• Subhash J Bhatt

Articles written in Proceedings – Mathematical Sciences

• Quotient-bounded elements in locally convex algebras. II

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 noncommutativeLp-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andLw-integral of a measurable field ofC*-algebras are discussed briefly.

• Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.

• On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019