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This is a problem I encountered in Jesse Peterson's Notes on Von Neumann Algebras. I want to show the following: given C* algebra A, suppose for any C* algebra B, every positive map from B to A is completely positive, then A is an abelian C* algebra.

I have no idea how to show this. Any help is greatly appreciated.

Thank you very much.

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See

Tomiyama, Jun, On the difference of n-positivity and complete positivity in C*-algebras, J. Funct. Anal. 49, 1-9 (1982). ZBL0497.46039.

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  • $\begingroup$ Thank you very much. $\endgroup$
    – user151245
    Jan 29, 2020 at 14:19

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