Timeline for Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$?
Current License: CC BY-SA 4.0
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| Oct 8, 2021 at 19:44 | comment | added | Robert Furber | If both inductive limits (i.e. colimits) are interpreted in the category of C$^*$-algebras, the answer is no, as already mentioned. However, if the colimit $\varinjlim A_n^{**}$ is interpreted in the category of W$^*$-algebras (with normal *-homomorphisms) the answer is yes because $({-})^{**}$ is the left adjoint to the forgetful functor from W$^*$-algebras to C$^*$-algebras. | |
| Oct 7, 2021 at 15:09 | comment | added | Yemon Choi | The answer to your new question is still no, and I claim that if you think about how @NarutakaOZAWA's example/hint works, you should see how to adapt it | |
| Oct 7, 2021 at 6:20 | comment | added | Math Lover | @NarutakaOZAWA: Thank you. But ig same result is true in category of operator spaces? | |
| Oct 7, 2021 at 6:17 | comment | added | Narutaka OZAWA | No. Almost never. Think about AF algebras. | |
| Oct 7, 2021 at 5:34 | history | asked | Math Lover | CC BY-SA 4.0 |