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Pseudodifferential operators with discontinuous symbols: $$K$$-theory and the index formula. (English. Russian original) Zbl 0879.58073
Funct. Anal. Appl. 26, No. 4, 266-275 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 45-56 (1992).
Let $${\mathcal A}$$ be the $$C^*$$-algebra generated by differential operators of order zero on a smooth compact boundaryless manifold $$M$$ of dimension $$n$$. Let $${\mathcal S} ={\mathcal A}/{\mathcal K}$$ be the quotient modulo the ideal of compact operators. $$K$$-theoretic properties allow the reduction of the computation of the index to an ideal $$I\subset {\mathcal S}$$. This ideal, since the symbols of the operators in $$\mathcal A$$ may have discontinuities, splits into a direct sum of two ideals, $$J$$ and $$I_{\text{cont}}$$, where $$I_{\text{cont}}$$ consists only of continuous symbols. The algebra $${\mathcal S}$$ has two series of representations allowing this splitting: the ideal $$I_{\text{cont}}$$ is associated with one-dimensional representations, while $$J$$ is connected with infinite-dimensional ones (in $$L_2(S_{n-1})$$). For each of these ideals the corresponding term in the index formula can be written as $$\langle\text{ch}_*\xi,\text{ch}^*\Psi\rangle$$, where $$\text{ch}_*:K_1({\mathcal L})\to HC_{\text{odd}}({\mathcal L})$$, $$\text{ch}^*:K^1({\mathcal L})\to HC^{\text{odd}}({\mathcal L})$$ are the Chern characters, $$HC$$ denotes cyclic (co)homology, $$\xi$$ is the class of a symbol in $$K_1({\mathcal L})$$ and $$\Psi$$ is the class of an extension in $$K^1({\mathcal L})$$. For $${\mathcal L}=I_{\text{cont}}$$ the class $$\Psi$$ is a classical pseudodifferential extension of the algebra of symbols and $$\text{ch}^*\Psi$$ reduces to a Todd class. For $${\mathcal L}=J$$ the authors find an expression for $$\text{ch}_*\xi$$, $$\text{ch}^*\Psi$$ using general techniques and obtain an analytic index formula. Passing to $$J$$ solves the index problem for odd-dimensional manifolds. There is a topological obstruction in the even-dimensional case. See also two other papers by the authors [Leningr. Math. J. 2, No. 5, 1085-1110 (1991); translation from Algebra Anal. 2, No. 5, 165-188 (1990; Zbl 0722.35096) and St. Petersbg. Math. J. 3, No. 5, 1089-1101 (1992); translation from Algebra Anal. 3, No. 5, 155-167 (1991; Zbl 0773.46035)].

##### MSC:
 58J22 Exotic index theories on manifolds 35S05 Pseudodifferential operators as generalizations of partial differential operators 19K56 Index theory 47G30 Pseudodifferential operators 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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##### References:
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