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.