Toeplitz operators and the Roe-Higson type index theorem in Riemannian surfaces. (English) Zbl 1365.19006

Seto, Tatsuki

Tokyo J. Math. 39, No. 2, 423-439 (2016).

The main theorem in the paper under review is stated as follows.
Let \(M\) be an oriented complete Riemannian manifold. Assume that \(M\) has dimension two and is a partitioned manifold in the sense that there is a closed hypersurface \(N\) in \(M\) such that \(M\) is decomposed into two submanifolds \(M^{\pm}\) by \(N\), so that \(M= M^+ \cup M^-\) and \(N\) is the intersection of \(M^{\pm}\) and is the boundary of both \(M^{\pm}\) (such as for instance, \(N= S^1\) the circle in \(M=S^2\setminus \{(0, \pm1)\}\) the sphere with the north and south poles removed). Let \(E\) be a \(\mathbb Z_2\)-graded spin bundle over \(M\) with a grading and \(D\) the Dirac operator on \(E\). Take \(\varphi\) a \(\mathrm{GL}_l(\mathbb C)\) (the general linear group over complex numbers)-valued, continuously differentiable map on \(M\), that is assumed to be bounded with its gradient bounded, and the inverse \(\varphi^{-1}\) is also bounded. Define an invertible \(u_{\varphi}\) as the diagonal sum of \(\varphi\) and the identity map, adjointed by the perturbed invertible for \(D\) by the identity, which belongs to the general linear group over the \(C^*\)-algebra containing the Roe algebra for \(M\) as a closed two-sided ideal, defined so as adding diagonal matrices with diagonal elements as bounded continuous functions on \(M\) to the Roe algebra. Then the Connes pairing between the \(K_1\)-theory class of \(u_{\varphi}\) and the Roe cocycle defined on a dense subalgebra of the Roe algebra for \(M\) is explicitly computed to be equal to the Fredholm index of the Toeplitz operator of \(\varphi\) restricted to \(N\), up to a scalar multiplication.
It may be viewed as a partial but non-trivial extension of the Roe-Higson index (trivial) theorem (by J. Roe [Lond. Math. Soc. Lect. Note Ser. 135, 187–228 (1988; Zbl 0677.58042)] and N. Higson [Topology 30, No. 3, 439–443 (1991; Zbl 0731.58065)]) to even-dimensional partitioned manifolds.

Reviewer: Takahiro Sudo (Nishihara)

MSC:

19K56	Index theory
46L87	Noncommutative differential geometry
47A53	(Semi-) Fredholm operators; index theories
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
58J20	Index theory and related fixed-point theorems on manifolds

Keywords:

index theory; noncommutative geometry; coarse geometry; Hilbert transform; Roe algebra; Dirac operator; Fredholm index; Toeplitz operator

Citations:

Zbl 0677.58042; Zbl 0731.58065