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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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