Notes on the Brown-Douglas-Fillmore Theorem

"The Brown-Douglas-Fillmore (BDF) Theorem is a remarkable theorem used in the study of C*-algebra and non-commutative geometry; it solved many open problems of operator theory. It was a revolutionary theorem that used new tools from algebraic topology to study many questions about extensions of commutative C*-algebra and related fields. The theorem also inspired many extensions of arbitrary C*-algebra, which later became important in non-commutative geometry and K theory. The book presents the original approach of BDF in pure form with some modifications and improvements in the exposition. Authors have emphasized the initial exposition of Brown, Douglas, and Fillmore appearing in the article, Unitary equivalence modulo the compact operators and extensions of C*-algebras, published in the Springer Lecture Notes 345. They have tried to preserve the crucial connection of BDF with concrete problems in operator theory. This connection is very relevant and many evolving research areas like Hilbert modules and Cowern- Douglas operator back this fact"--