# Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 235-253.

K-groups of Banach algebras and strongly irreducible decompositions of operators**Authors**: Yang Cao (1), Junsheng Fang (2), and Chunlan Jiang (2)

**Author institution:**(1) Department of Mathematics, Jilin University, Chang chun 130023, China

(2) Academia Sinica, Institute of Mathematics, Beijing 100080, China

(3) Department of Applied Mathematics, Hebei University of Technology, Tianjin 300130, China

**Summary:**A bounded linear operator $T$ on the Hilbert space $\sH$ is called strongly irreducible if $T$ does not commute with any nontrivial idempotent operator. One says that $T$ has a finite $\SI$ decomposition if $T$ can be written as the direct sum of finitely many strongly irreducible operators. In this paper, we use the ${\rm K}_0$-group of the commutant of operators to characterize operators with unique finite $\SI$ decomposition up to similarity. Also we show that the ${\rm K}_0$-group of $H^\infty(\Omega)$ is isomorphic to the integers, where $\Omega$ is simply connected.

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