$K$ Theory of Operators II, Higson's notes The 2019 Stack Overflow Developer Survey Results Are InSpecial elements in the $C^*$ algebra $A otimes mathcalK$.Understanding the map from $K_0(A)$ to homotopy class of maps,The classifying space of a gauge groupThe inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor productIn Kunneth Formula for Cohomology, the finitely generated condition is necessary.In what sense is every element of $H_2(G)$ “represented by a free action on some surface”Evaluating Gibbs state in the second quantized formalismExplaining the group structure in homotopy classes of map $[X,K(H)]$Induced $K$-theory maps between $C^*$ algebras.Understanding the map from $K_0(A)$ to homotopy class of maps,Special elements in the $C^*$ algebra $A otimes mathcalK$.$K$-Theory of operators I, Higson notes
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$K$ Theory of Operators II, Higson's notes
The 2019 Stack Overflow Developer Survey Results Are InSpecial elements in the $C^*$ algebra $A otimes mathcalK$.Understanding the map from $K_0(A)$ to homotopy class of maps,The classifying space of a gauge groupThe inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor productIn Kunneth Formula for Cohomology, the finitely generated condition is necessary.In what sense is every element of $H_2(G)$ “represented by a free action on some surface”Evaluating Gibbs state in the second quantized formalismExplaining the group structure in homotopy classes of map $[X,K(H)]$Induced $K$-theory maps between $C^*$ algebras.Understanding the map from $K_0(A)$ to homotopy class of maps,Special elements in the $C^*$ algebra $A otimes mathcalK$.$K$-Theory of operators I, Higson notes
$begingroup$
What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map.
Proposition 3.17: For any (ungraded) $C^*$-algebra $A$, the map
$$Phi:K_0(A) rightarrow [mathcalS, A otimes mathcalK ]$$
defined above* is an isomoprhism. Where $mathcalS=C_0(Bbb R)$, $mathcalK$ compact operators on the graded hilbert space $H=H_0oplus H_1$.
The map (*) is given as mapping an two idempotent matrix $p,q$ into
$$
f mapsto
beginpmatrix
p f(0) & 0 \
0 & q f(0)
endpmatrix
$$
where we regard $A otimes mathcalK cong M_2(A otimes mathcalK)$. More details of this map is here.
The details of inverse map can be found in here.
I could not really chase out how this maps are inverse - in particular, the map in 2. defines an element in $K_0(A otimes mathcalK)$, and here we use Morita equivalence to obtain an element,
$$K_0(A) cong K_0(A otimes mathcalK)$$
I would be grateful if someone can spell out more details. I believe, it would be easier for one to first read the original proof.
algebraic-topology operator-theory operator-algebras k-theory
asked Mar 30 at 17:55
CL.CL.
2,3222925
$endgroup$
add a comment |
$begingroup$
What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map.
Proposition 3.17: For any (ungraded) $C^*$-algebra $A$, the map
$$Phi:K_0(A) rightarrow [mathcalS, A otimes mathcalK ]$$
defined above* is an isomoprhism. Where $mathcalS=C_0(Bbb R)$, $mathcalK$ compact operators on the graded hilbert space $H=H_0oplus H_1$.
The map (*) is given as mapping an two idempotent matrix $p,q$ into
$$
f mapsto
beginpmatrix
p f(0) & 0 \
0 & q f(0)
endpmatrix
$$
where we regard $A otimes mathcalK cong M_2(A otimes mathcalK)$. More details of this map is here.
The details of inverse map can be found in here.
I could not really chase out how this maps are inverse - in particular, the map in 2. defines an element in $K_0(A otimes mathcalK)$, and here we use Morita equivalence to obtain an element,
$$K_0(A) cong K_0(A otimes mathcalK)$$
I would be grateful if someone can spell out more details. I believe, it would be easier for one to first read the original proof.
algebraic-topology operator-theory operator-algebras k-theory
asked Mar 30 at 17:55
CL.CL.
2,3222925
$endgroup$
add a comment |
$begingroup$
What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map.
Proposition 3.17: For any (ungraded) $C^*$-algebra $A$, the map
$$Phi:K_0(A) rightarrow [mathcalS, A otimes mathcalK ]$$
defined above* is an isomoprhism. Where $mathcalS=C_0(Bbb R)$, $mathcalK$ compact operators on the graded hilbert space $H=H_0oplus H_1$.
The map (*) is given as mapping an two idempotent matrix $p,q$ into
$$
f mapsto
beginpmatrix
p f(0) & 0 \
0 & q f(0)
endpmatrix
$$
where we regard $A otimes mathcalK cong M_2(A otimes mathcalK)$. More details of this map is here.
The details of inverse map can be found in here.
I could not really chase out how this maps are inverse - in particular, the map in 2. defines an element in $K_0(A otimes mathcalK)$, and here we use Morita equivalence to obtain an element,
$$K_0(A) cong K_0(A otimes mathcalK)$$
I would be grateful if someone can spell out more details. I believe, it would be easier for one to first read the original proof.
algebraic-topology operator-theory operator-algebras k-theory
asked Mar 30 at 17:55
CL.CL.
2,3222925
$endgroup$
What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map.
Proposition 3.17: For any (ungraded) $C^*$-algebra $A$, the map
$$Phi:K_0(A) rightarrow [mathcalS, A otimes mathcalK ]$$
defined above* is an isomoprhism. Where $mathcalS=C_0(Bbb R)$, $mathcalK$ compact operators on the graded hilbert space $H=H_0oplus H_1$.
The map (*) is given as mapping an two idempotent matrix $p,q$ into
$$
f mapsto
beginpmatrix
p f(0) & 0 \
0 & q f(0)
endpmatrix
$$
where we regard $A otimes mathcalK cong M_2(A otimes mathcalK)$. More details of this map is here.
The details of inverse map can be found in here.
I could not really chase out how this maps are inverse - in particular, the map in 2. defines an element in $K_0(A otimes mathcalK)$, and here we use Morita equivalence to obtain an element,
$$K_0(A) cong K_0(A otimes mathcalK)$$
I would be grateful if someone can spell out more details. I believe, it would be easier for one to first read the original proof.
algebraic-topology operator-theory operator-algebras k-theory
algebraic-topology operator-theory operator-algebras k-theory
asked Mar 30 at 17:55
CL.CL.
2,3222925
asked Mar 30 at 17:55
CL.CL.
2,3222925
edited Mar 30 at 19:31
CL.
asked Mar 30 at 17:55
CL.CL.
2,3222925
asked Mar 30 at 17:55
CL.CL.
2,3222925
asked Mar 30 at 17:55
CL.CL.
2,3222925
2,3222925
add a comment |
add a comment |
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