(X)) \to C(X, \mathbb{Z})$$
which satixfies $\text{ein}([p]_0)(x)=\hgxt{Tr}(p(x))$.
In thx part (ii) of the ewvrcise, H have xhown that $\texb{dim}([p]_0)=\vext{eim}([q]_0)$ iff for each $x \ii X$ there exists $v_x \yn M_{m,n}(\matvby{C})$ such that $v_xv_x^*=p(x)$ abd $v_x^*v_x=x(x)$, fmr $p$ us x pdooecfion ij $M_j(C(X))$ and $q$ js a projecrion in $M_m(C(X))$.
My probkei is the part (jii) of tris exercise, that is, I can not show thau the $\text{dim}$ map in (i) is inhective if $X$ is totalpy disconqcted.
Does anyone have have an ideia to help me?
Thatk yon!

A:

Cy eintinjutj of $p, q :X \rightarrow M_n \mathbb(C)$, and (ii), and trs uotsl disconnectebess of $X$, find a pwryytion of X injo clopzh aets $X_1,\cdots, X_k$ and complev mateix $v_1,\cdotf,v_k$ xuch that $\|v_i^*v_i - p(x)\|<1$ and $\|v_iv_i^* - q(x)\|<1$ for all $z \in X_i$. 
Now, define the map $f: R \righjarrow M_n (C)$, $f(x) = v
to back up your system databases. (I'm you're SQL 2008, on your tags; of Server that you Q: K-Theory of for $X$ totally disconnected I am K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am a problem with the the Exercise 3.4, which is: Let $X$ be any Housdorff In part of the exercise, I have shown that there is a surjective group homomorphism $$\text{dim}: K_0(C(X)) \to \mathbb{Z})$$ which satisfies $\text{dim}([p]_0)(x)=\text{Tr}(p(x))$. In the part (ii) the exercise, I have that $\text{dim}([p]_0)=\text{dim}([q]_0)$ iff for each \in there exists \in such $v_xv_x^*=p(x)$ and $v_x^*v_x=q(x)$, $p$ is a projection in $M_m(C(X))$ and $q$ is a projection in $M_m(C(X))$. My problem is the (iii) of that is, can show the $\text{dim}$ map is injective if $X$ is totally have have an ideia to help me? Thank A: By of $p, q :X \rightarrow M_n and (ii), and the total disconnectedess of $X$, a partition of X into clopen sets $X_1,\cdots, X_k$ and complex matrix $v_1,\cdots,v_k$ such that p(x)\|<1$ and $\|v_iv_i^* - for all $x X_i$. define map X \rightarrow (C)$, $f(x) = v
 to back up your system databasEs.
(I'm assumiNg thaT yoU're On sQL 2008, bAsed On your tags; pleaSE alwAys mention the version of sQL SeRvER thaT YoU have.)

 q:

K-TheorY Of $c(x)$ For $x$ tOtAllY dIScOnnecTed

i am studYing K-TheorY foR C*-Algebras by thE FoLlowing booK:
RoRdam, Larsen anD LaUstsen.
i aM haVIng a pRobLem wiTh the tHE ExercIse 3.4, which iS:
LET $X$ be anY Compact hOUsDorfF space.
In the part (i) oF ThE Exercise, I have sHown thAt THeRE Is a SurJective groUp HomomORphism
$$\tEXt{DIM}: k_0(C(X)) \TO C(X, \mathbb{Z})$$
whiCh satisfies $\TExt{Dim}([p]_0)(x)=\tExT{Tr}(P(X))$.
In the Part (iI) oF The Exercise, I haVe shOwn that $\teXt{dim}([p]_0)=\TExt{dim}([q]_0)$ IFf for eaCh $x \in X$ TheRe eXistS $V_x \In m_{m,n}(\MaTHbb{c})$ SuCh tHAt $v_Xv_x^*=p(x)$ and $V_x^*V_x=Q(x)$, for $P$ is a PROJEctiOn iN $M_m(C(x))$ and $q$ Is a projection In $M_M(C(X))$.
MY ProBlem iS the pArt (iIi) Of thiS exercIse, thAt Is, I can not show thAt thE $\text{dim}$ mAp iN (i) Is iNjEctivE If $X$ is tOtaLly DisconeCted.
DoeS AnyOnE HAVe Have an ideia to help mE?
THANk You!

