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Special Regular Rings with Involution

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A ring is regular if for every there exists an such that . Every field is a regular ring. A regular ring with an involution * is called *-regular if implies . The *-regular rings derived from W*-algebras have many special properties. This Demonstration considers three of them.

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(A) implies for any positive integer .

(A') implies .

(B) For all , there exists an element such that .

A implies A', but not conversely.

Let be the field of residues modulo a prime , where . This is *-regular if we take the identity map as involution. In this ring, the equality implies . So A' holds. But the equation has nontrivial solutions, as can be seen for . So A does not hold.

If a ring satisfies A' and B, then evidently it satisfies A.

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Contributed by: Izidor Hafner (January 2018)
Open content licensed under CC BY-NC-SA

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This Demonstration is based on [1, pp. 213–221], where special regular *-rings are studied. It is also shown that axioms A and B are mutually independent. If we take the field of complex numbers and the identity as the involution, the equation always has a solution, so B holds. But is satisfied for , where . So B holds; A does not.

If we take the ring of rational complex numbers with the complex conjugation as the involution, A holds and B does not.

Construction of regular *-rings from finite Baer *-rings (which include W*-algebras) is studied in [2, 3].

References

[1] N. Prijatelj and I. Vidav, "On Special *-Regular Rings," The Michigan Mathematical Journal, 18(3), 1971 pp. 213–221. doi:10.1307/mmj/1029000680.

[2] S. K. Berberian, Baer *-Rings, New York: Springer-Verlag, 1972.

[3] I. Hafner, "The Regular Ring and the Maximal Ring of Quotients of a Finite Baer *-Ring," The Michigan Mathematical Journal, 21(2), 1974 pp. 153–160. doi:10.1307/mmj/1029001260.

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