much of the structure of a division algebra can be garnered from the structure of its subspaces. for example, a division algebra of prime degree is cyclic if and only if it has a subspace each of whose elements have $p$-power in the center. we consider this and other conditions on the subspace, and describe the growth of $\{ \dim_k (kak)^i : i \geq 1 \}$, for a maximal separable subfield $k$ of a central simple algebra $a$ and $a \in a \setminus k.$ we tie this to brauer factor sets, the trace form, and the commutator question. this is joint work with matzri, saltman, and vishne, and in part with chapman.