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The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.

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In this paper we prove the following result. Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators on X , and let A ( X ) ⊂ L ( X ) be a standard operator algebra. Suppose we have a linear mapping D : A ( X ) → L ( X ) satisfying the relation D ( A 3 ) = D ( A ) A 2 + AD ( A ) A + A 2 D ( A ), for all AA ( X ). In this case D is of the form D ( A ) = ABBA , for all AA ( X ) and some BL ( X ). We apply this result, which generalizes a classical result of Chernoff, to semisimple H *-algebras.

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