K-group and similarity classification of operators

Abstract

Let H be a complex separable Hilbert space and L ( H ) denote the collection of bounded linear operators on H . An operator A in L ( H ) is said to be a Cowen–Douglas operator if there exist Ω , a connected open subset of complex plane C , and n , a positive integer, such that (a) Ω ⊂ σ ( A ) = { z ∈ C | A - z is not invertible } ; (b) ran ( A - z ) := { y | ( A - z ) x = y , x ∈ H } = H for z in Ω ; (c) ⋁ z ∈ Ω ker ( A - z ) = H ; and (d) dim ker ( A - z ) = n for z in Ω . In the paper, we give a similarity classification of Cowen–Douglas operators by using the ordered K -group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen–Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott–Gong in (Documenta Math. 7 (2002) 255; On the classification of simple inductive limit C * -algebras, II: The isomorphism theorem, preprint.) (also see (J. Funct. Anal. (1998) 1; Ann. Math. 144 (1996) 497; Amer. J. Math. (1996) 187)).