Almost commuting elements of real rank zero C∗-algebras

The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X; Y ] being small. Such operators are called almost commuting. How close is the pair X; Y to a pair of commuting operators X0; Y 0? In terms of one operator A = X + iY , suppose that the self-commutator [A;A] is small. How close is A to the set of normal operators?<br/>Our main result is a quantitative analogue of Huaxin Lin's theorem on almost<br/>commuting matrices. We prove that for every (nn)-matrix A with kAk 6 1 there<br/>exists a normal matrix A0 such that kA &amp;#x100000; A0k 6 Ck[A;A]k1=3. We also establish<br/>a general version of this result for arbitrary C-algebras of real rank zero assuming that A satises a certain index-type condition. For operators in Hilbert spaces, we obtain two-sided estimates of the distance to the set of normal operators in terms of k[A;A]k and the distance from A to the set of invertible operators.<br/>The technique is based on Davidson's results on extensions of almost normal<br/>operators, Alexandrov and Peller's results on operator and commutator Lipschitz<br/>functions, and a rened version of Filonov and Safarov's results on approximate<br/>spectral projections in C-algebras of real rank zero.<br/>In Chapter 4 we prove an analogue of Lin's theorem for nite matrices with<br/>respect to the normalized Hilbert{Schmidt norm. It is a renement of a previously<br/>known result by Glebsky, and is rather elementary.<br/>In Chapter 5 we construct a calculus of polynomials for almost commuting elements of C-algebras and study its spectral mapping properties. Chapters 4 and 5 are based on author's joint results with Nikolay Filonov.

Name dc:type.qualificationname

Doctor of Philosophy

Level dc:type.qualificationlevel

Doctoral Thesis

Grantor dc:publisher.institution

King's College London

Year dc:date.issued

2013

Author dc:creator

• Kachkovskiy, Ilya

Advisors dc:contributor.advisor

• Safarov, Yuri
• Pushnitski, Alexander

Language dc:language

eng

Identifier

oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb

OAI identifier oai:identifier

oai:kclpure.kcl.ac.uk:studenttheses/1c891b23-dba5-4395-99dc-2884cbeb3bfb