Algebraic K-theory assigns to a ring R a sequence of abelian groups K_i(R), measuring the complexity of the category of projective R-modules. The individual groups K_i(R) are the homotopy groups of a homotopical gadget called the K-theory spectrum K(R), which encodes extra gluing data between the different groups. When R is a commutative ring, the spectrum K(R) admits the structure of commutative ring spectrum, which homotopically encodes the tensor product operation on projective R-modules. As in classical algebra, a commutative ring spectrum S has a spectrum of units, or invertible elements, gl_1(S). In my talk, I will discuss a joint work with Kiran Luecke in which we study the spectra of the form gl_1(K(R)) for commutative rings R. Among other things we show that, Zariski locally, the first two homotopy groups of gl_1(K(R)) are not glued to the rest, so that gl_1(K(R)) splits into a product of a spectrum which has only zero and first non-trivial homotopy groups with a simply connected spectrum.