Large subalgebras

TL;DR

This paper introduces the concept of large and stably large subalgebras within simple unital C*-algebras, providing a unified framework and analyzing their properties and relationships, especially in the context of orbit breaking subalgebras.

Contribution

It unifies the theory of large subalgebras, proves their properties, and establishes their relationship with the parent algebra, including simplicity, trace bijections, and invariance of comparison properties.

Findings

Large subalgebras are simple and infinite dimensional.

Stably finite or purely infinite properties are preserved from subalgebra to algebra.

Trace and quasitrace structures are bijectively related between subalgebra and algebra.

Abstract

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.