ON GEOMETRIC PROPERTIES OF ℓP-SPACES ON UNITARY DUALS OF COMPACT GROUPS

Abstract

In this paper, we investigate the Banach space geometry of the noncommutative spaces ℓ^p_sch(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, ℓ^p_sch(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 L^p - and ℓ^p -spaces. This provides a unified geometric description of this noncommutative scale