projective unitary stable $G$-equivariant bundle $-P$ associated to the additive inverse of $P$ using the isomorphism from Theorem 4.8 of [@barcenasespinozajoachimuribe] to the Borel cohomology group and form similarly the class $P+ -P$. The classification of Projective Unitary stable $G$-equivariant bundles [@barcenasespinozajoachimuribe] gives:

There exists a Hilbert bundle $E$ over X such that the zero class $P+-P $ admits a representative given by the principal bundle associated to the Bundle $L_{HS}(E_{P},{E_{P}}^*)$ of Hilbert-Schmidt homomorphisms of $E_{P}$ to the dual bundle ${{E}_P}^*$.

\[definitiongeometricktheory\] Let $X$ be a connected $G$-space and $P$ a projective unitary stable $G$-equivariant bundle over $X$, with associated Hilbert space ${\mathcal{H}}$.

Denote by ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ the space of $*$-strong continuous, Fredholm operators on ${\mathcal{H}}$ and recall that the group of projective unitary operators $P{\mathcal{U}}({\mathcal{H}})$ in the ${}^*$-strong topology acts continuously by conjugation on ${\ensuremath{{\mathrm{Fred}}}}( {\mathcal{H}})$. The twisted equivariant $K$-Theory groups denoted by $K^{-p}_G(X;P)$ are defined in [@barcenasespinozajoachimuribe] as the $p-th$ homotopy groups (based a at a suitable section $s$) of the space of $G$-invariant sections of the bundle with fibre ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ and structural group $ P {\mathcal{U}}( {\mathcal{H}})$ associated to $P$. In symbols,

$$K^{-p}_G(X;P):= \pi_p \left( \Gamma(X;{\ensuremath{{\mathrm{Fred}}}}({P})_{s^*})^G, s \right).$$

Given a $G$-CW pair $(X,A)$, one defines the relative twisted 