# Explaining the group structure in homotopy classes of map $[X,K(H)]$

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 $$K(H) = M_\infty(\Bbb C)$$.

• Note that $K(H)$ is a vectorspace and contractible. Thus $[X,K(H)]$ is trivial if you mean this to just be continuous maps up to homotopy equivalence. – s.harp Mar 4 at 15:40
• Then from $C_0(X)$ to $K(H)$? – s.harp Mar 4 at 16:08
• Ok, so i tried to make the question more general, let me edit to the original case. Thanks a lot for pointing out. – CL. Mar 4 at 16:17

## 1 Answer

(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).

• Hi MaoWao, in retrospect, I sitll don't understand hwo the group structure, may you spell out an example? Thank you so much. I would be happy to give a bounty. – CL. Apr 14 at 20:39