The bar construction of an algebra as an E-infinite Hopf algebra

We prove that the bar construction of an $E_{\mathrm{\infty }}$ algebra forms an $E_{\mathrm{\infty }}$ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt-Eccles operad. (The surjection operad and the Barratt-Eccles operad are classical $E_{\mathrm{\infty }}$ operads.)