Mixing
tracial states on a finite dimensional $C^*$algebra
Clash Royale CLAN TAG#URR8PPP
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If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.
operator-theory operator-algebras c-star-algebras
asked Jul 21 at 18:04
mathrookie
437211
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If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.
operator-theory operator-algebras c-star-algebras
asked Jul 21 at 18:04
mathrookie
437211
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.
operator-theory operator-algebras c-star-algebras
asked Jul 21 at 18:04
mathrookie
437211
If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.
operator-theory operator-algebras c-star-algebras
asked Jul 21 at 18:04
mathrookie
437211
asked Jul 21 at 18:04
mathrookie
437211
asked Jul 21 at 18:04
mathrookie
437211
asked Jul 21 at 18:04
mathrookie
437211
437211
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
A finite-dimensional C$^*$-algebra is of the form
$$
A=bigoplus_j=1^m M_k_j(mathbb C).
$$
The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form
$$
phi(bigoplus_j=1^m x_j)=sum_j=1^m t_j, tau_k_j(x_j),
$$
where $tau_k_j$ is the tracial state on $M_k_j(mathbb C)$ and $t_1,ldots, t_m$ are convex coefficients.
answered Jul 21 at 19:42
Martin Argerami
116k1071164
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A finite-dimensional C$^*$-algebra is of the form
$$
A=bigoplus_j=1^m M_k_j(mathbb C).
$$
The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form
$$
phi(bigoplus_j=1^m x_j)=sum_j=1^m t_j, tau_k_j(x_j),
$$
where $tau_k_j$ is the tracial state on $M_k_j(mathbb C)$ and $t_1,ldots, t_m$ are convex coefficients.
answered Jul 21 at 19:42
Martin Argerami
116k1071164
add a comment |Â
up vote
1
down vote
accepted
A finite-dimensional C$^*$-algebra is of the form
$$
A=bigoplus_j=1^m M_k_j(mathbb C).
$$
The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form
$$
phi(bigoplus_j=1^m x_j)=sum_j=1^m t_j, tau_k_j(x_j),
$$
where $tau_k_j$ is the tracial state on $M_k_j(mathbb C)$ and $t_1,ldots, t_m$ are convex coefficients.
answered Jul 21 at 19:42
Martin Argerami
116k1071164
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A finite-dimensional C$^*$-algebra is of the form
$$
A=bigoplus_j=1^m M_k_j(mathbb C).
$$
The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form
$$
phi(bigoplus_j=1^m x_j)=sum_j=1^m t_j, tau_k_j(x_j),
$$
where $tau_k_j$ is the tracial state on $M_k_j(mathbb C)$ and $t_1,ldots, t_m$ are convex coefficients.
answered Jul 21 at 19:42
Martin Argerami
116k1071164
A finite-dimensional C$^*$-algebra is of the form
$$
A=bigoplus_j=1^m M_k_j(mathbb C).
$$
The number of states is indeed uncountable. Traces are precisely "convex combinations" of the traces in each block. That is, any trace on $A$ is of the form
$$
phi(bigoplus_j=1^m x_j)=sum_j=1^m t_j, tau_k_j(x_j),
$$
where $tau_k_j$ is the tracial state on $M_k_j(mathbb C)$ and $t_1,ldots, t_m$ are convex coefficients.
answered Jul 21 at 19:42
Martin Argerami
116k1071164
answered Jul 21 at 19:42
Martin Argerami
116k1071164
answered Jul 21 at 19:42
Martin Argerami
116k1071164
answered Jul 21 at 19:42
Martin Argerami
116k1071164
116k1071164
add a comment |Â
add a comment |Â
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