Conditional Expectations for Unbounded Operator Algebras

Inoue, Atsushi; Ogi, Hidekazu; Takakura, Mayumi

doi:https://doi.org/10.1155/2007/80152

International Journal of Mathematics and Mathematical Sciences

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2007 | Article ID 080152 | https://doi.org/10.1155/2007/80152

Conditional Expectations for Unbounded Operator Algebras

Atsushi Inoue,¹Hidekazu Ogi,²and Mayumi Takakura¹

Academic Editor: Manfred H. Moller

Received18 Dec 2006

Revised20 Mar 2007

Accepted19 May 2007

Published25 Jun 2007

Abstract

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of an O∗-algebra into the Hilbert space on which the O∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear map ℰ of an O∗-algebra ℳ onto a given O∗-subalgebra 𝒩 of ℳ. Here the domain D(ℰ) of ℰ does not equal to ℳ in general, and so such a conditional expectation is called unbounded.

References

M. Takesaki, “Conditional expectations in von Neumann algebras,” Journal of Functional Analysis, vol. 9, no. 3, pp. 306–321, 1972.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Takesaki, “Theory of Operator Algebras. I,” Springer, New York, NY, USA, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Takesaki, Theory of Operator Algebras. II, vol. 125 of Encyclopaedia of Mathematical Sciences, vol. 6 of Operator Algebras and Non-commutative Geometry, Springer, Berlin, Germany, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. P. Gudder and R. L. Hudson, “A noncommutative probability theory,” Transactions of the American Mathematical Society, vol. 245, pp. 1–41, 1978.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H.-J. Borchers, “On structure of the algebra of field operators,” Nuovo Cimento, vol. 24, pp. 214–236, 1962.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, vol. 1699 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. T. Powers, “Self-adjoint algebras of unbounded operators,” Communications in Mathematical Physics, vol. 21, no. 2, pp. 85–124, 1971.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, vol. 37 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Inoue, “On a class of unbounded operator algebras,” Pacific Journal of Mathematics, vol. 65, no. 1, pp. 77–95, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J.-P. Antoine, A. Inoue, and C. Trapani, Partial $*$ -Algebras and Their Operator Realizations, vol. 553 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Dixmier, Von Neumann Algebras, vol. 27 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1981.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. P. Gudder, “A Radon-Nikodým theorem for $*$ -algebras,” Pacific Journal of Mathematics, vol. 80, no. 1, pp. 141–149, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Inoue, “A Radon-Nikodým theorem for positive linear functionals on $*$ -algebras,” Journal of Operator Theory, vol. 10, no. 1, pp. 77–86, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Gudder and D. Strawther, “Orthogonally additive and orthogonally increasing functions on vector spaces,” Pacific Journal of Mathematics, vol. 58, no. 2, pp. 427–436, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2007 Atsushi Inoue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

298

Downloads

759

Citations