Abstract
In this paper, we investigate the Banach space geometry of the noncommutative spaces ℓpsch(Ĝ) associated with the unitary dual Ĝ of a compact group G. These spaces consist of operator-valued families whose norms are defined through the Schatten–von Neumann classes corresponding to irreducible unitary representations of G. We establish basic structural results, including H¨oldertype inequalities, duality relations, and an exact complex interpolation formula. Using these tools, we prove Clarkson type inequalities and modified Clarkson inequalities with constants depending only on p. Consequently, ℓpsch(Ĝ) has the Kadec–Klee property and is uniformly convex and uniformly smooth for every 1 < p < ∞. We also derive quantitative power-type estimates for the associated moduli of convexity and smoothness, including optimal-order estimates in the appropriate ranges of p. Finally, we show that these spaces have type min{2, p} and cotype max{2, p}. Thus, their geometric properties closely parallel those of the classical Lp - and ℓp -spaces. This provides a unified geometric description of this noncommutative scale