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We consider the analogues of the Horn inequalities in finite von Neumannalgebras, which concern the possible spectral distributions of sums $a+b$ ofself--adjoint elements $a$ and $b$ in a finite von Neumann algebra. It is anopen question whether all of these Horn inequalities must hold in all finitevon Neumann algebras, and this is related to Connes' embedding problem. Foreach choice of integers $1\le r\le n$, there is a set $T^n_r$ of Horn triples,and the Horn inequalities are in one-to-one correspondence with $\cup_{1\ler\le n}T^n_r$. We consider a property P$_n$, analogous to one introduced byTherianos and Thompson in the case of matrices, amounting to the existence ofprojections having certain properties relative to arbitrary flags, whichguarantees that a given Horn inequality holds in all finite von Neumannalgebras. It is an open question whether all Horn triples in $T^n_r$ haveproperty P$_n$. Certain triples in $T^n_r$ can be reduced to triples in$T^{n-1}_r$ by an operation we call {\em TT--reduction}. We show that propertyP$_n$ holds for the original triple if property P$_{n-1}$ holds for the reducedone. We then characterize the TT--irreducible Horn triples in $T^n_3$, forarbitrary $n$, and for those LR--minimal ones (namely, those havingLittlewood--Richardson coefficient equal to 1), we perform a construction ofprojections with respect to flags in arbitrary von Neumann algebras in order toprove property P$_n$ for them. This shows that all LR--minimal triples in$\cup_{n\ge3}T^n_3$ have property P$_n$, and so that the corresponding Horninequalities hold in all finite von Neumann algebras.

On a reduction procedure for Horn inequalities in finite von Neumann algebras