AI 中文总结
本文解决了Dixmier关于单C*-代数谱的1967年问题，证明了在某些单C*-代数（如CAR代数与SL_3(Z)的约化交叉积）之间不存在Borel可定义的单射，并推广到可数amenable群与自由群的酉对偶，同时给出核非I型代数谱的Borel可定义双射存在性。
AI 中文摘要
我们解决了Dixmier在1967年提出的关于单C*-代数谱的问题。当两个谱之间的函数由标准Borel空间上的酉表示之间的Borel映射诱导时，该函数被称为Borel可定义函数。设Γ=SL_3(Z)，K是Γ关于模2^n的同余核的完备化，A是典范反对易关系(CAR)代数，B=C(K)⋊_r Γ是约化交叉积。代数A和B都是单的、可分的、含单位的、精确的且反初等的；A是核的，而B是非核的。不存在从B的谱到A的谱的Borel可定义单射。事实上，在B的纯态空间上存在一个概率测度，使得从B的谱到A的谱的任意Borel可定义函数的每个Borel提升几乎处处取值于单个酉等价类中。回答Simon Thomas的一个问题，我们还证明了：对于每个可数amenable群H，不存在从自由群F_∞（在无限多个生成元上）的酉对偶到H的酉对偶的Borel可定义单射。在F_∞的一族无限维不可约表示上存在一个固定的概率测度，使得任意Borel可定义函数的每个Borel提升几乎处处取值于单个酉等价类中。我们还证明了，与此相反，任意两个可分的核的非I型C*-代数的谱之间存在Borel可定义的双射。
英文摘要
We solve Dixmier's 1967 problem about spectra of simple $C^*$-algebras. A function between spectra is called Borel-definable when it is induced by a Borel map between standard Borel spaces of unitary representations. Let $Γ=\mathrm{SL}_3(\mathbb{Z})$, let $K$ be its completion with respect to the congruence kernels modulo $2^n$, let $A$ be the canonical anticommutation relations (CAR) algebra, and let $B=C(K)\rtimes_rΓ$ be the reduced crossed product. The algebras $A$ and $B$ are simple, separable, unital, exact, and antiliminary; $A$ is nuclear, whereas $B$ is nonnuclear. There is no Borel-definable injection from the spectrum of $B$ to the spectrum of $A$. In fact, there is a probability measure on the pure-state space of $B$ such that every Borel lift of a Borel-definable function from the spectrum of $B$ to the spectrum of $A$ takes values in a single unitary-equivalence class almost everywhere. Answering a question of Simon Thomas, we also prove that, for every countable amenable group $H$, there is no Borel-definable injection from the unitary dual of the free group $F_\infty $ on infinitely many generators to the unitary dual of $H$. There is a fixed probability measure on a family of infinite-dimensional irreducible representations of $F_\infty$ such that every Borel lift of a Borel-definable function takes values in a single unitary-equivalence class almost everywhere. We also show that, in contrast, the spectra of any two separable nuclear non-type-I $C^*$-algebras admit a Borel-definable bijection.
Comments33 pages
