The Thom Isomorphism in Gauge-Equivariant $$K$$-Theory of $$C^*$$-Bundles

For a bundle of compact Lie groups $$p\colon {\cal G} \to B$$ over a compactum $$B$$ (with the structure group of automorphisms of the corresponding group), we introduce the gauge-equivariant $$K$$ -theory group $$K_{{\cal G}}^{0}(X; {\mathcal A} )$$ of a bundle $$\pi_{X}\colon X \to B$$ endowed with a continuous action of $${\cal G}$$ constructed using bundles $$E\to X$$ with the typical fiber being a projective finitely generated module over a unital $$C^*$$ -algebra $$ {\mathcal A} $$ . The index of a family of gauge-invariant (= $${\cal G}$$ -equivariant) Fredholm operators naturally takes values in these groups. We introduce and study products and use them to define the Thom homomorphism in gauge-equivariant $$K$$ -theory and prove that this homomorphism is an isomorphism. DOI 10.1134/S106192082460168X