Uniform Tracial Ultrapower

The uniform tracial ultrapower of a unital -algebra with nonempty tracial state space is the quotient of bounded sequences in by the ideal of sequences whose uniform tracial 2-seminorm tends to zero along a free ultrafilter . The seminorm is

A limit trace has the form with . The uniform tracial completion consists of bounded sequences that are Cauchy in this seminorm, modulo sequences tending to zero in it. Each trace on has a designated extension to , and is defined with these extensions. This quotient differs from the norm ultrapower, whose defining ideal uses operator norms.