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Characterization of the Pedersen–Baire envelope via Baire measurable affine functions

Prove that, under the canonical pairing with the state space, the Pedersen–Baire envelope A^∞ of a C*-algebra A is identified with the bounded affine Baire measurable functions on the compact state space States(A).

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Background

The paper embeds A∞ into the space of bounded Baire measurable functions on the state space via evaluation. The conjecture proposes a precise identification of A∞ with bounded affine Baire measurable functions, offering a state-space characterization of the Pedersen–Baire envelope.

References

Conjecture The right arrow in~eq:pb_state_diag identifies $A{\infty}$ with those bounded affine functions on $States(A)$ that are Baire measurable.

— Categories of abstract and noncommutative measurable spaces (2504.13708 - Fritz et al., 18 Apr 2025) in Conjecture (Section on Pedersen–Baire envelopes)