$K_{0}$-groups and strongly irreducible decompositions of operator tuples

An operator tuple $\mathbf{T}=(T_{1},\ldots,T_{n})$ is called strongly irreducible (SI), if the joint commutant of $\mathbf{T}$ does not any nontrivial idempotent operator. In this paper, we study the uniqueness of finitely strong irreducible decomposition of operator tuples up to similarity by $K$-theory of operator algebra, and give the algebraically similarity invariants of the Cowen-Douglas tuple with index 1 by using $K_{0}$-group of the commutant of operator tuples. As an application, we calculate $K_{0}$-groups of some multiplier algebras, and describe the similarity of backwards multishifts on Drury-Arveson space by means of inflation theory.