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

We study the von Neumann algebra \mathcal{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 \mathcal{M} decomposes into abelian parts and a measurable field of copies of M_2(\mathbb{C}). For \mu-almost every \theta \in \Theta, the self-adjoint part of the fiber M_2(\mathbb{C}), equipped with the commutator i[\cdot,\cdot], carries a three-dimensional real Lie algebra structure isomorphic to \mathfrak{su}(2). This provides a purely algebraic, fiberwise mechanism for the appearance of \mathfrak{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, \mathfrak{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.