For a Banach space X, let \(X^*\) be the dual, B(X) the Banach algebra of bounded linear operators, and \(B_ F(X)\), \(B_ K(X)\) and \(B_ I(K)\) the ideals of finite rank, compact, and integrable operators, respectivly. \textit{A. Grothendieck} [Mem. Am. Math. Soc. 16, 140 p. (1955; Zbl 0064.355)] showed that there is a natural isometric linear isomorphism \(\theta:B(X)\to B_ K(X)^{**}\) whenever X is reflexive and satisfies the approximation property. The paper under review uses the Arens products on \(B_ K(X)^{**}\) to give a converse to an extension of Grothendieck's theorem. In particular the following are among eight conditions shown to be equivalent: (1) \(B_ F(X)\) is dense in \(B_ K(X)\) and there is an isometric algebra isomorphism \(\Theta\) of \(B(X^{**})\) onto \(B_ K(X)^{**}\) with respect to the first Arens product such that \(\Theta(K^{**})\) equals the image of K under the natural map of \(B_ K(X)\) into \(B_ K(X)^{**}\) for each \(K\in B_ K(X);\) (2) there is a left identity element of norm one for the bidual of the closure of \(B_ F(X)\) with the first Arens product; (3) the first Arens representation of the bidual of the closure of \(B_ F(X)\) on \(X^{**}\) is an isometry onto \(B(X^{**});\) (4) \(X^*\) has the metric approximation property and \(B_ F(X^*)\) is dense in \(B_ I(X^*).\) When these conditions hold, a formula is given for the second Arens product which shows that \(B_ K(X)\) is Arens regular if and only if X is reflexive. These results extend and clarify results of \textit{J. Hennefeld} [Pac. J. Math. 29, 551-563 (1969; Zbl 0182.169) and Ill. J. Math. 23, 681-686 (1979; Zbl 0458.46032)]. Condition (4) is closely related, but not identical, to requiring that \(X^*\) have the Radon-Nikodym property as well as the metric approximation property.