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Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product. Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$?

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    $\begingroup$ Because $A^{**}$ is a von Neumann algebra, it has no outer derivations, so its first Hochschild cohomology must be $0$. This is stated in a paper of Kadison and Ringrose here projecteuclid.org/download/pdf_1/euclid.acta/1485889690 with reference to citations 9, 10, and 13. $\endgroup$
    – Ashwin Trisal
    Commented Feb 18, 2019 at 23:22

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