2005/05/12 by Cintia C. Silva, Silva, Cintia C.
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.math/0505261
This is a PhD Thesis supervised by Severino T. Melo
arxiv created 2005/05/12 · arxiv updated 2009/12/01
Let A denote the smallest C*-subalgebra of the algebra of all bounded operators on L2(R) containing: (i) all multiplications a(M) by functions a in C[-∞,+∞], (ii) all multiplications eijM, j in Z, and (iii) all operators of the form F-1b(M)F, where F denotes the Fourier transform and b is in C[-∞,+∞]. 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(S1,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 K0(A) is isomorphic to Z and that K1(A) is isomorphic to Z2. The isomorphism from E/K into C(S1,K) can be to extended to a C*-homomorphism γfrom A into the direct sum of two copies of C(S1,B), where B denotes the algebra of all bounded operators on L2(Z). We prove that the image of γis isomorphic to the direct sum of two copies of the crossed product of C[-∞,+∞] 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 K0(A) is isomorphic to Z and that K1(A) is isomorphic to Z2.