Paul S. Muhly, Baruch Solel
We study conjugacy invariants for completely positive maps that are inspired by the concept of curvature introduced for commuting d-tuples of contractions by Arveson
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In the recent preprint (OA/0006224) S. Parrott proved the equality between the Arveson`s curvature and the Fredholm index of a "pure" contraction with finite defect numbers. In the present note one derives a similar formula in the "non-pure" case.
It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the inde…
We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold, we define a secondary higher index class. Non-vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature. In the special case where one metric is induced from the other by…
The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature
该研究构建了Gromov-Lawson相对指标的粗版本并确立其基本性质，通过在Roe代数K-理论上构造反向映射，建立了由任意余维数嵌入子流形推导一致正数量曲率阻碍的抽象机制。
A numerical index is introduced for semigroups of completely positive maps of $\Cal B(H)$ which generalizes the index of E_0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E_0-semigroup, provided that the dilation is minimal.
Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimen…
Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
本文研究局部紧群适当作用流形上具有一致正标量曲率的完备不变度量存在性，利用等变 Callias 型指标定理给出障碍，并在连通李群情形下推广 Lawson-Yau 构造消除障碍。
Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity
本文利用Roe代数商的相对高阶指标理论，研究非紧流形无穷远处正数量曲率度量的存在性障碍，提出相对粗Baum-Connes猜想并验证其部分情形，应用于扩展图乘积。
Let M be a complete Riemannian manifold, D a Dirac-type operator on M whose Weitzenbock curvature is uniformly positive on the complement of a subset Z of M. We show that the coarse index of D is localized to the K-theory of the coarse C*-algebra of Z. Applications are discussed, including a coarse form of the relative index theorem.
Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature
本文为李群等距作用的流形建立等变椭圆算子及 Callias 型算子指标理论，并由此给出非余紧流形上正标量曲率度量的存在性障碍。
In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $λ$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $λ$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $λ$-Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open ques…
It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus …
Quantitative partitioned index theorem and noncompact band-width
本文提出量化分割流形索引理论，用于非紧致超曲面，并证明了Gromov带宽猜想的非紧致Riemann带版本。
Let $P$ be a completely positive map on $M_n(\mathbb{C})$ and let $E_P$ be the associated \emph{GNS}-$C^*$-correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of $E_P$, $\mathcal{O}(E_P)$, is strongly Morita equivalent to the Cuntz algebra $\mathcal{O}_{d(P)}$, where $d(P)$ is the index of $P$.
Filling Radius, Quantitative $K$-theory and Positive Scalar Curvature
该研究借助定量算子$K$-理论与指标理论，为一致正标量曲率的完备spin流形的填充半径建立了定量上界。
Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact Kähler manifolds
本文在紧Kähler流形上建立了Dolbeault-Dirac算子的$\mathrm{L}^p$-Banach非交换几何框架，证明了其闭实现是双扇形的且具有有界$\mathrm{H}^\infty$泛函演算，并得到$\mathrm{L}^p$-Hodge分解和Fredholm算子的指标等于全纯Euler示性数。
Let A be a local conformal net of factors on the circle with the split property. We provide a topological construction of soliton representations of the tensor product of n copies of A, that restrict to true representations of subnet inviant under cyclic permutations (cyclic orbifold). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the the cyclic orbifold has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector the…
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