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If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M,\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• An elementary proof would be ideal, but I also take a textbook reference.
• The comments after my earlier question indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M\cap\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• An elementary proof would be ideal, but I also take a textbook reference.
• The comments after my earlier question indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M,\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• The comments after my earlier question indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M,\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• An elementary proof would be ideal, but I also take a textbook reference.
• The comments after my earlier question indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M,\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• The comments after my earlier question "https://mathoverflow.net/questions/442854/intersection-of-type-i-von-neumann-algebra-factors" indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M,\mathcal N$ a type-I von-Neumann algebra factor?

Notes:

• The comments after my earlier question indicate that this is not the case for an intersection of infinitely many factors, but they crucially rely on the "infinite" part of it.
• I know that the intersection of two von-Neumann algebras is a von-Neumann algebra.
• I don't know if the intersection of two factors is a factor.