Spectral and Polar Decomposition in AW*-Algebras

The spectral decomposition of normal linear (bounded) operators and the polar decomposition of arbitrary linear (bounded) operators on Hilbert spaces have been interesting and technically useful results in operator theory [3, 9, 13, 20]. The development of the concept of von Neumann algebras on Hilbert spaces has shown that both these decompositions are always possible inside of the appropriate von Neumann algebra [14]. New light on these assertions was shed identifying the class of von Neumann algebras among all C*-algebras by the class of C*-algebras which possess a pre-dual Banach space, the W*algebras. The possibility of C*-theoretical representationless descriptions of spectral and polar decomposition of elements of von Neumann algebras (and may be of more general C*-algebras) inside them has been opened up. Steps to get results in this direction were made by several authors. The W*-case was investigated by S. Sakai in 1958-60, [18, 19]. Later on J. D. M. Wright has considered spectral decomposition of normal elements of embeddable AW*-algebras, i.e., of AW*-algebras possessing a faithful von Neumann type representation on a self-dual Hilbert A-module over a commutative AW*-algebra A (on a so called Kaplansky–Hilbert module), [23, 24]. But, unfortunately, not all AW*-algebras are embeddable. In 1970 J. Dyer [5] and O. Takenouchi [21] gave (∗-isomorphic) examples of type III, non-W*, AW*-factors, (see also K. Saitô [15]). Polar decomposition inside AW*-algebras was considered by I. Kaplansky [12] in 1968 and by S. K. Berberian [3] in 1972. They have shown the possibility of polar decomposition in several types of AW*algebras, but they did not get a complete answer. In the present paper the partial result of I. Kaplansky is used that AW*-algebras without direct commutative summands and with a decomposition property for it’s elements like described at Corollary 5 below allow polar decomposition inside them, [3, §21: Exerc. 1] and [12, Th. 65]. For a detailled overview on these results we refer to [3].