We now take up the general problem mentioned in the introduction: to what e x t e n t does an algebra of operators on a ttilbert space determine the structure of the C*-algebra it generates? More precisely, let ~ be a Hilbert space and let A be a subalgebra of L(~) which contains the identity operator. The meaning of this question can be illustrated in terms of invariants. Let us say a property of C*(•) is invariant (relative to ~ ) if, for e v e r y operator algebra A1 which is completely isometrically isomorphic to ,~, C*(A1) has the property when, and only when,
C*(A)
has it. Accordingly, if there are enough invariant properties to determineC*(,,4)
to within .-isomorphism, t h e n in an obvious sense ,~ deter- mines the structure of its generated C*-algebra.I t is not obvious t h a t invariant properties should exist at all. We will show, however, t h a t certain irreducible representations of C*(A) (the boundary representations) are A-invariant in the above sense. This leads to a body of general results, relating to analogues of Silov boundaries and the problem of implementing certain linear maps of operator algebras with .-isomorphisms (Section 2.2).
I n sections 2.3 and 2.4, we obtain a characterization of boundary representations which is more useful for specific applications, a number of which are t a k e n up later in Chapter 3.
2.1.
Boundary Representations.
L e t A be a linear subspace of a C*-algebra B, such t h a tB=C*(A).
We assume, throughout this chapter, t h a t such an A always contains the168 ~ ] T N , T A M B, ARV]~SON
identity of B. I f eo is a n y representation of B, t h e n e0[A has just one multipllcative com- pletely positive extension to B (namely co); in general, however, there m a y be other linear completely positive extensions of ~o[A.
De]inition 2.1.1. An irreducible representation eo of B is called a boundary representa- tion for A if eo I A has a unique completely positive linear extension to B.
W h e n B = C(X) (for X a compact Hausdorff space), the irreducible representations correspond to point evaluations; a n d if A is a separating linear subspace of C(X) t h e n the b o u n d a r y representations correspond to points of X which have unique representing measures (relative to A). This is one of the characteristic properties of points in the Choquet b o u n d a r y of X relative to A [16]. W h e n dealing with one-dimensional b o u n d a r y representa- tions of a general C*-algebra B, the analogy with Choquet b o u n d a r y points carries over quite well (cf. section 3.1). General b o u n d a r y representations, on the other hand,, can possess properties for which there is no c o m m u t a t i v e counterpart (cf. section 3.5), a n d one should p r o b a b l y not t r y to push the analogy too far.
The very useful feature of b o u n d a r y representations of B is their invariance relative to A, as described in the following theorem.
THEOREM 2.1.2. Let B and B t be C*-algebras and let A and A t be linear subspaces o/
B and B1, respectively. Assume B=C*(A) and BI=C*(At). Let q~ be a completely isometric linear map o / A on A t such that q~(e)=e. Then/or every boundary representation eoo/ B (relative to A) there exists a boundary representation co t o~ B 1 (relative to A1) such that 0) 1 o~(a) = co(a), aEA.
Proo/. We m a y assume t h a t B1 acts on a Hilbert space ~. B y 1.2.9, ~ m a y be extended to a completely positive linear m a p of B into L(~), which we denote b y the same letter ~.
:Now the m a p q~(a)-+co(a), aEA, is a completely contractive linear m a p of A 1 into c o ( A ) ~ L ( ~ ) which takes the identity to the identity, ~ being the Hilbert space on which ~o(B) acts, so b y 1.2.9, there exists a completely positive linear m a p ~: B I - ~ L ( ~ ) such t h a t ~oT(a)=w(a), aEA. We will show t h a t such a ~ m u s t be a representation of B 1.
This will complete the proof, for two representations which agree on ~ ( A ) = A 1 m u s t agree on C*(At)= B1, a n d such a ~ m u s t be irreducible because Q(BI)=C*(Qo?(A))=C*(~o(A))=
~o(B) a n d ~o is irreducible; thus ~ is the required b o u n d a r y representation of B1, relative to A 1.
