a c*-algebra a for which ext(a) is not a group

Annals of Mathematics

A C*-algebra A for which Ext(A) is not a GroupAuthor(s): Joel AndersonSource: Annals of Mathematics, Second Series, Vol. 107, No. 3 (May, 1978), pp. 455-458Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1971124 .

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Annals of Mathematics, 107 (1978), 455-458

A C*-algebra (d for which Ext () is not a group

By JOEL ANDERSON

In [4] and [5] Brown, Douglas and Fillmore introduced the classifying structure Ext((G) in their study of extensions of the compact operators by a separable C*-algebra Cf. As defined in [5], Ext(G) is a commutative semi- group. Brown, Douglas and Fillmore showed [4] that if a is commutative, then Ext((G) is a group. Subsequently, Voiculescu showed in [11] that Ext(af) always contains a neutral element and Choi and Effros proved in [8] that Ext((G) is a group if Cf is a nuclear unital separable C*-algebra. An elegant and unified proof of these results has recently been given by Arveson [3].

Our purpose here is to show that Ext((G) is not always a group. Our example is closely related to C*(F2), the C*-algebra generated by the left regular representation of F2, the free group on the generators (a, b}. The proof is based on an argument of Choi and Effros [9, Lemma 4.2] and, in fact, in the course of the proof it will be shown that the Calkin algebra is not separably injective as was conjectured in [9].

We shall follow the notation of [3]. In particular, the Calkin algebra C(XC) = 3(JC)/IX is by definition the quotient of 3(XC), the bounded linear operators on a complex separable Hilbert space SC by X, the compact operators on XC. T shall denote the coset in C(XC) that contains the operator T, and if p is a *-homomorphism of a C*-algebra into 3(ZC), then we write p- 7 ap, where 7w denotes the natural map of 93(X) onto C(JC). If af is a unital separable C*-algebra, an extension of X by af is a unital *-mono- morphism of af into C(XY) and Ext(al) is by definition all equivalence classes of extensions. (The extensions a, and U2 are equivalent if a,(.) = U62(.) U* for a unitary operator U.) An extension a of X by af is liftable if there is a completely positive map * of af into 3(XJ) such that a = j. Arveson showed [3, Section 4] that Ext(al) is a group if and only if each extension of X by a is liftable. We shall construct a separable C*-algebra and an extension that is not liftable.

Let XC = C1 0 XC2 (E ... denote a decomposition of the Hilbert space

0003-486X/78/0107-0003 $00.20 (? 1978 by Princeton University Press

For copying information, see inside back cover.



456 JOEL ANDERSON

XJC into orthogonal subspaces of dimension ms,< K< (the mn's will be specified below) and write OI = E (d i(XJC). If 611 is a free ultrafilter on the positive integers, then the formula

rz(M) = z21(C E M.) = limq trn(Mn)

defines a trace on OI and , ={Me OIL: z-(M*M) = O} is a maximal two- sided ideal in OR. Wassermann has shown [12, 1.6] that for a certain choice of the m, 's, there are unitary operators U = E E U, and V = E 1 V, in OIL such that for each fixed reduced word w(a, b) =A 1 in F2, the corresponding word w(U, V) = E D w(U,, V") in OI has the property that the sequence {tr,,(w(U,,,, V,,))} is finitely nonzero. Wassermann used this fact to conclude that (C*(U, V) + JGu)1gf and C*(F2) are *-isomorphic for any free ultrafilter qt. (C*(U, V) denotes the C*-algebra generated by U, V and the identity.) It also follows that for each A = D A,, in C*(U, V) the sequence {tr,,(A,)} converges.

Fix a free ultrafilter G1. By Wassermann's result C*(F2) and C*( U, V)/3 are *-isomorphic, where 3 = C*(U, V) ln . Define a state f on $(XC) by f(T) = rq[( E. TE,), where E,, is the projection of X onto Zn,,. Then f is OIL-central in the sense that f(MT) = f(TM) for all T in B(TC) and all M in Oil and f induces a representation Wf of 3(XC) into 3(ThXf) with kernel XJC. Thus 7 o 7r' is a *-isomorphism of 7rf(3(Z7)) onto C(IC). Also, f = fo7U-' is an OR-central state on C(ZC).

Our example consists of a certain separable subalgebra d of 7rf($(XC)) together with the extension a given by the restriction of 7w o 7c 1 to Cf. Be- fore defining (G we show the existence of a projection in OR with special properties.

PROPOSITION. There is a projection P in Oil such that f(P) > 1/2 and PJJ KC.

Proof. Since 3 is separable, there is a strictly positive element A in 3 and (A)'/k, k = 1, 2, - * *, is an approximate identity for 3 [1]. It follows that it suffices to find a projection P such that f(P) ? 1/2 and PA e X. Now A == E D A,, A,, e 7(C,), and as noted above, tr,,(A,,) converges. Since A e 3c, lim tr,,(A,,) = 0. Let (a,,, ... , am,,,,} denote the eigenvalues of A,,. Then a, = {i: as, < 2 tr,,(A,,)} has cardinality at least m,,/2. For otherwise,

M., tr,,(A,) = E ai_ > -'ai, > (m,,/2)2 tr,,(A,) ,

where E' denotes the sum over those i's not in a,,. If P,, denotes the projection of C,,, onto the span of the eigenvectors associated with {ai,,: i C a,,}, then 0 < PnAn = 2 tr,,(A,,)P,, and if P = E D P,,, then PA = E (D PA,, is



A C*-ALGEBRA a FOR WHICH EXT(a) IS NOT A GROUP 457

compact. Furthermore tr.(P.) = m-1(cardinality of ac) > 1/2 so that f(P) = f(P) = limq, tr,(Pn) 2 1/2. Q.E.D.

