AI 中文总结
构造一个可分单幺剩余有限维 $C^*$-代数 $A$，其极大张量平方含真等距，从而 $A$ 拟对角线而 $A\ max A$ 非拟对角线，通过性质 $(T)$ 子群与膨胀共轭实现。
AI 中文摘要
我们构造了一个可分、单幺且剩余有限维的 $C^*$-代数 $A$，其极大张量平方包含一个真等距。特别地，$A$ 是拟对角线的，而 $A\ max A$ 不是。该代数由从一个显式可数仿射群的同余商获得的有限正则表示生成。一个性质 $(T)$ 子群在极大张量平方中确定了一个对角 Kazhdan 投影。通过膨胀的共轭作用，该投影等价于一个真子投影。严格性由一个拟正则表示检测，该表示的两个因子限制下降到指定的代数 $A$。我们给出了该下降所需的范数估计，并显式构造了真等距。
英文摘要
We construct a separable unital residually finite-dimensional $C^*$-algebra $A$ whose maximal tensor square contains a proper isometry. In particular, $A$ is quasidiagonal whereas $A\tmax A$ is not. The algebra is generated by finite regular representations obtained from congruence quotients of an explicit countable affine group. A property~$(T)$ subgroup determines a diagonal Kazhdan projection in the maximal tensor square. Conjugation by a dilation makes this projection equivalent to a proper subprojection. Strictness is detected by a quasi-regular representation whose two factor restrictions descend to the specified algebra $A$. We give the norm estimate needed for this descent and construct the proper isometry explicitly.
Comments8 pages, mostly generated by an LLM. The Author claims no novelty, but improved readability
