Inner derivations on ultraprime normed real algebras (Q1359071)

We show that the set of all inner derivations of an ultraprime real Banach algebra is closed within all bounded derivations. More concretely, we show that for such an algebra \(A\) there exists a positive number \(\gamma\) (depending only on the ``constant of ultraprimeness'' of \(A\)) satisfying \(\gamma|a+Z(A)|\leq|D_a|\) for all \(a\) in \(A\), where \(Z(A)\) denotes the centre of \(A\) and \(D_a\) denotes the inner derivation on \(A\) induced by \(a\). This result is an extension of the corresponding complex version obtained by \textit{M. Cabrera} and \textit{J. Martínez} [Proc. Am. Math. Soc. (to appear)]. The proof relies on the following theorem: ultraproducts of a family of central ultraprime real Banach algebras with a unit and with constant of ultraprimeness greater than or equal to a fixed positive constant \(K\) are central ultraprime Banach algebras with a unit. This fact is obtained via a general result for real Banach algebras that reads as follows: If \(A\) is a central real Banach algebra with a unit \textbf{1}, then for every \(a\) in \(A\) satisfying \(|\mathbf{1}+{\mathbf a}^2|<1\) we have \[ \left[1+ \sqrt{1- |\mathbf{1}+a^2|}\right]^2\leq 2(|I+M_a|+|D_a|), \] where \(M_a\) denotes the two-sided multiplication operator by \(a\) on \(A\).