In this paper Aityah, Bott and Shapiro give an alternative definition of (relative) $K$-theory groups $K(X,Y)$ using sequences of bundles (this group is denoted by $L_n(X,Y)$ where $n$ is the length of such sequence). This treatment deal with compact pairs, i.e. it gives an alternative description of $K(X,Y)$ where $X$ is compact and $Y \subset X$ is closed. In locally compact setting, when $Y=\emptyset$, the $K$-theory group $K(X)$ is defined as $\tilde{K}(X^+)$ (reduced K-theory of one point compactification of $X$). In the book "Spin Geometry" by Lawson and Michelsohn one can find the appropriate definition of $L_n(X)$ when $X$ is only locally compact compact: this is formed using sequence of bundles which are exact out of some compact set.
Is it true that if $X$ is only locally compact then $K(X)$ and $L_n(X)$ are isomorphic? If so, do the standard "difference element" construction (explained in the above paper for compact pairs) do the job?
