论文标题
单一元素的扩张
Dilations of unitary tuples
论文作者
论文摘要
我们使用扩张理论和矩阵范围研究了所有$ d $ tublase $ u =（u_1，\ ldots，u_d）$的空间。给定了两个$ d $ -tuples $ u $和$ v $生成c*-algebras $ \ mathcal a $和$ \ mathcal b $，我们寻求最小的扩张常数$ c = c（u，v）$，因此$ u \ u \ prec prec cv $，我们的意思是，我们的意思是$ u $是$*$*$*$ - isomorphic $ cv a $ cv a $ cv a $ cv $ cv。这引起了公制\ [d_d（u，v）= \ log \ max \ {c（u，v），c（v，u）\} \] $*$*$*$ - 单位单元的等效类。我们还考虑了metric \ [d_ {hr}（u，v）= \ inf \ weft \ {\ | U'-v'\ |： d_ {hr}（u，v）\ leq k d_d（u，v）^{1/2}。 \ \]令$u_θ$为通用的单一元组$（u_1，\ ldots，u_d）$满足$ u_ \ el u_ \ ell u_k = e^{iθ_{k，\ ell}} u_k u_ \ u_ \ el ell我们发现$ c（u_θ，u_ {θ'}）\ leq e^{\ frac {1} {4} \ | |θ-θ-θ'\ |} $。从此，我们恢复了Haagerup-rordam和gao的结果，该结果存在映射$θ\ mapSto u（θ）\ b（h）^d $中的结果，这样$ u（θ）\ simu_θ$和\ simu_θ$和\ [\ | u（θ）-u（θ）-u（= us） - {θ'}'}'}）特殊利益的\是：非承诺的Unitaries $ {\ Mathrm u} $的通用$ D $ - $ D $ -D $ -TUPLE，免费的Haar Unitaries $ U_F $和通用$ D $ d $ - 通勤Unitaries unitaries $ u_0 $。我们获得界限\ [2 \ sqrt {1- \ frac {1} {d}}} \ leq c（u_f，u_0）\ leq 2 \ sqrt {1- \ frac {1} {1} {2d}}}}}}。 \]从中，我们恢复了通用Unitaries $ c（{\ Mathrm u}，u_0）的通行器上限\ leq \ sqrt {2d} $。在情况下，$ d = 3 $，我们获得了新的下限$ c（{\ mathrm u}，u_0），u_0）\ geq 1.858 $在先前已知的下限$ c（{\ mathrm u}，u_0），u_0）\ geq \ geq \ sqrt {3} $上进行改进。
We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric \[ d_D(u,v)=\log\max\{c(u,v),c(v,u)\} \] on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_Θ$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{iθ_{k,\ell}} u_k u_\ell$, where $Θ=(θ_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_Θ, u_{Θ'})\leq e^{\frac{1}{4}\|Θ-Θ'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $Θ\mapsto U(Θ)\in B(H)^d$ such that $U(Θ)\sim u_Θ$ and \[ \|U(Θ)-U({Θ'})\|\leq K\|Θ-Θ'\|^{1/2}. \] Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] From this, we recover Passer's upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.
