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''BDF theory''

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|self-adjoint operator]] $T = T ^ { * }$ on an infinite-dimensional separable [[Hilbert space|Hilbert space]] $\mathcal{H}$ is determined up to compact perturbations, modulo unitary equivalence, by its essential spectrum. (The essential spectrum is the spectrum $\sigma ( \pi ( T ) )$ of the image $\pi ( T )$ of $T$ in the Calkin algebra $\cal Q ( H ) = B ( H ) / K ( H )$; it is also the spectrum of the restriction of $T$ to the orthogonal complement of the eigenspaces of $T$ for the eigenvalues of finite multiplicity; cf. also [[Spectrum of an operator|Spectrum of an operator]].) In other words, unitary equivalence modulo the compacts $\mathcal{K} ( \mathcal{H} )$ washes out all information about the [[Spectral measure|spectral measure]] of $T$, and only the essential spectrum remains. This result was extended to normal operators (cf. also [[Normal operator|Normal operator]]) by I.D. Berg [[#References|[a4]]] and W. Sikonia [[#References|[a12]]], working independently. However, the theorem is not true for operators that are only essentially normal, in other words, for operators $T$ such that $T T ^ { * } - T ^ { * } T \in \mathcal{K} ( \mathcal{H} )$. Indeed, the  "unilateral shift"  $S$ satisfies $S ^ { * } S = 1$ and $S S ^ { * } = 1 - P$, where $P$ is a rank-one projection, yet $S$ cannot be a compact perturbation of a normal operator since its Fredholm index (cf. also [[Fredholm-operator(2)|Fredholm operator]]; [[Index of an operator|Index of an operator]]) is non-zero. In [[#References|[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 $T$ in $\mathcal{B} ( \mathcal{H} )$ is a compact perturbation of a normal operator if and only if $T$ is essentially normal and $\operatorname{ind}( T - \lambda ) = 0$ for every $\lambda \notin \sigma ( \pi ( T ) )$.

However, they went considerably further, by putting this theorem in a $C ^ { * }$-algebraic context in [[#References|[a2]]] and [[#References|[a3]]]. An operator $T$  "up to compact perturbations"  defines an injective $*$-homomorphism from a [[C*-algebra|$C ^ { * }$-algebra]] $A$ (the closed subalgebra of $\mathcal{Q} ( \mathcal{H} )$ generated by $\pi ( T )$ and $\pi ( T ^ { * } )$) to $\mathcal{Q} ( \mathcal{H} )$, and the $C ^ { * }$-algebra $A$ is Abelian if and only if $T$ is essentially normal. More generally, an extension of a separable $C ^ { * }$-algebra $A$ is an injective $*$-homomorphism $A \hookrightarrow \mathcal{Q} ( \mathcal{H} )$, since this is equivalent to a commutative diagram with exact rows:

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027040.png"/></td> </tr></table>

BDF defined a natural equivalence relation (basically unitary equivalence) and an addition operation on such extensions, giving a commutative [[Monoid|monoid]] $\operatorname { Ext } ( A )$, whose $0$-element is represented by split extensions (those for which there is a lifting $A \rightarrow \cal B ( H )$). (The essential uniqueness of the split extensions was shown in [[#References|[a14]]].) It was shown by M.D. Choi and E.G. Effros [[#References|[a6]]] (see also [[#References|[a1]]]) that this monoid is a [[Group|group]] whenever $A$ is nuclear (cf. also [[Nuclear space|Nuclear space]]). (BDF originally worked only with Abelian $C ^ { * }$-algebras $A = C ( X )$, for which this is automatic, and they used the notation $\operatorname { Ext } ( X )$ for $\operatorname { Ext } ( A )$.) BDF showed that $X \mapsto \operatorname { Ext } ( X )$ behaves like a generalized homology theory in $X$ (cf. also [[Generalized cohomology theories|Generalized cohomology theories]]), and in fact for finite CW-complexes (cf. also [[CW-complex|CW-complex]]) coincides with $K _ { 1 } ( X )$, where $K_*$ is the homology theory dual to complex [[K-theory|$K$-theory]]. This was extended in [[#References|[a7]]], where it was shown that $\operatorname { Ext } ( X )$ is canonically isomorphic to $K _ 1 ^ { S } ( X )$, Steenrod $K$-homology (cf. also [[Steenrod–Sitnikov homology|Steenrod–Sitnikov homology]]), for all compact metric spaces $X$, and in [[#References|[a5]]], where it was shown that on a suitable category of $C ^ { * }$-algebras, $\operatorname { Ext } ( A )$ fits into a short [[Exact sequence|exact sequence]]

\begin{equation*} 0 \rightarrow \operatorname { Ext } _ { \mathbf{Z} } ^ { 1 } ( K _ { 0 } ( A ) , \mathbf{Z} ) \rightarrow \operatorname { Ext } ( A ) \rightarrow \end{equation*}

\begin{equation*} \rightarrow \operatorname{Hom}_{\mathbf{Z}} ( K _ { 1 } ( A ) , \mathbf{Z} ) \rightarrow 0. \end{equation*}

