On the diagonals of projections in matrix algebras over von Neumann algebras / Soumyashant Nayak.

Format:

Book

Manuscript

Thesis/Dissertation

Author/Creator:

Nayak, Soumyashant, author.

Contributor:

Kadison, Richard V., 1925- degree supervisor, degree committee member.

Gressman, Philip, degree committee member.

Grant, Gregory, degree committee member.

University of Pennsylvania. Department of Mathematics, degree granting institution.

Language:

English

Subjects (All):

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Local Subjects:

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Physical Description:

ix, 65 leaves : illustrations ; 29 cm

Production:

[Philadelphia, Pennsylvania] : University of Pennsylvania, 2016.

Summary:

The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter's Theorem) studied by R. Kadison. A combinatorial lemma, which characterizes the permutation polytope of a vector in Rn in terms of majorization, plays an important role in a proof of the Schur-Horn theorem. The Pythagorean theorem and its converse follow from this as a special case. In the quest for finding a non-commutative version of the lemma alluded to above, the notion of C*-convexity looks promising as the correct generalization for convexity. We make generalizations and improvements of some results known about C*-convex sets.

We prove the Douglas lemma for von Neumann algebras and use it to prove some new results on the one-sided ideals of von Neumann algebras. As a useful technical tool, a non-commutative version of the Gram-Schmidt process is proved for finite von Neumann algebras. A complete characterization of the diagonals of projections in full matrix algebras over an abelian C*-algebra is provided in chapter 5. In chapter 6, we study the problem in the case of M2(Mn(C)), the full algebra of 2 x 2 matrices over Mn( C). The example gives us hints regarding the possibility of extracting an underlying notion of convexity for C*-polytopes, which are not necessarily convex.

Notes:

Ph. D. University of Pennsylvania 2016.

Department: Mathematics.

Supervisor: Richard V. Kadison.

Includes bibliographical references.

OCLC:

969574373