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 $T = T ^ { * }$ 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