[Paper Review] Almost normal operators mod Hilbert-Schmidt and the K-theory of the Banach algebras $E\Lambda(\Omega)$

Dan-Virgil Voiculescu|arXiv (Cornell University)|Dec 21, 2011

Advanced Operator Algebra Research12 references3 citations

TL;DR

This paper introduces the Banach ∗-algebras $E\Lambda(\Omega)$ to study almost normal operators modulo Hilbert-Schmidt operators, establishing a link between their K-theory and the Pincus g-function. It shows that $E\Lambda(\Omega)$ is isometrically isomorphic to the bidual of its ideal $K\Lambda(\Omega)$, and that the corona algebra $E\Lambda(\Omega)/K\Lambda(\Omega)$ is a C∗-algebra. The key result is that a mod Hilbert-Schmidt BDF-type theorem is equivalent to the K-theory of $E\Lambda(\Omega)$ being isomorphic to $L^1_{\text{re}}(\Omega, \lambda)$ via the g-function.

ABSTRACT

Is there a mod Hilbert-Schmidt analogue of the BDF-theorem, with the Pincus g-function playing the role of the index ? We show that part of the question is about the K-theory of certain Banach algebras. These Banach algebras, related to Lipschitz functions and dirichlet algebras have nice Banach space duality properties. Moreover their corona algebras are C*-algebras.

Motivation & Objective

• To establish a mod Hilbert-Schmidt analogue of the BDF-theorem, with the Pincus g-function replacing the index.
• To investigate the K-theory of the Banach algebras $E\Lambda(\Omega)$, which model operators with trace-class self-commutator arising from compressions of normal operators.
• To clarify the role of normal dilation in the classification of almost normal operators modulo Hilbert-Schmidt operators.
• To show that the corona algebra $E\Lambda(\Omega)/K\Lambda(\Omega)$ is a C∗-algebra, despite the underlying algebra being non-C∗.
• To explore whether integrality properties of $K_0(E\Lambda(\Omega))$ could obstruct normal dilation, thus providing a potential negative answer to the dilation problem.

Proposed method

• Define $E\Lambda(\Omega)$ as the algebra of operators in $L^2(\Omega, \lambda)$ whose commutators with multiplication operators by Lipschitz functions are Hilbert-Schmidt.
• Establish that $K\Lambda(\Omega)$, the ideal of compact operators in $E\Lambda(\Omega)$, is a Dirichlet algebra and is isometrically isomorphic to the predual of $E\Lambda(\Omega)$.
• Use the duality pairing $\mathrm{Tr}(T x + [Z, T] y)$ to identify $E\Lambda(\Omega)$ with the bidual of $K\Lambda(\Omega)$.
• Apply results from norm-ideal perturbations of Hilbert space operators to construct a bounded approximate unit of projections in $K\Lambda(\Omega)$.
• Analyze the K-theory of $E\Lambda(\Omega)$ and $E\Lambda(\Omega)_0$, the inductive limit over bounded $\Omega$, using the Pincus g-function as a candidate isomorphism to $L^1_{\text{re}}(\Omega, \lambda)$.
• Demonstrate that the corona algebra $E\Lambda(\Omega)/K\Lambda(\Omega)$ is a C∗-algebra, despite $E\Lambda(\Omega)$ not being a C∗-algebra.

Experimental results

Research questions

• RQ1 Is there a BDF-type classification theorem for almost normal operators modulo Hilbert-Schmidt operators, with the Pincus g-function playing the role of the index?
• RQ2 Does the K-theory of $E\Lambda(\Omega)$, particularly $K_0(E\Lambda(\Omega))$, reflect the structure of the g-function as an $L^1$-function?
• RQ3 Can the normal dilation problem for almost normal operators be obstructed by integrality properties of $K_0(E\Lambda(\Omega))$?
• RQ4 What is the structure of the corona algebra $E\Lambda(\Omega)/K\Lambda(\Omega)$, and why is it a C∗-algebra despite the non-C∗ nature of $E\Lambda(\Omega)$?
• RQ5 How do bi-Lipschitz maps between Borel sets $\Omega_1$ and $\Omega_2$ induce isomorphisms on $E\Lambda(\Omega)$ and its corona algebra?

Key findings

• The algebra $E\Lambda(\Omega)$ is isometrically isomorphic to the bidual of $K\Lambda(\Omega)$, the ideal of compact operators in $E\Lambda(\Omega)$, via the duality pairing $\mathrm{Tr}(T x + [Z, T] y)$.
• The corona algebra $E\Lambda(\Omega)/K\Lambda(\Omega)$ is a C∗-algebra, even though $E\Lambda(\Omega)$ is not a C∗-algebra.
• The problem of a mod Hilbert-Schmidt BDF-theorem is equivalent to the K-theory of $E\Lambda(\Omega)_0$ being isomorphic to $L^1_{\text{re}}(\Omega, \lambda)$ via the Pincus g-function.
• The ideal $K\Lambda(\Omega)$ is a Dirichlet algebra, with a Markovian semigroup of completely positive contractions induced by the derivation $\partial a = [X, a] \oplus [Y, a]$.
• Bi-Lipschitz maps between Borel sets $\Omega_1$ and $\Omega_2$ induce $*$-isomorphisms between $E\Lambda(\Omega_1)$ and $E\Lambda(\Omega_2)$, and between their respective corona algebras.
• The center of the corona algebra $(E/K)\Lambda(\Omega)$ is an open question, though it is analogous to the Calkin algebra bicommutant problem.