Quotient-bounded elements in locally convex algebras. II
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Consideration of quotient-bounded elements in a locally convexGB^{*}-algebra leads to the study of properGB^{*}-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB^{*}-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC^{*}-algebra and two other representation theorems forb^{*}-algebras (also calledlmc^{*}-algebras), one representinga b^{*}-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL^{p}-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL^{w}-integral of a measurable field ofC^{*}-algebras are discussed briefly.
Subhash J Bhatt^{1}
Volume 130, 2020
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