Projective convex combinations in \(C^ *\)-algebras with the unitary factorization property. (Q556571)

A unital \(C^*\)-algebra \(\mathcal{A}\) is said to have the unitary factorization property (written as \(\mathcal A \in(\text{UF})\)) if each unitary \(u\) is a finite product of symmetries (self-adjoint unitaries). For example, each unital purely infinite simple \(C^*\)-algebra \(A\) belongs to \((\text{UF})\). A von Neumann algebra \(\mathcal{M}\) belongs to \((\text{UF})\) if and only if \(\mathcal{M}\) has no direct summands of finite type \(\text{I}\). For a unital \(C^*\)-algebra \(\mathcal{A}\), the authors define the following increasing sequence \[ {\mathcal L}_1={\mathcal A}^{\text{pr}},\;\dots, {\mathcal L}_{n+1} =\{ap+bp^\bot:a,b \in {\mathcal L}_n, p \in {\mathcal A}^{\text{pr}}\},\;\dots, \] where \({\mathcal A}^{\text{pr}}\) is the set of all projections from \(\mathcal A\), and put \({\mathcal L} = \bigcup_{n=1}^\infty {\mathcal L}_n\). The main result is the following Theorem: If \(\mathcal A\in(\text{UF})\), then \(\mathcal A = \mathcal L\). As a corollary, they obtain that any operator from the \(C^*\)-algebra \(\mathcal A \in (\text{UF})\) can be represented as a finite sum of finite products of its projections.