We say that a \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-ring \documentclass[12pt]{minimal}
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\begin{document}$ R $\end{document} is a generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-ring
if, for each nonempty subset \documentclass[12pt]{minimal}
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\begin{document}$ S $\end{document} of \documentclass[12pt]{minimal}
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\begin{document}$ R $\end{document}, the right annihilator \documentclass[12pt]{minimal}
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\begin{document}$ r_{R}(S^{n}) $\end{document} is generated
as a right ideal by a projection for some positive integer \documentclass[12pt]{minimal}
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\begin{document}$ n $\end{document} depending
on \documentclass[12pt]{minimal}
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\begin{document}$ S $\end{document}.
Each nonempty set of projections in a generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-ring is a complete lattice.
We study the properties of the \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings.
We show that abelian generalized
Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings are well behaved with respect to finite direct products and
certain triangular matrix extensions.
We give some algebraic examples
that are generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings but not Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings.
We obtain the classes of
both finite and infinite dimensional Banach \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-algebras which are generalized
Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings but not Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings.
We define a generalized \documentclass[12pt]{minimal}
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\begin{document}$ AW^{*} $\end{document}-algebra as a \documentclass[12pt]{minimal}
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\begin{document}$ C^{*} $\end{document}-algebra
that is a generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-ring.
The concept of generalized \documentclass[12pt]{minimal}
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\begin{document}$ AW^{*} $\end{document}-algebra is a generalization of \documentclass[12pt]{minimal}
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\begin{document}$ AW^{*} $\end{document}-algebra,
an algebraic extension of a \documentclass[12pt]{minimal}
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\begin{document}$ W^{*} $\end{document}-algebra.
We show that for semicommutative \documentclass[12pt]{minimal}
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\begin{document}$ C^{*} $\end{document}-algebras the notions of
generalized \documentclass[12pt]{minimal}
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\begin{document}$ AW^{*} $\end{document}-algebra and \documentclass[12pt]{minimal}
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\begin{document}$ AW^{*} $\end{document}-algebra coincide.
