2 added 25 characters in body

By a result of Schochet, the category of C*-algebras with homotopy equivalences and "Schochet fibrations" is a pointed category of fibrant objects whose homotopy category is the ordinary homotopy category of C*-algebras.

It was observed by Andersen and Grodal that the above pointed category of fibrant objects is not the full subcategory of fibrant objects of a Quillen model category.

Here is a recent reference reviewing both results: O. Uuye: Homotopy Theory for C*-algebras

1

By a result of Schochet, the category of C*-algebras with homotopy equivalences and "Schochet fibrations" is a pointed category of fibrant objects whose homotopy category is the ordinary homotopy category of C*-algebras.

It was observed by Andersen and Grodal that the above pointed category of fibrant objects is not the full subcategory of fibrant objects of a Quillen model category.

Here is a recent reference: O. Uuye: Homotopy Theory for C*-algebras