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''BDF theory''
 
''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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302701.png" /> on an infinite-dimensional separable [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302702.png" /> is determined up to compact perturbations, modulo unitary equivalence, by its essential spectrum. (The essential spectrum is the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302703.png" /> of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302704.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302705.png" /> in the Calkin algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302706.png" />; it is also the spectrum of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302707.png" /> to the orthogonal complement of the eigenspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302708.png" /> for the eigenvalues of finite multiplicity; cf. also [[Spectrum of an operator|Spectrum of an operator]].) In other words, unitary equivalence modulo the compacts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b1302709.png" /> washes out all information about the [[Spectral measure|spectral measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027010.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027012.png" />. Indeed, the  "unilateral shift"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027013.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027016.png" /> is a rank-one projection, yet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027019.png" /> is a compact perturbation of a normal operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027020.png" /> is essentially normal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027021.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027022.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027023.png" />-algebraic context in [[#References|[a2]]] and [[#References|[a3]]]. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027024.png" /> "up to compact perturbations"  defines an injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027025.png" />-homomorphism from a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027026.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027027.png" /> (the closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027028.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027030.png" />) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027031.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027032.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027033.png" /> is Abelian if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027034.png" /> is essentially normal. More generally, an extension of a separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027036.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027037.png" /> is an injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027038.png" />-homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027039.png" />, since this is equivalent to a commutative diagram with exact rows:
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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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027040.png" /></td> </tr></table>
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<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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027042.png" />, whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027043.png" />-element is represented by split extensions (those for which there is a lifting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027044.png" />). (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027045.png" /> is nuclear (cf. also [[Nuclear space|Nuclear space]]). (BDF originally worked only with Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027046.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027048.png" />, for which this is automatic, and they used the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027050.png" />.) BDF showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027051.png" /> behaves like a generalized homology theory in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027052.png" /> (cf. also [[Generalized cohomology theories|Generalized cohomology theories]]), and in fact for finite CW-complexes (cf. also [[CW-complex|CW-complex]]) coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027054.png" /> is the homology theory dual to complex [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027055.png" />-theory]]. This was extended in [[#References|[a7]]], where it was shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027056.png" /> is canonically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027057.png" />, Steenrod <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027059.png" />-homology (cf. also [[Steenrod–Sitnikov homology|Steenrod–Sitnikov homology]]), for all compact metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027060.png" />, and in [[#References|[a5]]], where it was shown that on a suitable category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027061.png" />-algebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027062.png" /> fits into a short [[Exact sequence|exact sequence]]
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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]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027063.png" /></td> </tr></table>
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\begin{equation*} 0 \rightarrow \operatorname { Ext } _ { \mathbf{Z} } ^ { 1 } ( K _ { 0 } ( A ) , \mathbf{Z} ) \rightarrow \operatorname { Ext } ( A ) \rightarrow \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027064.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027065.png" />-algebra extensions. One type of generalization (see [[#References|[a13]]]) involves replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027066.png" /> by the algebra of  "compact"  operators of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027068.png" /> factor (cf. also [[Von Neumann algebra|von Neumann algebra]]). Another sort of generalization involves replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027069.png" /> by an algebra of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027071.png" /> is another separable (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027072.png" />-unital) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027073.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027077.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027078.png" /> is nuclear, this coincides [[#References|[a8]]] with Kasparov's bivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027079.png" />-functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027080.png" />.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,  "Notes on extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027081.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027082.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027083.png" />-algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027084.png" />-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. ChoiE.G. Effros,   "The completely positive lifting problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027086.png" />-algebras"  ''Ann. of Math. (2)'' , '''104''' :  3  (1976)  pp. 585–609</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. KaminkerC. Schochet,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027087.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027088.png" />-functor and extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027089.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027090.png" />-extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027091.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027092.png" />-extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027093.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130270/b13027094.png" />-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>
+
<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. ChoiE.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. PimsnerS. 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>

Latest revision as of 17:44, 1 July 2020

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 $T = T ^ { * }$ on an infinite-dimensional separable 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.) In other words, unitary equivalence modulo the compacts $\mathcal{K} ( \mathcal{H} )$ washes out all information about the spectral measure of $T$, 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 $T$ 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; 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 $T$ 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 [a2] and [a3]. An operator $T$ "up to compact perturbations" defines an injective $*$-homomorphism from a $C ^ { * }$-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:

BDF defined a natural equivalence relation (basically unitary equivalence) and an addition operation on such extensions, giving a commutative 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 [a14].) It was shown by M.D. Choi and E.G. Effros [a6] (see also [a1]) that this monoid is a group whenever $A$ is nuclear (cf. also 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), and in fact for finite CW-complexes (cf. also CW-complex) coincides with $K _ { 1 } ( X )$, where $K_*$ is the homology theory dual to complex $K$-theory. This was extended in [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), for all compact metric spaces $X$, and in [a5], where it was shown that on a suitable category of $C ^ { * }$-algebras, $\operatorname { Ext } ( A )$ fits into a short 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 [a13]) involves replacing $\mathcal{K} ( \mathcal{H} )$ by the algebra of "compact" operators of a $\mathrm{II} _ { \infty }$ factor (cf. also 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 [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 $\operatorname { Ext } ( A , B )$, and when $A$ is nuclear, this coincides [a8] with Kasparov's bivariant $K$-functor $K K ^ { 1 } ( A , B )$.

References

[a1] W. Arveson, "Notes on extensions of $C ^ { * }$-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 $C ^ { * }$-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 $C ^ { * }$-algebras and $K$-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 $C ^ { * }$-algebras" Ann. of Math. (2) , 104 : 3 (1976) pp. 585–609
[a7] 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
[a8] 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
[a9] 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
[a10] 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
[a11] J. Rosenberg, C. Schochet, "Comparing functors classifying extensions of $C ^ { * }$-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
How to Cite This Entry:
Brown-Douglas-Fillmore theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brown-Douglas-Fillmore_theory&oldid=22195
This article was adapted from an original article by Jonathan Rosenberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article