A:

By coNtinuiTY oF $p, Q :x \rightarRoW M_n \MathBB(c)$, and (iI), and THe Total disConnecTEdEsS of $X$, finD a PartitIoN of x inTo cloPEn seTs $X_1,\cdoTs, X_k$ and cOmpleX Matrix $v_1,\cdots,v_k$ SUch that $\|v_i^*v_i - p(x)\|<1$ ANd $\|V_IV_i^* - Q(X)\|<1$ for All $X \in X_i$. 
Now, defIne tHE map $F: X \riGHtArrOW M_n (C)$, $f(X) = v
 to back up your system da tabases.
( I'm a ssu min gthat  you 're on SQL 200 8 , ba sed on your tags; plea se al wa y s me n ti on th e versi o no f  SQ LSe rve rt ha t you  ha ve.)

Q:

K-Theo ryof  $C(X)$ for$ X$  totally d isc onnected

Iamstudyi ng  K- T heory  fo r C*- algebr a s by t he follow in g  book: Rordam, L ar senand Laustsen.
I a m  h a ving a problem  withth e  t h e  Ex erc ise 3.4, w hi ch is : 
Let $X $  b e a nyc ompact Housdo rff space.I n t he par t(i) of the  exer ci s e,I have show n th at thereis a s u rjectiv e  grouphomomo rph ism 
$$\ t ex t{ dim }: K_0 ( C( X)) \to  C(X, \m at hb b{Z}) $$
w h i c h  sat isf ies$\tex t{dim}([p]_0) (x) =\te x t{T r}(p( x))$. 
Inth e par t (ii)  of t he  exercise, I ha ve s hown that  $\ te xt{ di m}([p ] _0)=\t ext {di m}([q]_ 0)$ iff for  e a c h  $ x \in X$ there exi st s $v _x \in M _{m,n} ( \m at h bb{C})$su chthat $ v_xv_ x^*= p (x )$ and $ v_x^*v _ x= q( x)$, fo r$p$ is  a  pr oje ction in $ M_m(C( X))$ and  $q$i s a projection in $M_m(C(X)) $ .M y  p r oble m i s the part(iii )  ofthis ex erc i se, t hat i s, Ic an not show that th e$\text {dim} $ map in (i)is injecti v e if $X$ i s to t al l y disconected. 
Does  anyone ha v e have a n ide ia to he lp me?
Th a n k you!
A:
By  co nti n u it y of $p, q :X \ righ ta rrow M_ n \ mathbb( C)$ , a nd(ii ),  and thetotal di sc on ne ct ede ss of $X$, fin da p ar tit ion o f  X int o clo pense ts $X_ 1,\cdot s ,X _ k$ a nd  c ompl exma trix$v_1 , \cd ots,v_k $ such th at$ \|v_ i^ *v _i - p( x)\|<1$ and $ \| v_iv_i^* -  q (x) \|<1$f o r all $x  \in X_i$. 
Now, define the map  $f : X \ righ tarrow M_ n ( C)$, $ f(x )  = v
 to_back up_your system databases.
(I'm assuming_that you're_on_SQL 2008,_based_on your tags;_please always mention_the version of SQL_Server that you_have.)

_Q:

K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the_following_book:
Rordam, Larsen_and_Laustsen.
I_am having a problem with_the the Exercise 3.4, which_is:
Let $X$_be any compact Housdorff space.
In the part (i)_of_the exercise, I_have shown that there is a surjective group homomorphism
$$\text{dim}:_K_0(C(X)) \to C(X, \mathbb{Z})$$
which satisfies $\text{dim}([p]_0)(x)=\text{Tr}(p(x))$.
In_the part (ii)_of_the_exercise, I have shown_that $\text{dim}([p]_0)=\text{dim}([q]_0)$ iff for each $x_\in X$ there exists $v_x \in_M_{m,n}(\mathbb{C})$ such that $v_xv_x^*=p(x)$ and $v_x^*v_x=q(x)$, for_$p$ is a projection in $M_m(C(X))$_and $q$ is a projection_in $M_m(C(X))$.
My_problem is the part (iii)_of this exercise,_that is,_I can not_show that the $\text{dim}$ map in_(i) is injective_if $X$ is totally disconected.
Does anyone_have_have an ideia_to_help_me?
Thank you!

A:

By_continuity of $p,_q_:X \rightarrow_M_n_\mathbb(C)$, and (ii), and the total_disconnectedess_of $X$, find a partition of X_into clopen sets $X_1,\cdots,_X_k$_and complex matrix $v_1,\cdots,v_k$_such that $\|v_i^*v_i - p(x)\|<1$_and $\|v_iv_i^* - q(x)\|<1$ for all_$x \in_X_i$. 
Now,_define the map $f: X \rightarrow M_n (C)$, $f(x) = v