It is now (as of 2000) known that BDF theory is just a special case of a more general theory of $C ^ { * }$-algebra extensions. One type of generalization (see [[#References|[a13]]]) involves replacing $\mathcal{K} ( \mathcal{H} )$ by the algebra of  "compact"  operators of a $\mathrm{II} _ { \infty }$ factor (cf. also [[Von Neumann algebra|von Neumann algebra]]). Another sort of generalization involves replacing $\mathcal{K} ( \mathcal{H} )$ by an algebra of the form $B \otimes \mathcal{K} ( \mathcal{H} )$, where $B$ is another separable (or $\sigma$-unital) $C ^ { * }$-algebra. Theories of this sort were worked out in [[#References|[a9]]], [[#References|[a10]]] and in [[#References|[a8]]], though the theory of [[#References|[a9]]], [[#References|[a10]]] turns out to be basically a special case of Kasparov's theory (see [[#References|[a11]]]). Kasparov's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027076.png"/>-theory gives rise to a bivariant functor $\operatorname { Ext } ( A , B )$, and when $A$ is nuclear, this coincides [[#References|[a8]]] with Kasparov's bivariant $K$-functor $K K ^ { 1 } ( A , B )$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  W. Arveson,   "Notes on extensions of $C ^ { * }$-algebras"  ''Duke Math. J.'' , '''44''' :  2  (1977)  pp. 329–355</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L.G. Brown,   R.G. Douglas,   P.A. Fillmore,   "Unitary equivalence modulo the compact operators and extensions of $C ^ { * }$-algebras" , ''Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N.S., 1973)'' , ''Lecture Notes in Mathematics'' , '''345''' , Springer  (1973)  pp. 58–128</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L.G. Brown,   R.G. Douglas,   P.A. Fillmore,   "Extensions of $C ^ { * }$-algebras and $K$-homology"  ''Ann. of Math. (2)'' , '''105''' :  2  (1977)  pp. 265–324</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  I.D. Berg,   "An extension of the Weyl–von Neumann theorem to normal operators"  ''Trans. Amer. Math. Soc.'' , '''160'''  (1971)  pp. 365–371</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  L.G. Brown,   "The universal coefficient theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027085.png"/> and quasidiagonality" , ''Operator Algebras and Group Representations I (Neptun, 1980)'' , ''Monographs Stud. Math.'' , '''17''' , Pitman  (1984)  pp. 60–64</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M.D. Choi,   E.G. Effros,   "The completely positive lifting problem for $C ^ { * }$-algebras"  ''Ann. of Math. (2)'' , '''104''' :  3  (1976)  pp. 585–609</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J. Kaminker,   C. Schochet,   "$K$-theory and Steenrod homology: applications to the Brown–Douglas–Fillmore theory of operator algebras"  ''Trans. Amer. Math. Soc.'' , '''227'''  (1977)  pp. 63–107</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  G.G. Kasparov,   "The operator $K$-functor and extensions of $C ^ { * }$-algebras"  ''Math. USSR Izv.'' , '''16'''  (1981)  pp. 513–572  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''44''' :  3  (1980)  pp. 571–636; 719</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Pimsner,   S. Popa,   D. Voiculescu,   "Homogeneous $C ^ { * }$-extensions of $C ( X ) \otimes \mathcal{K} ( H )$. I"  ''J. Oper. Th.'' , '''1''' :  1  (1979)  pp. 55–108</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Pimsner,   S. Popa,   D. Voiculescu,   "Homogeneous $C ^ { * }$-extensions of $C ( X ) \otimes \mathcal{K} ( H )$. II"  ''J. Oper. Th.'' , '''4''' :  2  (1980)  pp. 211–249</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J. Rosenberg,   C. Schochet,   "Comparing functors classifying extensions of $C ^ { * }$-algebras"  ''J. Oper. Th.'' , '''5''' :  2  (1981)  pp. 267–282</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  W. Sikonia,   "The von Neumann converse of Weyl's theorem"  ''Indiana Univ. Math. J.'' , '''21'''  (1971/72)  pp. 121–124</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  G. Skandalis,   "On the group of extensions relative to a semifinite factor"  ''J. Oper. Th.'' , '''13''' :  2  (1985)  pp. 255–263</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  D. Voiculescu,   "A non-commutative Weyl–von Neumann theorem"  ''Rev. Roum. Math. Pures Appl.'' , '''21''' :  1  (1976)  pp. 97–113</td></tr></table>