Semicontinuity for unbounded operators affiliated with operator algebras

Hyoungsoon Kim, Purdue University


Let A be a C*-algebra and A** its enveloping von Neumann algebra. Pedersen and Akemann developed four concepts of lower semicontinuity for elements of A**. Later, Brown suggested using only three classes: strongly lsc, middle lsc, and weakly lsc. In this thesis, we generalize those semicontinuity concepts to the unbounded operators affiliated with A**. In order to deal with unbounded operators, we use certain Mobius maps, and the theory of quadratic forms which was developed by Kato, Robinson, Davies, and Simon. First, we identify our generalized strongly lsc elements with weak* lsc affine ($-\infty,\infty$) -valued functions vanishing at 0 on the quasistate space Q(A), and then we generalize various results of the theory of strong semicontinuity. Also, we discuss some interpolation problems for unbounded operators. Secondly, we generalize the concept of middle semicontinuity by making use of the theory of double centralizers of Pedersen's ideal which was given by Lazar, Taylor, and Phillips. Then we obtain a Dauns-Hofmann type theorem and another interpolation theorem.




Brown, Purdue University.

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