Does the following metric metrize the weak*topology on the state space?

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Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply that the energy metric metrizes the weak* topology on S(A)?




functional-analysis operator-algebras noncommutative-geometry




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    Would you mind introducing this metric here? Thank you.
    – Tomek Kania
    Aug 1 at 14:46






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    Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
    – s.harp
    2 days ago










  • The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
    – TerryL
    yesterday














up vote
0
down vote

favorite
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Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply that the energy metric metrizes the weak* topology on S(A)?




functional-analysis operator-algebras noncommutative-geometry




share|cite|improve this question















  • 2




    Would you mind introducing this metric here? Thank you.
    – Tomek Kania
    Aug 1 at 14:46






  • 1




    Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
    – s.harp
    2 days ago










  • The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
    – TerryL
    yesterday












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply that the energy metric metrizes the weak* topology on S(A)?




functional-analysis operator-algebras noncommutative-geometry




share|cite|improve this question











Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply that the energy metric metrizes the weak* topology on S(A)?





functional-analysis operator-algebras noncommutative-geometry





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 1 at 14:01









TerryL

335




335







  • 2




    Would you mind introducing this metric here? Thank you.
    – Tomek Kania
    Aug 1 at 14:46






  • 1




    Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
    – s.harp
    2 days ago










  • The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
    – TerryL
    yesterday












  • 2




    Would you mind introducing this metric here? Thank you.
    – Tomek Kania
    Aug 1 at 14:46






  • 1




    Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
    – s.harp
    2 days ago










  • The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
    – TerryL
    yesterday







2




2




Would you mind introducing this metric here? Thank you.
– Tomek Kania
Aug 1 at 14:46




Would you mind introducing this metric here? Thank you.
– Tomek Kania
Aug 1 at 14:46




1




1




Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
– s.harp
2 days ago




Seconding @Tomek's remark. However there is one more thing to be said, if your algebra $A$ is finite dimensional (in the sense that it is a finite dimensional vector space) then all metrics induce the same topology on the dual. So if your energy metric extends to the entire dual in a way that addition and scalar multiplication are continuous, then it metrizises the weak* topology on $S(A)$. All this is by virtue of $A$ being finite dimensional.
– s.harp
2 days ago












The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
– TerryL
yesterday




The C*-algebra does happen to be finite-dimensional! What do you recommend as a good reference for this fact?
– TerryL
yesterday















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