A NOTE ON ESSENTIALLY NORMAL OPERATORS

Abstract

Associated with every compact metric space X, there is a classifying structure Ext(X) for the equivalence classes of extensions of C(X) by the 7*-algebra of compact operators. In this note we give a new proof that Ext(X) is a group, based on operator-theoretic techniques. In a recent paper [3], Brown, Douglas and Fillmore have classified essentially normal operators up to compact perturbations. More generally, for each compact metric space X, they consider the class of all (separably acting) separable C*-algebras s/, which contain the algebra W(t) of all compact operators on the underlying Hilbert space X, and for which the quotient C*-algebra s//(Jt') is isomorphic to 0(X). This family of algebras is associated with an abelian group Ext(X) which classifies the algebra to an appropriate equivalence. When X is a subset of the complex plane, this reduces to con sidering the class of essentially normal operators having X as their essential spectrum, with respect to the relation unitary equivalence modulo compact perturbations. In this case, the Fredholm index gives rise to a homomorphism of Ext(X) into the free abelian group having one generator for each hole of X, and the classification result alluded to in the first sentence asserts that this homomorphism is injective. Now in the general case, it is not very hard to show that Ext(X) is a commutative semigroup with zero, and one of the main results of Brown Douglas-Fillmore is that Ext(X) is in fact a group: i.e. has inverses. The proof is quite indirect, and it may be of interest to have an alternate proof which is independent of their machinery. The purpose of this note is to point out how such a proof can be based on a lifting theorem of Vesterstrom [5] and Andersen [1] (see also Ando [2]), together with some elementary considerations from dilation theory. For simplicity, we only consider the case where X is a compact subset of the complex plane C, but the reader may easily see that everything goes through for arbitrary compact metric spaces. Let X c C be compact. We define EN(X) to be the class of all essentially normal operators A (i.e., A*A AA* is compact), such that A acts on a * This research was partially supported by a grant from the National Science Foundation. This content downloaded from 157.55.39.102 on Sun, 25 Dec 2016 06:47:05 UTC All use subject to http://about.jstor.org/terms 144 Proceedings of the Royal Irish Academy separable infinite dimensional Hilbert space 9A and has X as its essential spectrum (the essential spectrum of A is written a6(A), and as usual is defined as the spectrum of the image of A in the Calkin algebra). Note that the Hilbert SpaceX'A is allowed to vary with A. If A and B are two elements of EN(X), then so is the direct sum A 03 B, as well as every compact perturbation A + K, K E %'(tA). Two elements A,B E EN(X) are said to be equivalent (A B) if there is a unitary operator U: 1A' A--> such that UA BU is compact (equivalently, UA U* B E r(ArB)). Ext(X) is defined as the set of all equivalence classes EN(X)/r..-i. If a,b e Ext(X), one may define a + b, as the equivalence class of any operator of the form A (0 B, where A and B are arbitrarily chosen elements of a and b respectively. It is easy to see that + is well-defined, and is a commutative associative binary operation on Ext(X). Now one of the preliminary results of [3] implies that if A,N E EN(X) and N is normal, then A 0 N A. By taking A to be normal we see that N XN A A (D N A, so that the normal elements of EN(X) determine a single class in Ext(X); moreover, the preceding implies further that this class functions as a zero for the additive semigroup Ext(X). Thus, to show that Ext(X) is a group, it suffices to show that for every a e Ext(X), the equation a + x = 0 has a solution x e Ext(X). This is the content of the following: Theorem A. For every A e EN(X), there eXists B e EN(X) such that A 0 B hs he form N + K, where N is nommal and K is compact. To prove Theorem A we shall make use of the following special case of the lifting theorem [1], [5] mentioned above (note, incidentally, that liftings with similar properties are discussed in [2]): Theorem B. Let 9 be a separable C*-algebra with unit, let 9 be a closed two ided idal in 91 such that W/93 iscommutative, and et 7r: W W/% be the canonical projection. Then there is a positive linear map #: 9/931 -?91, which preserves identities, such that vr o k is the identity map of W/93. Now let A e Y(*) be essentially normal with ae(A) = X. We will produce an operator B with similar properties such that A 0f B is normal plus compact. Let 4 be the C*-algebra generated by A, I and 'C(t): dI = 0*(A) + e(a). Sinoe the image A of A in sIf(*') is normal, generates d/() (along with 1) as a 0*-algebra, and has X as its spectrum, it follows that d/W(t) is isomorphic with 0(X) in such a way that A corresponds to the independent variable function z E 0(X). Combining the positive map of Theorem B with the inverse of this isomorphism, we obtain a positive linear map e: 0(X) -v SI having the pro perties