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 (depending only on the "constant of ultraprimeness" of A) satisfying r// a+Z(A)/// D_a //for all a in A, where Z(A) denotes the centre of A and Da denotes the inner derivation on A induced by a. This result is an extension of the corresponding complex version obtained by the authors in [Proc. Amer. 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 1, then for every a in A satisfying // 1 +a^2//< 1 we have[1+1-//1 + a^2//]}^2 <2(// I+M_a//+// D_a// ) where Ma denotes the two-sided multiplication operator by a on A.
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机译:我们表明,超素实Banach代数的所有内部派生集在所有有界派生中都是封闭的。更具体地说,我们证明对于这样的代数A,存在一个满足r // a + Z(A)// / D_a //的正数(仅取决于A的“超素常数”)。在A中,Z(A)表示A的中心,Da表示由a诱导的A的内推导。此结果是作者在[Proc。阿米尔。数学。 Soc。,出现]。证明依赖于以下定理:具有单位且超素常数大于或等于固定正常数K的中心超素实Banach代数族的超积是带单位的中心超素Banach代数。这个事实是通过实Banach代数的一般结果得出的,该一般结果如下:如果A是一个中心的实Banach代数,单位为1,那么对于A中的每个满足// 1 + a ^ 2 // 1的a [1 + 1-// 1 + a ^ 2 //]} ^ 2 <2(// I + M_a // + // D_a //)其中,Ma用A表示a的双面乘法算子。
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