Symmetric amenability and the nonexistence of Lie and Jordan derivations
Symmetric amenability and the nonexistence of Lie and Jordan derivations
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DOI：
10.1017/s0305004100075010
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发表时间：
1996-10
影响因子：
0.8
通讯作者：
NEl 1RU
中科院分区：
文献类型：
--
作者：
NEl 1RU
A·M·辛克莱证明了：如果是半单Banach代数，则从Into到的每一个连续Jordan导子都是导子([12，定理3·3]；‘Jordan导子’在下面的第6节中命名)。如果是Banach-双模，人们可以考虑从Into到的Jordan导子，然后问辛克莱的定理是否仍然成立。这一领域的更新工作见[1]。简单的例子表明，它不能对所有的模和所有的半单代数都成立。然而，对于更受限的代数类，包括C*-代数，我们得到了一个肯定的结果，我们发展了两种方法。第一个依赖于对称顺从性，这是我们在第2、3和4节中首次提出的顺从Banach代数理论的发展。如果一个Banach代数有一个由对称张量组成的近似对角线，则它是对称顺从的。大多数(但不是全部)顺从性Banach代数是对称顺从性的，人们可以证明对称顺从性的结果类似于[8]中关于顺从性的结果。然而，与舒适性不同，对称性舒适性似乎没有简洁的同源特征。我们的一个结果[定理6·2]是：如果对称服从，则A-双模的每一个连续Jordan导子都是导子。特殊的技巧使得这个结果可以推广到其他代数，例如所有的C*-代数。乔丹派生的这种方法在第6节中介绍。
A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.