Why is $U_n^+(A)/U_n^+(A)_0\simeq U_n(A^+)/U_n(A^+)_0$ for a unital C*-algebra $A$?

Question

Was reading Wegge-Olsen's K-theory and C*-algebras and in chapter 4 they state that $U_n^+(A)/U_n^+(A)_0\simeq U_n(A^+)/U_n(A^+)_0$, to show that he says that $(a_{ij})+1_n$ is invertible (unitary) if and only if $(a_{ij})$ is invertible (unitary). Given the product in $A^+$ that doesn't even follow for $n=1$, is he already using the isomorphism there is between $A^+$ and $A\oplus \mathbb{C}$? If that's the case, how would the argument above translate to the usual product in $A^+$?

Answer 1

If $A$ is unital, the isomorphic between $A^+$ and $A\oplus \mathbb{C}$ is $$(a,c)\mapsto(a{\color{red}{+c1_A}})\oplus c.$$ Where the element in $A^+$ is of form $(a,c)(a\in A,c\in\mathbb{C})$ and the product is $(a,c)(a',c')=(aa'+ac'+ca',cc')$; the element in $A\oplus \mathbb{C}$ is of form $a\oplus c(a\in A,c\in\mathbb{C})$ and the product is $(a\oplus c)(a'\oplus c')=aa'\oplus cc'$.