Geodesic
The Haagerup problem on subadditive weights on $W^*$-algebras.~II
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2013), pp. 72-76.

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In year 1975 U. Haagerup has posed the following question: whether every normal subadditive weight on a $W^*$-algebra is $\sigma$-weakly lower semicontinuous? In year 2011 the author has positively answered this question in a particular case of abelian $W^*$-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional $W^*$-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic $W^*$-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic $W^*$-algebras.
Keywords: $W^*$-algebra, subadditive weight, normal functional, projection, atom, normed solid space, bounded linear operator, Hilbert space. 
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A. M. Bikchentaev. The Haagerup problem on subadditive weights on $W^*$-algebras.~II. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2013), pp. 72-76. http://geodesic.mathdoc.fr/item/IVM_2013_12_a6/

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