Homotopy classification of projections in the corona algebra of a non-simple $C\sp *$-algebra

TL;DR

This paper investigates the classification of projections in the corona algebra of certain non-simple $C^*$-algebras, providing conditions for liftability and equivalence, and illustrating the complexity of their $K$-theoretic relationships.

Contribution

It introduces a homotopy classification framework for projections in the corona algebra of non-simple $C^*$-algebras, extending previous theories and providing explicit examples of non-coinciding equivalence notions.

Findings

Conditions for liftability of projections to the multiplier algebra.

Characterizations of projection equivalences in $K_0$, Murray-von Neumann, unitary, and homotopic senses.

Examples demonstrating the non-coincidence of different equivalence notions.

Abstract

We study projections in the corona algebra of $C(X)\otimes K$ where $X=[0,1],[0,\infty),(-\infty,\infty)$, and $[0,1]/\{0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. Then we also characterize the conditions for two projections to be equal in $K_0$-group, Murray-von Neumann equivalent, unitarily equivalent, and homotopic from the weakest to the strongest. In light of these characterizations, we construct examples showing that any two equivalence notions do not coincide, which serve as examples of non-stable K-theory of $C\sp*$-algebras.