解析簇的算子代数
Operator algebras for analytic varieties
摘要：
研究了不可约完备Pick核乘子代数的同构问题。这些正是Drury-Arveson空间的乘子代数$\mathcal M$对单位球$\mathbb{B}_d$的全纯子簇$V$的限制$\mathcal M_V$。我们发现$\mathcal M_V$与$\mathcal M_W$完全等距同构当且仅当$W$是$V$在球的双全纯自同构下的像.在这种情况下，同构是酉实现的。这是加强，然后表明，当$d<\infty$，每个等距同构是完全等距。当两个这样的代数（代数）同构的问题进行了研究。当$V$和$W$都是不可约簇的有限并集和$\mathbb{B}_d$与$d<\infty$中的离散簇时，则$\mathcal M_V$和$\mathcal M_W$之间的同构确定了簇之间的双全纯（具有乘子坐标）;并且同构是与此函数的复合。这些映射自动是弱$*$连续的。我们提出了一些例子表明，相反的失败在几个方面。我们讨论了几种特殊的情况，其中逆不成立-特别是，平滑曲线和Blaschke序列。我们还讨论了与簇相关的范数闭代数，并指出了一些不同之处。
英文摘要：
We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\mathcal M_V$ of the multiplier algebra $\mathcal M$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball $\mathbb{B}_d$. We find that $\mathcal M_V$ is completely isometrically isomorphic to $\mathcal M_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when $d<\infty$, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When $V$ and $W$ are each a finite union of irreducible varieties and a discrete variety in $\mathbb{B}_d$ with $d<\infty$, then an isomorphism between $\mathcal M_V$ and $\mathcal M_W$ determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-$*$ continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold---particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.
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