Wigner’s Theorem states that bijections of the set [Formula: see text] of one-dimensional projections on a Hilbert space [Formula: see text] that preserve transition probabilities are induced by either a unitary or an anti-unitary operator on [Formula: see text] (which is uniquely determined up to a phase). Since elements of [Formula: see text] define pure states on the C*-algebra [Formula: see text] of all bounded operators on [Formula: see text] (though typically not producing all of them), this suggests possible generalizations to arbitrary C*-algebras. This paper is a detailed study of this problem, based on earlier results by R. V. Kadison (1965), F. W. Shultz (1982), K. Thomsen (1982), and others. Perhaps surprisingly, the sharpest known version of Wigner’s Theorem for C*-algebras (which is a variation on a result from Shultz, with considerably simplified proof) generalizes the equivalence between the hypotheses in the original theorem and those in an analogous result on (anti-)unitary implementability of Jordan automorphisms of [Formula: see text], and does not yield (anti-)unitary implementability itself, far from it: abstract existence results that do give such implementability seem arbitrary and practically useless. As such, it would be fair to say that there is no Wigner Theorem for C*-algebras.

中文翻译：

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(否) C*-代数的 Wigner 定理

维格纳定理指出，保持转移概率的一维投影在希尔伯特空间 [公式：参见文本] 上的集合 [公式：参见文本] 的双射是由 [公式：参见text]（由一个阶段唯一确定）。由于 [Formula: see text] 的元素定义了 [Formula: see text] 上所有有界运算符的 C*-algebra [Formula: see text] 上的纯状态（尽管通常不会产生所有这些），这表明可能的泛化为任意 C*-代数。本文是对这个问题的详细研究，基于 RV Kadison (1965)、FW Shultz (1982)、K. Thomsen (1982) 等人的早期结果。也许令人惊讶的是，已知最清晰的 C*-代数 Wigner 定理版本（这是 Shultz 结果的变体，用相当简化的证明）概括了原始定理中的假设与[公式：见文本]的 Jordan 自同构的（反）酉可实现性的类似结果中的假设之间的等价性，并且不产生（反）酉可实现性本身，远非如此：确实赋予这种可实现性的抽象存在结果似乎是任意的，实际上是无用的。因此，可以公平地说 C*-代数不存在 Wigner 定理。确实提供了这种可实现性的抽象存在结果似乎是任意的，实际上是无用的。因此，可以公平地说 C*-代数不存在 Wigner 定理。确实提供了这种可实现性的抽象存在结果似乎是任意的，实际上是无用的。因此，可以公平地说 C*-代数不存在 Wigner 定理。