View source for Brown-Douglas-Fillmore 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" />.

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:

<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>

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]]

<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>

<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>

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" />.

====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. Choi,   E.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. Kaminker,   C. 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>