The story of Brown–Douglas–Fillmore theory begins with the Weyl–von Neumann theorem, which, in one of its formulations, says that a bounded self-adjoint operator on an infinite-dimensional separable Hilbert space is determined up to compact perturbations, modulo unitary equivalence, by its essential spectrum. (The essential spectrum is the spectrum of the image of in the Calkin algebra ; it is also the spectrum of the restriction of to the orthogonal complement of the eigenspaces of for the eigenvalues of finite multiplicity; cf. also Spectrum of an operator.) In other words, unitary equivalence modulo the compacts washes out all information about the spectral measure of , and only the essential spectrum remains. This result was extended to normal operators (cf. also Normal operator) by I.D. Berg [a4] and W. Sikonia [a12], working independently. However, the theorem is not true for operators that are only essentially normal, in other words, for operators such that . Indeed, the  "unilateral shift"  satisfies and , where is a rank-one projection, yet cannot be a compact perturbation of a normal operator since its Fredholm index (cf. also Fredholm operator; Index of an operator) is non-zero. In [a2], L.G. Brown, R.G. Douglas and P.A. Fillmore (known to operator theorists as  "BDF" ) showed that this is the only obstruction: an operator in is a compact perturbation of a normal operator if and only if is essentially normal and for every .

However, they went considerably further, by putting this theorem in a -algebraic context in [a2] and [a3]. An operator  "up to compact perturbations"  defines an injective -homomorphism from a -algebra (the closed subalgebra of generated by and ) to , and the -algebra is Abelian if and only if is essentially normal. More generally, an extension of a separable -algebra is an injective -homomorphism , since this is equivalent to a commutative diagram with exact rows: BDF defined a natural equivalence relation (basically unitary equivalence) and an addition operation on such extensions, giving a commutative monoid , whose -element is represented by split extensions (those for which there is a lifting ). (The essential uniqueness of the split extensions was shown in [a14].) It was shown by M.D. Choi and E.G. Effros [a6] (see also [a1]) that this monoid is a group whenever is nuclear (cf. also Nuclear space). (BDF originally worked only with Abelian -algebras , for which this is automatic, and they used the notation for .) BDF showed that behaves like a generalized homology theory in (cf. also Generalized cohomology theories), and in fact for finite CW-complexes (cf. also CW-complex) coincides with , where is the homology theory dual to complex -theory. This was extended in [a7], where it was shown that is canonically isomorphic to , Steenrod -homology (cf. also Steenrod–Sitnikov homology), for all compact metric spaces , and in [a5], where it was shown that on a suitable category of -algebras, fits into a short exact sequence

It is now (as of 2000) known that BDF theory is just a special case of a more general theory of -algebra extensions. One type of generalization (see [a13]) involves replacing by the algebra of  "compact"  operators of a factor (cf. also von Neumann algebra). Another sort of generalization involves replacing by an algebra of the form , where is another separable (or -unital) -algebra. Theories of this sort were worked out in [a9], [a10] and in [a8], though the theory of [a9], [a10] turns out to be basically a special case of Kasparov's theory (see [a11]). Kasparov's -theory gives rise to a bivariant functor , and when is nuclear, this coincides [a8] with Kasparov's bivariant -functor .

References

[a1]	W. Arveson,   "Notes on extensions of -algebras"  Duke Math. J. , 44 : 2  (1977)  pp. 329–355
[a2]	L.G. Brown,   R.G. Douglas,   P.A. Fillmore,   "Unitary equivalence modulo the compact operators and extensions of -algebras" , Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) , Lecture Notes in Mathematics , 345 , Springer  (1973)  pp. 58–128
[a3]	L.G. Brown,   R.G. Douglas,   P.A. Fillmore,   "Extensions of -algebras and -homology"  Ann. of Math. (2) , 105 : 2  (1977)  pp. 265–324
[a4]	I.D. Berg,   "An extension of the Weyl–von Neumann theorem to normal operators"  Trans. Amer. Math. Soc. , 160  (1971)  pp. 365–371
[a5]	L.G. Brown,   "The universal coefficient theorem for and quasidiagonality" , Operator Algebras and Group Representations I (Neptun, 1980) , Monographs Stud. Math. , 17 , Pitman  (1984)  pp. 60–64
[a6]	M.D. Choi,   E.G. Effros,   "The completely positive lifting problem for -algebras"  Ann. of Math. (2) , 104 : 3  (1976)  pp. 585–609
[a7]	J. Kaminker,   C. Schochet,   "-theory and Steenrod homology: applications to the Brown–Douglas–Fillmore theory of operator algebras"  Trans. Amer. Math. Soc. , 227  (1977)  pp. 63–107
[a8]	G.G. Kasparov,   "The operator -functor and extensions of -algebras"  Math. USSR Izv. , 16  (1981)  pp. 513–572  Izv. Akad. Nauk. SSSR Ser. Mat. , 44 : 3  (1980)  pp. 571–636; 719
[a9]	M. Pimsner,   S. Popa,   D. Voiculescu,   "Homogeneous -extensions of . I"  J. Oper. Th. , 1 : 1  (1979)  pp. 55–108
[a10]	M. Pimsner,   S. Popa,   D. Voiculescu,   "Homogeneous -extensions of . II"  J. Oper. Th. , 4 : 2  (1980)  pp. 211–249
[a11]	J. Rosenberg,   C. Schochet,   "Comparing functors classifying extensions of -algebras"  J. Oper. Th. , 5 : 2  (1981)  pp. 267–282
[a12]	W. Sikonia,   "The von Neumann converse of Weyl's theorem"  Indiana Univ. Math. J. , 21  (1971/72)  pp. 121–124
[a13]	G. Skandalis,   "On the group of extensions relative to a semifinite factor"  J. Oper. Th. , 13 : 2  (1985)  pp. 255–263
[a14]	D. Voiculescu,   "A non-commutative Weyl–von Neumann theorem"  Rev. Roum. Math. Pures Appl. , 21 : 1  (1976)  pp. 97–113