If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension $\widetilde{SA}$ can then be identified with the algebra of continuous $a:[0,1] \to B$ such that $a(0)=a(1) \in \mathbb{C} \subset B$ where $B$ is either $A$ or $\tilde A$ according as $A$ is already unital or not.

In C*-algebra K-theory, we have the isomorphism $K_1(A) \to K_0(SA)$. This is not the isomorphism generally equated with Bott periodicty, but it is still a significant isomorphism (note: I am thinking of the definition of $K_1$ in terms of unitaries). In the commutative context, this isomorphism comes from a clutching construction for vector bundles. We could formulate it like this:

For $X = (X,x_0)$ a pointed compact Hausdorff space, let $\mathrm{Clutch}_n(X)$ denote the set of homotopy classes of maps $X \to U(n)$ and let $\mathrm{Vect}_n(X)$ denote the set of isomorphism classes of complex vector bundles over $X$ with $n$-dimensional fibre over $x_0$. Then, there is a bijection $$ \mathrm{Clutch}_n(X) \to \mathrm{Vect}_n(SX). $$ Here, $SX$ is the pointed suspension got by forming the cylinder $X \times I$ and collapsing $(\{x_0\} \times I) \cup (X \times \{0,1\})$ to a point. $I$ is the closed unit interval.

Now, we reformulate for the C*-algebraic setting:

For $A$ be a C*-algebra, let $\mathrm{Clutch}_n(A)$ denote the set of homotopy classes of unitaries in $M_n(\tilde A)$. Let $\mathrm{Vect}_n(A)$ denote the set of equivalence classes of projections in $M_\infty(\tilde A)$ whose scalar part is a rank-$n$ projection. Here $M_\infty(\tilde A)$ means the $*$-algebra of infinite matrices with finitely many nonzero entries in $\tilde A$ and equivalence of projections in $M_\infty(\tilde A)$ is given, for example, by homotopies which stay stay inside one of the finite matrix algebras. There is a map $$ \mathrm{Clutch}_n(A) \to \mathrm{Vect}_n(SA) $$ given as follows. Given $u \in U_n(\tilde A)$, choose $w : [0,1] \to U_{2n}(\tilde A)$ such that $w(0) = 1_{2n}$ and $w(1) = \left( \begin{smallmatrix} u & 0 \\ 0 & u^* \end{smallmatrix} \right)$. We send the class of $u$ to the class represented by the projection $p_u = w 1_n w^* \in M_{2n}(\widetilde{SA})$.

The above construction is just the usual one appearing in most introductory books on K-theory, see e.g. Rørdam, Larsen & Laustsen or Wegge-Olsen's books. Standard arguments will show you that the map described above does not depend on choice of $u$ or $w$, and as well that the map is surjective. For example, this is done Theorem 7.3.5 of Wegge-0lsen's book in all gory detail. But, at least in the references I've looked at, it is only shown that, if $p_u \sim p_v$ for $u,v \in \mathrm{Clutch}_n(A)$, then $u \oplus 1_k \sim v \oplus 1_k$ for some $k$, i.e. $u$ and $v$ are stably-homotopic. This is enough to get the K-theory isomorphism, but...

Question: I would like to see an an example where the map described above is not injective.

The fact that you get a bijection in the classical commutative setting is rather nice, and I'm very curious what goes wrong in general.


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