The main result states that every special AJW-algebra can be
decomposed into the direct sum of totally irreversible and reversible subalgebras. In turn,
every reversible special AJW-algebra decomposes into a direct sum of two subalgebras, one of which
has purely real enveloping real von Neumann algebra, and the second one contains an ideal, whose
complexification is a C*-algebra and the annihilator of this complexification
in the enveloping C*-algebra of this subalgebra is equal to zero.
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