Direct Integral Decomposition of Two-Projection Algebras and Fiberwise Lie Algebra Structure of su(2)

We study the von Neumann algebra M = \P, Q\'' generated by two orthogonal projections on a separable Hilbert space. Using the Halmos direct integral decomposition in the sense of Dixmier and Takesaki, we show that M decomposes into abelian parts and a measurable field of copies of M₂ (C). For -almost every, the self-adjoint part of the fiber M₂ (C), equipped with the commutator i, , carries a three-dimensional real Lie algebra structure isomorphic to su (2). This provides a purely algebraic, fiberwise mechanism for the appearance of su (2) from minimal non-commutative data. We then consider a quantum dynamical semigroup in the standard GKSL form with noise operators P and Q. Using the Frigerio–Evans theorem, we prove that its fixed-point algebra is \P, Q\', which is always abelian. Consequently, su (2) does not appear in the fixed-point algebra; instead, dissipative dynamics induce a collapse from the non-commutative fiber structure to classical abelian observables. This work provides the rigorous mathematical foundation for the non-commutative projection framework developed in the author’s previous preprints and clarifies the distinction between algebraic generation and dissipative selection.