Abstract While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements a and b operator commute, while a 2 and b do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner. These include for instance the spaces of self-adjoint elements of unital C⁎-algebras. We show that elements a and b in a JB-algebra operator commute if and only if they generate an associative subalgebra of mutually operator commuting elements, and hence operator commutativity in JB-algebras is as well-behaved as it can be. Letting U a denote the quadratic operator of a, we also show that positive a and b operator commute if and only if U a b 2 = U b a 2 . We use this result to conclude that the unit interval of a JB-algebra is a sequential effect algebra as defined by Gudder and Greechie.

中文翻译：

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Jordan 算子代数中的交换性

摘要 虽然 Jordan 代数是可交换的，但算子交换性对于一般的非结合 Jordan 代数表现不佳。当两个元素的乘积算子交换时，称这些元素为算子交换。在某些 Jordan 代数中，运算符对易可能表现不佳，例如，元素 a 和 b 运算符可交换，而 a 2 和 b 不运算符可交换。在这篇论文中，我们研究 JB-algebras，真正的 Jordan 代数，它们也是 Banach 空间以兼容的方式。这些包括例如单位 C⁎-代数的自伴随元素的空间。我们证明 JB 代数算子中的元素 a 和 b 交换当且仅当它们生成一个相互算子交换元素的关联子代数，因此 JB 代数中的算子可交换性是表现良好的。让 U a 表示 a 的二次算子，我们还表明正 a 和 b 算子交换当且仅当 U ab 2 = U ba 2 。我们使用这个结果得出结论，JB 代数的单位区间是由 Gudder 和 Greechie 定义的序列效应代数。