Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras. It's well known double dual of $C^*$-algebra is again a $C^*$ algebra.
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$
Can someone give me any reference?
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8$\begingroup$ No. Almost never. Think about AF algebras. $\endgroup$– Narutaka OZAWACommented Oct 7, 2021 at 6:17
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1$\begingroup$ @NarutakaOZAWA: Thank you. But ig same result is true in category of operator spaces? $\endgroup$– Math LoverCommented Oct 7, 2021 at 6:20
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1$\begingroup$ 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 $\endgroup$– Yemon ChoiCommented Oct 7, 2021 at 15:09
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$\begingroup$ 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. $\endgroup$– Robert FurberCommented Oct 8, 2021 at 19:44
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