Abstract
The semigroup of all linear maps on a vector space is regular, but the semigroup of continuous linear maps on a Hilbert space is not, in general, regular; nor is the product of two regular elements regular. In this chapter, we show that in those types of von Neumann algebras of operators in which the lattice of projections is modular, the set of regular elements do form a (necessarily regular) semigroup. This is done using the construction of a regular biordered set (as defined in Nambooripad, Mem. Am. Math. Soc. 22:224, 1979, [9]) from a complemented modular lattice (as in Patijn, Semigroup Forum 21:205–220, 1980, [11]).
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Nambooripad, K.S.S. (2015). Regular Elements in von Neumann Algebras. In: Romeo, P., Meakin, J., Rajan, A. (eds) Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2488-4_3
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DOI: https://doi.org/10.1007/978-81-322-2488-4_3
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