Lattice operations on Rickart ∗-rings under the star order

Abstract

Various authors have investigated properties of the star order (introduced by Drazin in 1978) on algebras of matrices and of bounded linear operators on a Hilbert space. Rickart involution rings (∗-rings) are a certain algebraic analogue of von Neumann algebras, which cover these particular algebras. In 1983, Janowitz proved, in particular, that, in a star-ordered Rickart ∗-ring, every pair of elements bounded from above has a meet and also a join. However, the latter conclusion seems to be based on some wrong assumption. We show that the conclusion is nevertheless correct, and provide equational descriptions of joins and meets for this case. We also present various general properties of the star order in Rickart∗-rings, give several necessary and sufficient conditions (again, equational) for a pair of elements to have a least upper bound of a special kind, and discuss the question when a star-ordered Rickart ∗-ring is a lower semilattice.

Keywords:

• involution ring
• linear operator
• lattice operations
• Rickart ∗-ring
• star order

AMS Subject Classifications:

• 06A06
• 16W10
• 47A05
• 47L30

Acknowledgements

This work has been supported by the Latvian Science Council, Grant No. 271/2012. The author is thankful to the anonymous referee, whose numerous comments helped to improve the presentation. The handling editor has turned the author’s attention to the related paper [21].

Notes

This work has been supported by the Latvian Science Council [grant number 271/2012].