Now clearly C*(q~(B))~ C*(q~(A))= B1, so there is, b y 1.2.3, a completely positive m a p
~: C*(~(B))-~L(~o) such t h a t ~ =~ on B r B y 1.1.1, there is a representation ~ of C*(cp(B)) on ~ a n d an operator VEL(~o, ~) such t h a t ~(x) = V*:~(x) V, xEC*(~(B)), a n d V ~ is cyclic for ~(C*(~(B)): F o r aEA, eo(a)=~oq~(a)=V*TeoqJ(a)V, so t h a t xEB-+V*~o~(x)V is a
completely positive extension of co [A to B. Since (o is a b o u n d a r y representation, we con- clude t h a t co(x)= V*zo~o(x)V, for all x E B.
We claim now t h a t V is unitary. Indeed, V*V = g*~oqJ(e) V =w(e) = I , so V is isometric, a n d it suffices to show t h a t [ V ~ ] = R . B u t [ V ~ ] is cyclic for ~(C*(~(B)))=C*(Tcocf(B)), and so [ V ~ ] = R follows if we prove t h a t the self-adjoint family of operators z o ~ ( B ) leaves [ V ~ ] invariant. Choose a u n i t a r y element u in B. Then for ~ E ~ , we h a v e
II ov(u) W (u) ll 2 = II ov(u) V ll 2 - 2 Re (V*zov(u) V~, r ][r 2
= II ov(u) V ll 2 - II (u) ll 2 = I l i o n ( u ) V ll II ll ll ll II [I = 0, since V*~oq)(u) V=co(u), o)(u) is unitary, a n d HT~o~(u)] I 4 1 . Thus, 7~o~0(u) V ~ = Vw(u)~E [ V ~ ] , for every ~ E ~ , and hence ~ o T ( u ) leaves [ V ~ ] invariant. Since B is the norm- closed linear span of its u n i t a r y elements, we see t h a t uo~0(B) leaves [ V ~ ] invariant;
b y the above comments, V is unitary.
Thus ~ = V-I~V is a representation of C*(q)(B)), a n d hence ~ = ~ ] B 1 is a representation of B r T h a t completes the proof.
Note t h a t the proof shows somewhat more t h a n we have claimed, when B~ acts on a Hilbert space ~; namely, for every completely positive extension ~ of ~ to B, there exists a unique completely positive m a p p: C*(~(B))->L(~), which is necessarily a .-representa- tion, such t h a t ~o~(x)=w(x), xEB.
We shall give a n u m b e r of applications of this theorem, a basic result of this paper, in the following two sections a n d in Chapter 3. Chapter 3 also contains a n u m b e r of examples of b o u n d a r y representations.
L e t X be a compact Hausdorff space a n d let A be a linear subspaee of C(X) which contains the constants and separate points of X. T h e n there is a smallest closed subset K of X such t h a t every function in A achieves its m a x i m u m modulus on K, called the Silov boundary of X relative to A [16]. We now introduce a n o n - c o m m u t a t i v e generaliza- tion of the Silov boundary.
De/inition 2.1.3. L e t A be a linear subspace of a C*-algebra B such t h a t A contains the identity and generates B as a C*-algebra. A closed (two-sided) ideal J in B is called a boundary ideal for A if the canonical quotient m a p q: B-~B/J is completely isometric on A. A b o u n d a r y ideal is called the Silov boundary for A if it contains every other bound- a r y ideal.
I f B=C(X) a n d K is a closed subset of X, t h e n J=(/EC(X):/(K) = 0 } is a closed ideal in B, a n d the quotient n o r m in BIg is given b y ]l/] KH =supxEK ]/(X)], for /E C(X). Thus, J is a b o u n d a r y ideal for A iff K is a b o u n d a r y for A in the sense of the discussion preceding
170 Wn.T3"a~r B. A R V E S O N
2.1.3 (here we use the fact t h a t a constant-preserving isometric linear m a p between sub- spaces of abelian C*-algebras is completely isometric, b y 1.2.11). The known correspondence between closed subsets of X a n d ideals in C(X) now shows t h a t 2.1.3 reduces to the usual definition of the Sflov b o u n d a r y for subspaces of c o m m u t a t i v e C*-algebras.
Note t h a t the Silov b o u n d a r y of A _ B is unique, whenever it exists. W h e t h e r or not it always exists under the general conditions of 2.1.3 is, however, still an open question.
I n the c o m m u t a t i v e case B = C(X), it is known t h a t the closure of the set of all Choquet b o u n d a r y points is the Silov boundary. We will show in the n e x t section t h a t a similar fact is true for "admissible" subspaces of a r b i t r a r y U*-algebras, b u t t h a t is the best general result we now know. F o r reasons brought out clearly in the n e x t section, this question has significance in the development of an a b s t r a c t t h e o r y of (non self-adjoint) operator algebras.
