Spectral Fusion Deformations for Locally Compact Quantum Groups

We develop a deformation framework for $C^{\ast }$-algebras equipped with a coaction of a locally compact quantum group, formulated intrinsically at the level of spectral subspaces determined by the coaction. The construction is defined algebraically on a finite spectral core and extended by continuity to a natural Fréchet $\ast$-algebra completion under mild analytic regularity assumptions. Deformations are governed by scalar fusion data assigning phases to fusion channels of irreducible corepresentations. Associativity and $\ast$-compatibility are characterized by explicit algebraic identities. The framework recovers a range of known deformation procedures, including Rieffel, Kasprzak, and Drinfeld-type constructions, and also yields genuinely new deformations that do not arise from dual $2$–cocycles or crossed-product methods. At the $C^{\ast }$-level, we identify a minimal reduced setting in which the deformed algebra admits a canonical completion, formulated in terms of boundedness of the deformed left regular action on the Haar–GNS space. This separates algebraic coherence from analytic implementability and clarifies the precise role of higher-order fusion data in deformation theory for locally compact quantum groups. In particular, the framework exhibits explicit associator-level deformations governed by fusion $3$–cocycles that cannot arise from any dual $2$–cocycle or crossed-product construction.