Cintia C. Silva | Searcharxiv

arXiv2005

K-Theory of pseudodifferential operators with semi-periodic symbols

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2π-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol σand an operator-valued symbol γ. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to σand to γ. For the second computation, we prove that the image of γis isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

math.OA↗

arXiv2005

K-teoria de operadores pseudodiferenciais na reta com simbolos semiperiodicos (in Portuguese)

Let A denote the smallest C*-subalgebra of the algebra of all bounded operators on L^2(R) containing: (i) all multiplications a(M) by functions a in C[-\infty,+\infty], (ii) all multiplications e^{ijM}, j in Z, and (iii) all operators of the form F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. It is known that the principal symbol mapping extends to a surjective C*-homomorphism σfrom A into C(M), where M is a certain compactification of two copies of R. It is also known that E, the kernel of σ, contains the compact ideal K and that the quotient of E by K, is isomorphic to the direct sum of two copies of C(S^1,K). Using the explicit form of these two isomorphisms, we are able to compute the connecting mappings in the cyclic exact sequence in K-theory associated to the homomorphism σand to proof that K_0(A) is isomorphic to Z and that K_1(A) is isomorphic to Z^2. The isomorphism from E/K into C(S^1,K) can be to extended to a C*-homomorphism γfrom A into the direct sum of two copies of C(S^1,B), where B denotes the algebra of all bounded operators on L^2(Z). We prove that the image of γis isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by the translation-by-one automorphism. Using the Pimsner-Voiculescu exact sequence, we then compute the K-theory of the image of γ. That leads to a second proof that K_0(A) is isomorphic to Z and that K_1(A) is isomorphic to Z^2.

math.OA↗