Toeplitz operators and the Roe-Higson type index theorem
Toeplitz operators and the Roe-Higson type index theorem
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Toeplitz 算子和 Roe-Higson 型索引定理
DOI：
10.4171/jncg/287
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发表时间：
2014
影响因子：
0.9
通讯作者：
Tatsuki Seto
中科院分区：
文献类型：
--
作者：
Tatsuki Seto
设$M$是一个完备的黎曼流形，并假定$M$被一个超曲面$N$分割。本文在非紧流形上引入了一类新的函数C_{w}（M），它比Higson函数代数稍大.在属于$C_{\mathrm{w}}（M）$的$\phi$中，利用Kasparov积构造了$M$的Roe代数的$K_{1}$-群中的指标类$\mathrm{Ind}（\phi，D）$。它被认为是Roe的奇数索引类的对应物。最后证明了Connes的$\mathrm{Ind}（\phi，D）$与Roe的循环1 $-上循环的配对等于N$上Toeplitz算子的Fredholm指标.这是Roe-Higson指标定理在偶数维分块流形上的推广。
Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $\phi$ that belongs to $C_{\mathrm{w}}(M)$ we construct an index class $\mathrm{Ind}(\phi , D)$ in $K_{1}$-group of the Roe algebra of $M$ by using the Kasparov product. It is supposed to be a counterpart of Roe's odd index class. We finally prove that Connes' pairing of $\mathrm{Ind}(\phi , D)$ and Roe's cyclic $1$-cocycle is equal to the Fredholm index of a Toeplitz operator on $N$. This is an extension of the Roe-Higson index theorem to even-dimensional partitioned manifold.