Topologizing UCTs for unital extensions
Let \(A\) and \(B\) be C*-algebras and let \(K(H)\) be the C*-algebra of compact operators on a separable Hilbert space. Kasparov has identified his odd KK-groups \(\mathrm{KK}_1(A,B)\) with a certain group of unitary equivalence classes of extensions of \(B\otimes K(H)\) by \(A\). If \(A\) is unital, we may restrict to extensions that are unital. Depending on whether we use unitaries in the multiplier algebra of \(B\otimes K(H)\) or in the Calkin algebra, we get the relations of weak and strong unitary equivalence of such unital extensions. The Universal Coefficient Theorem computes KK-groups in terms of homomorphisms between K-theory groups and extensions. If \(A\) is unital and satisfies the Universal Coefficient Theorem, then the two unital Ext-groups may also be described in such terms, using the class of the unit in \(\mathrm{K}_0(A)\) as an extra ingredient. The KK-groups have been equipped with a natural, possibly non-Hausdorff topology. And it has been shown that the exact sequence in the Universal Coefficient Theorem is an exact sequence of topological groups when it applies. In the present article, the unital Ext-groups are equipped with suitable topologies, making them pseudo-Polish groups. It is shown that the exact sequences that relate the unital Ext-groups to K-theory are exact as sequences of topological groups.
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