A strong subadditivity-like inequality for quantum entropy in semifinite von Neumann algebras

Let $$\mathcal {M}$$ be a semifinite von Neumann algebra with a normal faithful semifinite trace $$\tau$$, and let $$\mathcal {A}$$, $$\mathcal {B}$$, $$\mathcal {R}$$ be its subalgebras such that $$\mathcal {R}\subset \mathcal {A}\cap \mathcal {B}$$ and that $$\tau$$ restricted to any of these subalgebras is semifinite. Denote by $$\mathbb {E}_\mathcal {A}$$, $$\mathbb {E}_\mathcal {B}$$, and $$\mathbb {E}_\mathcal {R}$$ the normal conditional expectations from $$\mathcal {M}$$ onto $$\mathcal {A}$$, $$\mathcal {B}$$ and $$\mathcal {R}$$, respectively, such that $$\tau$$ is invariant with respect to any of them. The quadruple $$\mathcal {M}$$, $$\mathcal {A}$$, $$\mathcal {B}$$, $$\mathcal {R}$$ is said to be a commuting square if

In this note, we show that the property of being a commuting square is characterized by a sort of the SSA-like (strong subadditivity of entropy) inequality for an arbitrary normal state $$\rho$$ on $$\mathcal {M}$$, where $$H(\varphi)$$ denotes the Segal entropy of the state $$\varphi$$. The situation when we have equality in the inequality above is also investigated, and various equivalent conditions are obtained, in an especially appealing form for finite von Neumann algebras. For such algebras, one more condition is obtained under the assumption of independence of the algebras $$\mathcal {A}$$ and $$\mathcal {B}$$.