I'm studying on the book of Douglas: "Banach algebra techniques in operator theory" and there is a passage I don't understand, and I hope you can give me a hand.
"A continuous function $f$ from $X$ to $\mathbb{T}$ (the circle group) determines first an element $\{f\}$ of $\pi^1(X)$ (the group of homotopy classes of continuous maps from $X$ to $\mathbb{T}$) and second, viewed as an invertible function on $X$, determines a coset $f + G_0$ of $\Lambda$. ($G_0$ is the connected component of the invertible functions that contains the $\mathbb{1}$ function and $\Lambda$ is the abstract index group for $C(X)$.)"
The part that I don't understand is the "coset $f + G_0$ of $\Lambda$."
$\Lambda$ is a group, not a subgroup (or is the subgroup of itself but this case is irrelevant.) The cosets are sets of the form: $element\ of \ the \ group(SUBGROUP)$. How can $f + G_0$ be a coset?
Please, try to clarify to me in what sense $f + G_0$ is a coset of $\Lambda$. Thank you very much.
If $f + G_0$ is intended as $fG_0$ everything does make sense, because this is an element of $\Lambda$. But it's not an orthographic error because this notation it's repeated in all the theorem, and the cosets of $G_0$ are denoted, as normal, with the notation $fG_0$ earlier in the book. I judge this a very bad choice of notation... :-(