The global dimension theorem for non-unital and certain other separable $C^*$-algebras

In this article we prove that the global homological dimension of a separable $C^*$-algebra containing a bi-ideal of finite codimension that cannot be complemented as a subalgebra is at least 2. As a consequence we also obtain this bound for the global dimension of separable $C^*$-algebras without an identity and for finite-dimensional separable $\operatorname{GCR}$-algebras.