THEOREM. If a = C*(2rf(U), Wf(V), 2rf(P)), then Ext(d) is not a group.

Proof. Let a denote the restriction of w o z-' to d. Then a is an extension of UC by a and by Arveson's result [3, Section 4] it suffices to show that a is not liftable. To this end, let RI denote the projection of fCf onto the closure of 7rf(g)XCf. Since lrf(g) is an ideal of wf(C*(U, V)), R reduces 7rf(C*( U, V)). Also lrf (P) < R because wf (P)f (g),JCf _wf (JC)XCf = {0}. Thus the map A e-* 2cf(A)R is a *-homomorphism of C*(U, V) whose kernel contains g. Therefore Rwf(C*(U, V)) IRXf is *-isomorphic to a quotient of C*(F2). But C*(F2) is simple [10], so there is a *-isomorphism q of C*(F2) onto Rzf(C*(U, V))IXf .

To complete the proof we use an argument of Choi and Effros [9, Lemma 4.2]. Let u = -1(7rf(U)R) and v = p-1(wf(V)R) denote the generators of C*(F2). It is shown in [7, end of Section 3], that there is a projection e in B(12(F2)) such that

e + ueu* > 1 and e + vev* + v2e(v2)* < 1.

Write 3 = C*(C*(F2), e). As B(R:Xf) is injective [2, p. 149], there is a completely positive extension *: 3 -+ B(RJCf) of 9. Write D = A(e). Since u and v are in the multiplicative domain of * (see [6, Theorem 3.1]) we get

D + 2rf(U)RD7Cf(U)*R > 'RxCf and

D + 1Tf(V)RDjrf(V)*R + 2cf(V2)RD1Tf(V2)*R R < 1,f .

Or, viewing D as an element of i(jJCf) which is 0 on R'JCf, we have

D + 2f(U)D=rf(U)* > R, D + 1rf(V)D7rf(V)* + irf(V2)D=rf(V2)* < R. Put 5F = C*(d, R, D) and suppose there were a completely positive map p of (i into B(XC) such that ( = a. Then, because 9(XC) is injective, there would be a completely positive extension 0: -T -? C(?(C) of p such that

(1) 0(D) + US(D) [* > 0(R) and (2) 0(D) + VO(D) V* + V20(D)( V2)* 0 @(R)

because 0(irf(U)) = T and 0(irf(V)) = V are in the multiplicative domain of 6. By applying f to (1) and (2) and using the fact that f is Sit-central, we obtain

2f(6(D)) >?(0(R)) > 3f(6(D))

so that f(0(R)) = 0. But R ? ifr(P) and so



458 JOEL ANDERSON

0(R) > 0(wf(P)) = a(zf(P)) = P.

Therefore f(0(R)) > f(P) > 1/2, a contradiction. Q.E.D.

Choi and Effros defined a C*-algebra 93 to be separably injective if given unital separable C*-algebras (a C: , any completely positive map A: (a 93 has a completely positive extension *: 3 -* $3 [9, Section 4]. Clearly, the proof of the theorem shows that, as Choi and Effros conjectured, the Calkin algebra is not separably injective.

THE PENNSYLVANIA STATE UNIVERSITY, UNIVERSITY PARK, PA.

REFERENCES

[1] J. AARNES and R. V. KADISON, Pure states and approximate identities, Proc. A.M.S. 21 (1969), 749-752.

[2] W. ARVESON, Subalgebras of C*-algebras, Acta Math. 123 (1969), 141-224. [3] , Notes on extension of C*-algebras, Duke Math. J. to appear. [4] L. BROWN, R. G. DOUGLAS and P. FILLMORE, Unitary equivalence modulo compact

operators and extensions of C*-algebras, Springer-Verlag, Lecture notes 345, 1973, Heidelberg.

[5] , Extensions of C*-algebras and K-homology, Ann. of Math. 105 (1977), 265-324. [6] M. D. CHOI, A Schwartz inequality for positive linear maps on C*-algebras, Ill. J. Math.

18 (1974), 565-574. [7] M. D. CHOI and E. G. EFFROS, Injectivity and operator spaces, J. Functional Anal. 24

(1977), 156-209. [8] , The completely positive lifting problem for C*-algebras, Ann. of Math. 104

(1976), 585-609. [9] , Lifting problems and the cohomology of operator algebras, to appear. [10] R. T. POWERS, Simplicity of the C*-algebra associated with the free group on two

generators, Duke Math. J. 42 (1975), 151-156. [11] D. VOICULESCU, A non-commutative Weyl-von Neumann theorem, Rev. Roum, Pures et

Appl. 21 (1976), 97-113. [12] S. WASSERMANN, On tensor products of certain group C*-algebras, J. Functional Anal. 23

(1976), 239-254.

(Received April 20, 1977)



a c*-algebra a for which ext(a) is not a group

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