Speaker: **N. Christopher Phillips** (Univ. Oregon)

Title: Large subalgebras of crossed product C^{*}-algebras

Date/Time: December 12 (Wed.), 2012, 4:30-6:00

Room: 118 Math. Sci. Building

Abstract: This is work in progress; not everything has been checked.

We define a "large subalgebra" and a "centrally large subalgebra"
of a C^{*}-algebra. The motivating example is what we now call the
"orbit breaking subalgebra" of the crossed product by a minimal
homeomorphism h of a compact metric space X. Let v be the standard
unitary in the crossed product C^{*} (Z, X, h). For a closed subset
Y of X, we form the subalgebra of C^{*} (Z, X, h) generated by C (X)
and all elements f v for f in C (X) such that f vanishes on Y. When
each orbit meets Y at most once, this subalgebra is centrally large in
the crossed product. Crossed products by smooth free minimal actions
of Z^{d} also contain centrally large subalgebras which are simple
direct limits, with no dimension growth, of recursive subhomogeneous
algebras.

If B is a large subalgebra of A, then the Cuntz semigroups of A and B are the almost the same: if one deletes the classes of nonzero projections, then the inclusion is a bijection on what is left. Also (joint work with Dawn Archey), if B is a centrally large subalgebra of A, and B has stable rank one, then so does A. Moreover, if B is a centrally large subalgebra of A, if B is Z-stable, and if A is nuclear, then A is Z-stable.