Generalized π-Baer *-rings

Document Type : Original Manuscript


1 Department of Mathematics, Dezful Branch, Islamic Azad University, Dezful, Iran.

2 Department of Computer Engineering, Shahed University, Tehran, Iran.


‎A *-ring‎ ‎$R$ is called a generalized $\pi$-Baer *-ring‎, ‎if for any projection invariant left ideal $Y$ of $R$‎, ‎the right annihilator of $Y^n$‎ ‎is generated‎, ‎as a right ideal‎, ‎by a projection, ‎for some positive integer $n$‎, ‎depending on $Y$‎. ‎In this paper, ‎we‎ ‎study some properties of generalized $\pi$-Baer *-rings‎. ‎We show that this notion is well-behaved with respect to‎ ‎polynomial extensions, full matrix rings, and several classes of triangular matrix rings‎.‎ ‎We indicate interrelationships between the generalized $\pi$-Baer *-rings and related classes of rings such as‎ generalized ‎$\pi$-Baer rings‎, generalized ‎Baer *-rings‎, generalized quasi-Baer *-rings, and ‎$‎\pi$-Baer \s-rings.
‎We obtain algebraic examples which are generalized‎ $‎\pi$-Baer $ \ast $-rings but are not $‎\pi$-Baer *-rings‎. ‎We show that for pre-C*-algebras these two notions are equivalent‎.
‎We obtain classes of Banach *-algebras‎ ‎which are generalized‎ $‎\pi$-Baer *-rings but are not $‎\pi$-Baer *-rings‎. We finish the paper by showing that for a locally compact‎
‎abelian group $G$‎, ‎the group algebra $L^{1}(G)$ is a‎ ‎generalized $‎\pi$-Baer $*$-ring‎, ‎if and only if so is the group C*-algebra‎ ‎$C^{*}(G)$‎, ‎if and only if $G$ is finite‎.


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