Inclusion into even, resp. odd indices is homotopic to identity?

Question

I came to this question, when i studied https://arxiv.org/pdf/math/0001094.pdf lemma 4.3. He states that for a Hilbert $B$-module $F$, the inclusions $j_i:F^\infty\to F^\infty$ into the even, repectively odd indices are homotopic to the identity. Here $F^\infty:=l^2(\mathbb{N})\otimes F$ is the Hilbert module of $F$ valued sequences bounded by the $l^2$-norm. With homotopic he means that there exists a path of isometries between them. The obvious homotopy $\sqrt{1-t^2}j_i+t^2id_{F^\infty}$ fails to be an isometry for $t\in (0,1)$. Can somebody help?