Abstract
In this paper, we show that for unital, separable C*-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor K⁎ are natural equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable C⁎-algebras of stable rank one, is a well-defined continuous functor from the category of C⁎-algebras of stable rank one to the category Cu_. We prove that this new functor and the functor K_ are naturally equivalent for unital, separable, K-pure C⁎-algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of C⁎-algebras of real rank zero.
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