Definition of $K_1(A)$ for a Banach algebra $A$

Question

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or take $\bigcup_{n\in\mathbb{N}}GL_n(A^+)$ and quotient by the component containing the identity. The two give the same thing. Besides giving some conveniences, is there a conceptual reason for taking the first option?