It is Higon's notes on Index Theory, Ex 3.16, pg 43 where he describes a group structure (where I suppose works in more general context) of the graded $*$-algebra homotopy classes of map $$ [S,K(H)]$$
Here $S=C_0(\Bbb R)$ is the graded $*$-algebra given by even and odd functions as grading.
$K(H)$ is the $C^*$ algebra of operators on a graded Hilbert space $H=H_0 \oplus H_1$ whose homogenous subspaces are both infinite dimensional.
He claims this structure is given by the map
$$K(H) \oplus K(H) \rightarrow K(H \oplus H) \cong K(H)$$
associated to a graded unitary isomorphism $$H \oplus H \cong H \quad (*)$$
My questions:
(0) Previously in the notes it is written that $K(H)$ is the space of compact operators. With the same notation, it is defined as space of graded $*$ operators of $H=H_0 \oplus H_1$. I believe compact condition is still required? Where is it used?
(i) how is this independent of choice of the a graded unitary isomoprhism at $(*)$
(ii) How does such an isomoprhism in $(*)$ exist?
(iii) What's wrong with simply defining $(f+g)_x:= f_x+g_x$ point wise?
@MaoWao, In retrospect I still don't understand how two maps compose, Suppose $$ f \mapsto \begin{pmatrix} f_1 & 0 \\ 0 & f_2 \end{pmatrix}, g \mapsto \begin{pmatrix} g_1 & 0 \\ 0 & g_2 \end{pmatrix} $$ where $f_1, f_2 , g_1, g_2 \in K(H)$ , can we define addition by $$ f_1 +g_2 = \begin{pmatrix} f_1 & \\ & f_2 \\ && g_1 \\ &&& g_2 \end{pmatrix} $$ simply regarding it as a larger matrix, since we regard $K(H) = M_\infty(\Bbb C)$.
(0) You could do the same for all bounded linear operators instead of the compact ones, but compact operators are much more relevant to $K$-theory (or rather $M_\infty(\mathbb{C})=\bigcup_n M_n(\mathbb{C})$, but this is not even a $C^\ast$-algebra, so one considers its completion $K(\ell^2)$ instead).
(i) If $u\colon H\to H$ is a graded unitary isomorphism, then the $\ast$-homomorphism $\alpha $ and $u^\ast \alpha(\cdot)u$ are homotopic through the $\ast$-homorphisms $u_t^\ast \alpha(\cdot)u_t$, where $(u_t)$ is a continuous path of graded unitaries connecting $u$ and $1$. You can use this to see that different graded unitaries in $(\ast)$ give rise to homotopic $\ast$-homomorphisms in the construction of the group structure.
(ii) This is "just" arithmetic with cardinal numbers. Remember that a unitary isomorphism of Hilbert spaces is just a bijection between fixed orthonormal bases. Thus, if $(e_i)_{i\in I}$ is an orthonormal basis of $H_0$, you need to find a bijection $I\to I\sqcup I$, and the same for $H_1$. This is possible because $I$ is assumed to be infinite. In the most important case when $H_0, H_1$ are separable, you can for example split the natural numbers into even and odd ones.
(iii) The sum of $\ast$-homomorphisms is usually not an $\ast$-homomorphism (you break multiplicativity).