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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJun 19th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNew entry [[affiliated operator]] of a $C^\ast$-algebra aka [[affiliated element]]. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics. I sent a version of that entry but the $n$Lab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy: ## Motivation Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics. ## Idea The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of "approximable" operators by means of operator algebra itself. One way to do this is to define the __affiliated elements__ of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the [[spectral theorem|spectral decomposition]] will supply the approximation. ## Literature * [[S. L. Woronowicz]], K. Napiórkowski, _Operator theory in $C^\ast$-framework_, Reports on Mathematical Physics __31__, Issue 3 (1992), 353-371, <a href="http://dx.doi.org/10.1016/0034-4877(92)90025-V">doi</a>, [pdf](https://http:www.fuw.edu.pl/~slworono/PDF-y/OP.pdf) * S. L. Woronowicz, _$C^\ast$-algebras generated by unbounded elements_, [pdf](https://www.fuw.edu.pl/~slworono/PDF-y/GENER.pdf) * wikipedia [affiliated operator](http://en.wikipedia.org/wiki/Affiliated_operator)

   New entry affiliated operator of a $C^\ast$-algebra aka affiliated element. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics.

   I sent a version of that entry but the $n$Lab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy:

   Motivation

   Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics.

   Idea

   The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of “approximable” operators by means of operator algebra itself. One way to do this is to define the affiliated elements of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the spectral decomposition will supply the approximation.

   Literature

   • S. L. Woronowicz, K. Napiórkowski, Operator theory in $C^\ast$-framework, Reports on Mathematical Physics 31, Issue 3 (1992), 353-371, doi, pdf
   • S. L. Woronowicz, $C^\ast$-algebras generated by unbounded elements, pdf
   • wikipedia affiliated operator