Morphisms of Extensions of C*-Algebras: Pushing Forward the Busby Invariant

Abstract

We study completions of diagrams of extensions of C*-algebras of the form 0 wwÄ v wwÄ v wwÄ v wwÄ 0 0 wwÄ v wwÄ v wwÄ v wwÄ 0 in which all three C*-algebras in one of the rows and either the ideal or the quotient in the other are given, along with the three morphisms between them. We find universal solutions to all four of these problems under restrictions of varying severity, on the given vertical maps and describe the solutions in terms of push-outs and pull-backs of certain diagrams. Our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products. This leads to new results about the K-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. We also obtain new results about extensions of matricial field C*-algebras, verifying partially a conjecture of Blackadar and Kirchberg. Finally, we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes.

Morphisms of extensions of Hilbert C*-modules

“…It is obvious that if W ′ ⊆ W is a submodule of W such that Φ(V ) is an ideal submodule of W ′ , then we have W ′ ⊆ I(Φ(V )). We can state now an analog of the Theorem 2.4 stated for a C * -algebra case in [8].…”

“…What about underlying morphisms of C * -algebras? The existence of a morphism ε : B → B 1 of C * -algebras is guaranteed by [8] and from the pullback construction we know that a morphism E is an ε-morphism.…”

Almost commuting orthogonal matrices

“…By [5,10] an almost commuting commuting pair of unitary matrices U, V is close to a commuting pair of unitary matrices if and only if the Bott index of the pair is zero. So long as we measure noncommutativity via the operator norm, use complex scalars, don't worry about algorithms or quantitative results, we can end here the story on almost commuting unitary matrices.…”

“…By results from [5,10] this means that close to any pair of almost commuting real orthogonal there is a pair of exactly commuting unitary (though not necessarily real orthogonal) matrices. Unlike in the case of self-dual unitaries, it turns out that there is no new obstruction to finding real orthogonal approximates.…”

“…Both of these questions can be studied by looking at the class of C * -algebras called noncommutive CW complexes (NCCW complexes). Glossing over the technical details, the general story about NCCW complexes and approximate relations in is as follow: If relations corresponds to a one-or two-dimensional NCCW complex, then complex matrices that almost satisfy these relations can be perturbated to ones that exactly satisfy the relations, if and only if the invariants coming from the K-theory of the 2-cells vanish ( [5]). …”