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Andrej Bauer
Andrej Bauer
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Existance Existence of conditional expectations map onto subalgebras

Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existanceexistence of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

1.But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

2.Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

3.For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

  1. But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

  2. Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

  3. For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $C^*$ - algebra inclusion?

Existance of conditional expectations map onto subalgebras

Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existance of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

1.But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

2.Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

3.For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $C^*$ - algebra inclusion?

Existence of conditional expectations map onto subalgebras

Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existence of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

  1. But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

  2. Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

  3. For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $C^*$ - algebra inclusion?

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Anupam Ah
Anupam Ah
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Existance of conditional expectations map onto subalgebras

Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existance of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

1.But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

2.Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

3.For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $C^*$ - algebra inclusion?