2005/11/04 by Severino T. Melo, Melo, Severino T., Cintia C. Silva +1
Mathematics · #46L80 #47G30 #Advanced Operator Algebra Research #Algebra over a field #Bounded function #Combinatorics #Discrete mathematics #FOS: Mathematics #K-Theory and Homology (math.KT) #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematical proof #Mathematics #Operator (biology) #Operator Algebras (math.OA) #Product (mathematics) #Pseudodifferential operators #Pure mathematics #Sequence (biology) #Spectral Theory in Mathematical Physics #Trigonometric functions #math.KT #math.OA #msc:46L80 #msc:47G30
paper · pdf · doi:10.48550/arxiv.math/0511098
arxiv created 2005/11/04 · openalex publication_date 2005/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a∈ C[-∞,+∞], (ii) all multiplications by 2π-periodic continuous functions and (iii) all Fourier multipliers F-1b(M)F, where F denotes the Fourier transform and b is in C[-∞,+∞]. 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⊕ 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[-∞,+∞] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.