ON COMPACT EXTENSIONS OF TRACIAL W∗-DYNAMICAL SYSTEMS

The paper’s Theorem A states and proves the equivalence of the three conditions (compact extension via almost periodicity; density of Γ-invariant conditional convolutions; density of Q-finitely generated Γ-invariant submodules) with a clear chain (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (1). The statement is explicitly given and proved in Section 3, with the equivalence spelled out on p. 3 and the proof steps detailed later: Definition 3.1–3.2 introduce relative almost periodicity and conditional precompactness; Theorem A is stated; and the proofs of (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (1) are provided, using the conditional Hilbert–Schmidt framework and Proposition 2.12 (conditional orthonormality) to pass from Γ-invariant kernels to finitely generated submodules and back. See Theorem A and its proof pieces in the text and preliminaries (Theorem A: ; Defs. 3.1–3.2: ; conditional convolution formalism: ; (1) ⇒ (2) proof: ; (2) ⇒ (3) and (3) ⇒ (1) proof sketch: ; closedness of AP2 and invariance: ). The candidate solution correctly proves several directions—(3) ⇒ (1), a plausible (3) ⇒ (2) via projections realized by kernels, and a direct (2) ⇒ (1). However, it never proves (1) ⇒ (3), and its “completion of the cycle” is logically circular: it asserts (1) ⇒ (2) via (1) ⇒ (3) ⇒ (2) while (1) ⇒ (3) is not established, and then asserts (2) ⇒ (3) via (2) ⇒ (1) ⇒ (3), again requiring (1) ⇒ (3). Thus, the model’s argument does not actually establish equivalence. Additionally, some technical steps (explicit finite-sum kernel for projections; left-boundedness hypotheses) are not justified, whereas the paper avoids these pitfalls by using the conditional Hilbert–Schmidt machinery.