局部紧量子群上的一致连续性
Uniform continuity over locally compact quantum groups
摘要：
对Kustermans-Vaes意义下的局部紧量子群$G$，定义了$L^\infty(G)$中左一致连续元的$LUC(G)$空间.这个定义涵盖了局部紧群上通常的左一致连续函数和Fourier代数上的Granirer一致连续泛函。证明了$LUC(G)$是一个包含$C^*$ -代数$C_0(G)$并包含在其乘子代数$M(C_0(G))$中的算子系统.我们用这个部分回答一个开放的问题Bedos-Tuset：如果$G$是合作的，那么存在一个左不变平均$M(C_0(G))$是足够的$G$是顺从的。进一步研究了$L^\infty(G)$的弱概周期元空间$WAP(G)$：它是包含$C_0(G)$的$L^\infty(G)$中的闭算子系统，并且--对于相容的$G$ --包含在$LUC(G)$中.最后，我们证明了-在一定的条件下，这是总是满足的，如果$G$是一个群体-运营商系统$LUC(G)$是一个$C^*$ -代数。
英文摘要：
We define, for a locally compact quantum group $G$ in the sense of Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements in $L^\infty(G)$. This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an operator system containing the $C^*$-algebra $C_0(G)$ and contained in its multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study the space $WAP(G)$ of weakly almost periodic elements of $L^\infty(G)$: it is a closed operator system in $L^\infty(G)$ containing $C_0(G)$ and--for co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain conditions, which are always satisfied if $G$ is a group--the operator system $LUC(G)$ is a $C^*$-algebra.
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