Set theory and C* algebras

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3. Tensor products

Add remark
Problem 3.1.
Does the number of C*-norms on $A\otimes _{alg}B$ for

$A=B=\mathcal{B}\left( H\right)$

$A=B=C^{\ast }\left( \mathbb{F}_{\infty }\right)$ , where $\mathbb{F}% _{\infty }$ is the free group on countably many generators

$A=B\left( H\right)$ and $B=\mathcal{Q}\left( H\right)$ where $% \mathcal{Q}\left( H\right)$ is the Calkin algebra

depend on the model of set theory?
Add remark

Problem 3.2.
What is the ideal structure of $\mathcal{B}\left( H\right) \otimes _{\min }\mathcal{Q}\left( H\right)$ ? Does it depend on the model of set theory?
Add remark

Problem 3.3.
Does $\otimes _{\min }$ or $\otimes _{\max }$ commute with forcing?

Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.