The relative radius of comparison of the crossed product of a non-unital C*-algebra by a finite group

2025/05/02 by Asadi-Vasfi, M. Ali, Elliott, George A.
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2505.00952

Abstract

In this paper, we prove results on the relative radius of comparison of C*-algebras and their crossed products, focusing on the non-unital setting. More precisely, let A be a stably finite simple non-type-I (not necessarily unital) C*-algebra, let G be a finite group, and let α\colon G → Aut (A) be an action which has the weak tracial Rokhlin property. Let a be a non-zero positive element in A^α⊗ K. Then we show that the radius of comparison of Cu (A^α) relative to [a] is bounded above by the radius of comparison of Cu (A) relative to [a]. If further A is exact and a is in the Pedersen ideal of A^α⊗ K, then the radius of comparison of Cu (A\rtimes_α G) relative to [a] is equal to its radius of comparison relative to [p⋅ a], scaled by 1/|G|, where p is the averaging projection in the multiplier algebra of (A ⊗ K) \rtimes_{α⊗ id} G. Moreover, the radius of comparison of Cu (A\rtimes_α G) relative to [a] is bounded above by 1/|G| times the radius of comparison of Cu (A) relative to [a]. We also prove that the inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed point of the purely positive part of Cu (A). An important consequence of our results is that they apply to non-unital C*-algebras and give new insights into comparison theory of C*-algebras and their crossed products.

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Related

• The Cuntz semigroup and the radius of comparison of the crossed product\n by a finite group
• An algebraic approach to the radius of comparison
• The radius of comparison of the tensor product of a C*-algebra with C (X)
• Some Permanence properties for crossed products by compact group actions with the tracial Rokhlin property
• Rokhlin dimension: duality, tracial properties, and crossed products