Prescribing the cardinality of the unitary dual in ZFC

For any cardinals 0 ≤ s ≤ c and 1 ≤ t ≤ c = 2^ℵ₀, we construct in ZFC a simple unital complex C*-algebra of norm density c with exactly s irreducible classes on separable Hilbert spaces and t irreducible classes on Hilbert spaces of density c, and no other irreducible classes. It admits a faithful representation on a separable Hilbert space and has a unique faithful trace, whose GNS weak closure is the hyperfinite II₁ factor. The construction retains Tanaka’s state-joining unitaries within the blocks of a partition of the CAR irreducible classes. We classify the irreducible representations for every partition and prove simplicity and the trace assertions directly. In particular, every positive cardinal at most c occurs as the size of the unitary dual of such an algebra. We also give a retained-family construction of these total dual sizes from OpenAI’s selective-bridge and iteration theorems, together with an account of their proof ideas. The sharp general bound for an algebra with a faithful representation on a separable Hilbert space is 2^c, attained even by the hyperfinite II₁ factor viewed as a simple C*-algebra. GitHub repository: https://github.com/eulogica/zfc-prescribed-irreducible-spectrum