Commuting Elements in Tensor Products of C*-Algebras
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I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen.
Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a continuous function from $mathbb{R}$ to $mathbb{C}$ satisfying some properties). In the exercise, it says
Then $u_1:= 1otimes u$ and $u_2:=operatorname{diag}(u’,1,dots, 1)otimes 1$ are commuting unitaries in $B:=M_n((SA)^sim)$.
Here $M_n((SA)^sim)=(SA)^sim otimes M_n$ where $M_n$ denotes the algebra of $ntimes n$ matrices, $SA$ is the suspension of $A$ and $(SA)^sim$ is $SA$ with a unit adjoined.
I cannot see how $u_1$ and $u_2$ are commuting. I suppose
$$M_n((SA)^sim)simeq (SA)^sim otimes M_n subset C(mathbb{T})otimes Aotimes M_n.$$
However, $uin M_n(A)$ and $operatorname{diag}(u’,1,dots, 1)in C(mathbb{T})otimes M_n$. That is to say, we should write
$$u_1=1otimes left(sum_{i,j} u_{i,j}otimes e_{i,j}right),$$
$$u_2=u’otimes e_{1,1}otimes 1 + sum_{knot =1} 1otimes e_{k,k}otimes 1.$$
But it does not appear to me that $u_1$ and $u_2$ are commuting?
tensor-products c-star-algebras k-theory
asked Nov 17 at 16:15
Fan
780313
add a comment |
up vote
1
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I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen.
Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a continuous function from $mathbb{R}$ to $mathbb{C}$ satisfying some properties). In the exercise, it says
Then $u_1:= 1otimes u$ and $u_2:=operatorname{diag}(u’,1,dots, 1)otimes 1$ are commuting unitaries in $B:=M_n((SA)^sim)$.
Here $M_n((SA)^sim)=(SA)^sim otimes M_n$ where $M_n$ denotes the algebra of $ntimes n$ matrices, $SA$ is the suspension of $A$ and $(SA)^sim$ is $SA$ with a unit adjoined.
I cannot see how $u_1$ and $u_2$ are commuting. I suppose
$$M_n((SA)^sim)simeq (SA)^sim otimes M_n subset C(mathbb{T})otimes Aotimes M_n.$$
However, $uin M_n(A)$ and $operatorname{diag}(u’,1,dots, 1)in C(mathbb{T})otimes M_n$. That is to say, we should write
$$u_1=1otimes left(sum_{i,j} u_{i,j}otimes e_{i,j}right),$$
$$u_2=u’otimes e_{1,1}otimes 1 + sum_{knot =1} 1otimes e_{k,k}otimes 1.$$
But it does not appear to me that $u_1$ and $u_2$ are commuting?
tensor-products c-star-algebras k-theory
asked Nov 17 at 16:15
Fan
780313
I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen.
Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a continuous function from $mathbb{R}$ to $mathbb{C}$ satisfying some properties). In the exercise, it says
Then $u_1:= 1otimes u$ and $u_2:=operatorname{diag}(u’,1,dots, 1)otimes 1$ are commuting unitaries in $B:=M_n((SA)^sim)$.
Here $M_n((SA)^sim)=(SA)^sim otimes M_n$ where $M_n$ denotes the algebra of $ntimes n$ matrices, $SA$ is the suspension of $A$ and $(SA)^sim$ is $SA$ with a unit adjoined.
I cannot see how $u_1$ and $u_2$ are commuting. I suppose
$$M_n((SA)^sim)simeq (SA)^sim otimes M_n subset C(mathbb{T})otimes Aotimes M_n.$$
However, $uin M_n(A)$ and $operatorname{diag}(u’,1,dots, 1)in C(mathbb{T})otimes M_n$. That is to say, we should write
$$u_1=1otimes left(sum_{i,j} u_{i,j}otimes e_{i,j}right),$$
$$u_2=u’otimes e_{1,1}otimes 1 + sum_{knot =1} 1otimes e_{k,k}otimes 1.$$
But it does not appear to me that $u_1$ and $u_2$ are commuting?
tensor-products c-star-algebras k-theory
asked Nov 17 at 16:15
Fan
780313
I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen.
Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a continuous function from $mathbb{R}$ to $mathbb{C}$ satisfying some properties). In the exercise, it says
Then $u_1:= 1otimes u$ and $u_2:=operatorname{diag}(u’,1,dots, 1)otimes 1$ are commuting unitaries in $B:=M_n((SA)^sim)$.
Here $M_n((SA)^sim)=(SA)^sim otimes M_n$ where $M_n$ denotes the algebra of $ntimes n$ matrices, $SA$ is the suspension of $A$ and $(SA)^sim$ is $SA$ with a unit adjoined.
I cannot see how $u_1$ and $u_2$ are commuting. I suppose
$$M_n((SA)^sim)simeq (SA)^sim otimes M_n subset C(mathbb{T})otimes Aotimes M_n.$$
However, $uin M_n(A)$ and $operatorname{diag}(u’,1,dots, 1)in C(mathbb{T})otimes M_n$. That is to say, we should write
$$u_1=1otimes left(sum_{i,j} u_{i,j}otimes e_{i,j}right),$$
$$u_2=u’otimes e_{1,1}otimes 1 + sum_{knot =1} 1otimes e_{k,k}otimes 1.$$
But it does not appear to me that $u_1$ and $u_2$ are commuting?
tensor-products c-star-algebras k-theory
tensor-products c-star-algebras k-theory
asked Nov 17 at 16:15
Fan
780313
asked Nov 17 at 16:15
Fan
780313
edited Nov 18 at 5:48
asked Nov 17 at 16:15
Fan
780313
asked Nov 17 at 16:15
Fan
780313
asked Nov 17 at 16:15
Fan
780313
780313
I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17
add a comment |
I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17
I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17
add a comment |
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I'm confused at the example for a standard unitary, $tmapsto exp(frac{it}{1+|t|})$ is not in $C_0(mathbb{R})^sim$ since the limit dose not exist when $|t|to +infty$. Where is my problem?
– C.Ding
Nov 18 at 1:27
I think $SA={fin C([0,1]to A): f(0)=f(1)=0}={fin C(mathbb{T}to A): f(1)=0}$.
– C.Ding
Nov 18 at 1:31
Besides, $(SA)^sim={fin C(mathbb{T}to A): f(1)=lambda, pi_mathbb{C}(f(z))=lambda,forall zin mathbb{T}}neq C(mathbb{T})otimes A$.
– C.Ding
Nov 18 at 2:02
@C.Ding yes you’re right
– Fan
Nov 18 at 5:13
@C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(mathbb{R})^sim$ with $C(mathbb{T})$
– Fan
Nov 18 at 6:17