Large subalgebras
  We define and study large and stably large subalgebras of simple unital C*-algebras. The basic example is the orbit breaking subalgebra of a crossed product by Z, as follows. Let X be an infinite compact metric space, let h be a minimal homeomorphism of X, and let Y be a closed subset of X. Let u be the standard unitary in C* (Z, X, h). The Y-orbit breaking subalgebra is the subalgebra of C* (Z, X, h) generated by C (X) and all elements f u for f in C (X) such that f vanishes on Y. If intersects each orbit of h at most once, then the Y-orbit breaking subalgebra is large in C* (Z, X, h). Large subalgebras obtained via generalizations of this construction have appeared in a number of places, and we unify their theory in this paper.   We prove the following results for an infinite dimensional simple unital C*-algebra A and a stably large subalgebra B of A:   B is simple and infinite dimensional.   If B is stably finite then so is A, and if B is purely infinite then so is A.   The restriction maps from the tracial states of A to the tracial states of B and from the normalized 2-quasitraces on A to the normalized 2-quasitraces on B are bijective.   When A is stably finite, the inclusion of B in A induces an isomorphism on the semigroups that remain after deleting from the Cuntz semigroups of A and B all the classes of nonzero projections.   B and A have the same radius of comparison. 
N. Christopher Phillips
math.OA