  We study the algebraic $K$-theory of the ring of continuous functions on a
compact Hausdorff space with values in a local division ring, e.g., a local
field: We compute its negative $K$-theory and show its $K$-regularity. The
complex case reproves the results of Rosenberg, Friedlander--Walker, and
Corti\~nas--Thom. Our consideration in the real case proves two previously
unconfirmed claims made by Rosenberg in 1990. The algebraic nature of our
methods enables us to deal with the nonarchimedean and noncommutative cases
analogously.
