Pullbacks, $C(X)$-algebras, and their Cuntz semigroup

By Ramon Antoine, F. Perera, Luis Santiago. 2011

46 citations recorded by the source corpus.

In this paper we analyse the structure of the Cuntz semigroup of certain $C(X)$-algebras, for compact spaces of low dimension, that have no $\mathrm{K}_1$-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C$^*$-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily simple C$^*$-algebra of stable rank one and vanishing $\mathrm{K}_1$ for each closed, two sided ideal. We apply our results to study a variety of examples.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/85ea1d75de940e1bf2ac091e6fdd7409bbc82093

Citations in the corpus

• Every classifiable simple C*-algebra has a Cartan subalgebra (2018, unknown): “ →Aˆ n+1 suchthatthecompositionwiththemapC([0,1],E n+1)։ C([0,1],Eq n+1)induced by the canonical projection E n+1 ։E q n+1, Aˆ n ϕˆn −→Aˆ n+1 →C([0,1],E n+1)։C([0,1],E q n+1), 5 is of the form (3) C([0,1],E n)⊇Aˆ n ∋f →u∗ V(f) D(f) u, where u is a continuous path of unitaries [0,1]→U(Eq n+1), V(f)”