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With the help of some structure results for locally compact groups, the second real continuous bounded cohomology group of a connected locally compact group is described and it is proved that the corresponding group is finite-dimensional for any almost connected locally compact group.

We describe Jordan measurable projections in terms of open projections. Then we introduce lower and upper envelopes of elements of the enveloping yon Neumann algebra and give a description of Pdemarm integrable Hermitian elements of A** =_ ~ in terms of these envelopes. Moreover, we construct the C*- algebra of elements of the enveloping yon Neumann algebra that are Riemann integrable with respect to the given state. 1.2. Constructing the Jordan measure Theorem 1. A projec~tion P ~ ~ is Jordan measurable with respect to ~ if and only if there exist open projections Q, R ~ A that satisfy the conditions Q 0 in A. It follows from the relations W(an) > ~(P) - 1In and 0~(an) _< ~(P) that we can also assume that the sequence w(an) is monotone increasing and has the limit ~(P). Let us form the elements (a,~) 1/m , m ~ Z; the inequality an _< P implies that an and P commute; in turn, the latter condition implies the inequality (an) i/'~ _< PI/" = P. Furthermore, since the properties of the functional calculus imply the inequality (an) 1/m

Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra A is the third dual of some Banach space, then the space A is isometrically isomorphic to the second dual of some von Neumann algebra A and the von Neumann algebra A is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of A (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders.