Close operator algebras and almost multiplicative maps
نویسندگان
چکیده
where A,B ⊂ B(H), and the distance between A⊗Mn and B ⊗Mn is measured in B(H) ⊗ Mn ∼= B(H). We also investigate the consequences of “complete closeness”, i.e. what can be said when dcb(A,B) is small? For example, if dcb(A,B) is small, then any projection p ∈ A ⊗ Mn can be suitably approximated by a projection q ∈ B ⊗ Mn, leading an isomorphism K0(A) → K0(B) which maps [p]0 to [q]0. This strategy is due to Khoshkam [8], who used it to obtain an isomorphism K∗(A) ∼= K∗(B) when d(A,B) is sufficiently small and A is nuclear. The Cuntz semigroup Cu(A) of a C∗-algebra A is a highly refined invariant obtained from equivalence classes of positive elements of A ⊗ K. In contrast to projections, where the class in K0 is invariant under small perturbations, the Cuntz class of a positive element a ∈ A ⊗ K is sensitive to small modifications in norm. Nevertheless, it is possible to extend Khoshkam’s isomorphism to this setting.
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