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Jon Bannon
Jon Bannon
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Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator operator closure of the convex hull of the conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator closure of the convex hull of the conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak operator closure of the convex hull of the conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

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Jon Bannon
Jon Bannon
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The existence Examples of propernon-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator closure of the convex hull of unitarythe conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

The existence of proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator closure of the convex hull of unitary conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

Examples of non-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator closure of the convex hull of the conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

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Jon Bannon
Jon Bannon
  • 7.1k
  • 6
  • 69
  • 112

The existence of proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak-operator closure of the convex hull of unitary conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.