gl. Introduction In the previous paper [7] we defined a weakly (resp. strictly) unbounded Erv#-algebra and obtained the following fact: If ut is an EPV#-algebra, then there exists a projection E in et, n ptS such that wrE is a weakly unbounded EPV#-algebra, Qti-E is a strictly unbounded EW#-algebra and 9.t equals the product utEÅ~ Qti-E of the EW'-algebras or. and 9J,-E. The primary purpose of this paper is to investigate linear functionals on a weakly unbounded EPV#-algebra. In g3, we shall

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... dy the general theory of weakly unbounded Erv"-algebras. First, we define the notation of a weakly unbounded EW#-algebra Q( associated with a family {utz}z.A of von Neumann algebras utz and show that the definition js equivalent to the definition of a weakly unbounded EPV"-algebra defined in [7]. Next, we define the locally convex topologies (; weak, ff-weak, locally aweak, strong, a-strong, locally c-strong and locally uniform topologies) on ut and the commutants, bicommutants of ut. Furthermore, we shall investigate the relation between the topologies and the commutants. In g4, we shall study the dual space Qt* (resp. ut*) of Qt with respect to the locally uniform topology (resp. a-weak topology). Then we have that E}I" (resp. ee ut*nQ(*) equals the direct sum 2 9tz" (resp. Z(utz)*) of the dual space aEA AEA utz* (resp. (9.tz).) of the von Neumann algebra uta with respect to the uniform topology (resp. a-weak topology), (Theorem 4.1). In g5, we shall obtain the structure of invariant subspaces of or*n9J*: Every closed left (resp. right) invariant subspace V of ut"n wr* is of the form; V =(ut* n M.)Eo (resp. V=Eo(ut" n ut,)) for some projection Eo in ut (Theorem 5.1). In g6, we shall define normal and singular linear functionals on ut and obtain the following fact: Every element ip of ut* is uniquely decomposed into