Let X be a compact
metric space of dimension d. In previous work, we have shown that for all
sufficiently large n, every element of the identity component U0(C(X) ⊗ Mn) of
the unitary group U(C(X) ⊗ Mn) is a product of at most 4 exponentials
of skewadjoint elements. On the other hand, if X is a manifold then some
elements of U0(C(X) ⊗ Mn) require at least about d∕n2 exponentials. Similar
qualitative behavior (with different bounds: 5 and d∕(2n2)) holds for the problem
of factoring elements of the identity component inv0(C(X) ⊗ Mn) of the
invertible group as products of exponentials of arbitrary elements of the algebra.
In this paper, we identify the sets of finite products of 10 other types of
elements of inv0(C(X) ⊗ Mn), and we show that the minimum lengths of
factorizations have the same qualitative behavior as the two exponential factorization
problems above (after a suitable minor modification in 3 of the 10 cases). We
obtain upper bounds for large n that range from 5 to 22, and lower bounds
approximately of the form rd∕n2 with r ranging from 1/16 to 2. The classes of
elements we consider all make sense in general unital C∗-algebras. They are:
unipotents, positive invertibles, selfadjoint invertibles, symmetries, ∗-symmetries,
commutators of elements of inv0(A) and U0(A), accretive elements, accretive
unitaries, and positive-stable elements (real part of spectrum positive). The
last three classes are the ones requiring the slight modification; without it,
lengths of factorization behave like exponential length rather than exponential
rank.
