Title:

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 selfadjoint 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 X',Y'? In terms of one operator A = X + iY, suppose that the selfcommutator [A,A*] is small. How close is A to the set of normal operators? Our main result is a quantitative analogue of Huaxin Lin's theorem on almost commuting matrices. We prove that for every (n x n)matrix A with A ≤ 1 there exists a normal matrix A' such that AA' ≤ C[A,A*]¹/³. We also establish a general version of this result for arbitrary C*algebras of real rank zero assuming that A satisfies a certain indextype condition. For operators in Hilbert spaces, we obtain twosided estimates of the distance to the set of normal operators in terms of [A,A*] and the distance from A to the set of invertible operators. The technique is based on Davidson's results on extensions of almost normal operators, Alexandrov and Peller's results on operator and commutator Lipschitz functions, and a refined version of Filonov and Safarov's results on approximate spectral projections in C*algebras of real rank zero. In Chapter 4 we prove an analogue of Lin's theorem for finite matrices with respect to the normalized HilbertSchmidt norm. It is a renement of a previously known result by Glebsky, and is rather elementary. 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.
