K-teoria de operadores pseudodiferenciais com simbolos semi-periodicos no cilindro (in Portuguese)

Let A denote the C*-algebra of bounded operators on L^2(RxS^1) generated by: (a) multiplications by smooth functions on S^1; (b) multiplications by continuous functions on the two point compactification of R; (c) multiplications by 2\pi-periodic continuous functions; (d) the operator L given by the inverse of the square root of the identity operator minus the Laplacian operator on RxS^1; and (e) operators of the form DL, where D is either the differencial operator on R or a first order differential operator on S^1 with smooth coefficients. Let \sigma be the complex-valued symbol on A that arises from the Gelfand map of the C*-algebra A/E, where E is the commutator ideal of A. This is the continuous extension of the usual principal symbol of pseudodifferential operators. It is known that E contains the compact ideal K of A and E/K is isomorphic to C(S^1, K)\oplus C(S^1, K), where here K is the algebra of all compact operators on ZxS^1. This isomorphism can be extended to a C*-homomorphism \gamma from A into C(S^1, B)\oplus C(S^1, B), where B denotes the algebra of all bounded operators on ZxS^1. We compute the index map in the six-term exact sequence associated to \sigma, using a Fedosov-Atiyah-Singer index formula. Given A\dagger generated by classes of operators in (a), (d) and (e), we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of A\dagger by an automorphism. We use the Pimsner-Voiculescu exact sequence to compute the K-theory of the crossed product. So, we can prove that K0(A) is isomorphic to Z^5 and K1(A) is isomorphic to Z^4.

💡 Research Summary

The paper undertakes a comprehensive K‑theoretic analysis of a C(^*)‑algebra (A) acting on the Hilbert space (L^{2}(\mathbb{R}\times S^{1})). The algebra is generated by five families of operators: (a) multiplication by smooth functions on the circle (S^{1}); (b) multiplication by continuous functions on the two‑point compactification (\overline{\mathbb{R}}=\mathbb{R}\cup{\pm\infty}); (c) multiplication by (2\pi)-periodic continuous functions; (d) the operator (L=(I-\Delta)^{-1/2}), where (\Delta) is the Laplacian on the cylinder; and (e) operators of the form (DL), where (D) is either the derivative with respect to the (\mathbb{R})‑coordinate or a first‑order differential operator on (S^{1}) with smooth coefficients.

The first step is to identify the commutator ideal (E) of (A). It is shown that (E) contains the compact ideal (\mathcal{K}) and that the quotient (E/\mathcal{K}) is isomorphic to the direct sum (C(S^{1},\mathcal{K})\oplus C(S^{1},\mathcal{K})). This reflects the fact that the “ends’’ of the cylinder ((\pm\infty) along the (\mathbb{R})‑direction) give rise to two independent copies of continuous compact‑operator valued functions on the circle.

Next, the Gelfand map (\sigma:A\to A/E) is used to define a complex‑valued symbol on (A). This symbol extends the usual principal symbol of pseudodifferential operators to a continuous function on the whole cylinder, incorporating the semi‑periodic behaviour in the (\mathbb{R})‑direction. The six‑term exact sequence in K‑theory associated with (\sigma) is written down, and the index map (\partial) is computed by means of a Fedosov‑Atiyah‑Singer index formula. The crucial observation is that the combination of (L) with the first‑order differential operators yields an elliptic symbol on (\mathbb{Z}\times S^{1}) whose topological degree can be read off directly; consequently (\partial) returns an integer equal to this degree.

To make the computation tractable, the authors introduce a subalgebra (A^{\dagger}) generated only by (a), (d) and (e). They construct a C(^*)‑homomorphism (\gamma:A\to C(S^{1},B)\oplus C(S^{1},B)), where (B) denotes the bounded operators on (\mathbb{Z}\times S^{1}). The image of (\gamma) is proved to be isomorphic to the direct sum of two copies of the crossed product (A^{\dagger}\rtimes_{\alpha}\mathbb{Z}), where (\alpha) is the automorphism induced by a unit translation in the (\mathbb{R})‑direction. This crossed‑product description captures the non‑commutative torus‑like structure inherent in the semi‑periodic symbols.

The K‑theory of the crossed product is then obtained via the Pimsner‑Voiculescu six‑term exact sequence. Because the automorphism (\alpha) acts trivially on K‑theory (the induced map (\alpha_{*}) is the identity), the sequence splits, yielding
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