2.2. Admissible subspaces o/C*-algebras. I n this section, we show t h a t the Silov b o u n d a r y exists for "admissible" subspaces, a n d we obtain some consequences; toward the end of the section we discuss a sufficient condition for admissibility.
L e t A be a linear subspace of a C*-algebra B such t h a t B = C*(A). We remind the reader t h a t A is always assumed to contain the identity.
De]inition 2.2.1. A is called an admissible subspace of B if the intersection of the kernels of the b o u n d a r y representations (for A) is a b o u n d a r y ideal for A.
L e t bd A denote the class of all b o u n d a r y representations for A (to avoid set-theoretic difficulties, one should regard bd A as a set of representatives, one t a k e n from each uni- t a r y equivalence class of b o u n d a r y representations: we shall be deliberately casual a b o u t this kind of distinction). The reader can easily see t h a t A is admissible if, a n d only if, it satisfies the condition: ]or every integer N >~ 1 and every N • N matrix (a,) over A , one has
II(a.)H=
sup II(w(a,,))ll,r
where the n o r m of (a~j) is inherited f r o m B | Note also that, since a representation of B is always completely contractive, one need only check the inequality ~<. I t is significant t h a t admissibility is an i n v a r i a n t for completely isometric linear maps:
THEOREM 2.2.2. Let A (resp. A1) be a linear subspace o / a C*-algebra B (resp. B1) such that B = C * ( A ) (resp. B 1 =C*(AO) and suppose there is a completely isometric linear map
o / A on A 1 such that qJ(e) =e. I / A is admissible then so is A 1.
Proo/. We will show t h a t A 1 satisfies the condition of the preceding paragraph. L e t
~V be a positive integer a n d let (btj) be an N • N m a t r i x over A r T h e n there exists elements
a~j6A such t h a t ~0(a~j)=b~. For each 0)Ebd A, let 0)1 be the element of bd A 1 satisfying 0)1o~ =0) (by 2.1.2). Since A is admissible a n d ~ is completely isometric, we h a v e
H (b,j)[[ = II (a~)II = s u p AI[ (0)(a,J))ll = sup A ]] (0)~ (b~J)) II < ~,,sup [1(0), (b~,)) H, a n d the proof is complete.
P R O P O S I T I O ~ 2.2.3. Let A be a subspace o[ B such that C*(A) = B, let J be a boundary ideal/or A, and let 0)Ebd A. Then J c_ker 0).
Proo/. L e t q be the quotient m a p of B into B/J. T h e n q] A is completely isometric, q(e)=e, a n d C*(q(A))=q(C*(A))=q(B)=B/J. B y 2.1.2, there exists a representation 0)x of B / J such t h a t 0)1oq =0) on A. Since 0)1oq a n d 0) are ,-homomorphisms a n d A generates B, we have eo 1 o q =0) on B. Thus, if x E J = K e r q, t h e n 0)(x)=0)10q(x)= 0; thus J~_ ker o~, completing the proof.
T }t ~ o R E M 2.2.3. Let A be an admissible subspace o/B, and let K be the intersection o/all kernels o/boundary representations. Then K is the Silov boundary ideal/or A.
Proo/. B y hypothesis, K is a b o u n d a r y ideal. I f J is a n y other b o u n d a r y ideal a n d if 0)Ebd A, t h e n b y 2.2.2 we have J _ K e r 0); hence, Jc_K, a n d we are done.
Now let A be an admissible subspaee of B, let K = N o , b a K e r 0) be the Silov b o u n d a r y for A, a n d let q be the quotient m a p of B onto B/K. The process of passing from A to q(A) c_ B / K is analogous to passing from a subspace A of C(X) to the space of restrictions A [~X_~ C(aX), a n d in dealing with " a b s t r a c t " admissible subspaees, it is convenient to do this. Note, for example, t h a t we have the following.
PROPOSITION 2.2.4. Let A be an admissible subspace o/ a C*-algebra B such that B=C*(A), let K be the Silov boundary ideal/or A, and let q be the quotient map o / B in B/K.
Then q(A) is an admissible 8ubspace o / B / K which has {0} as its Silov boundary.
Proo/. The admissibility of q(A) is evident from 2.2.2. B y 2.1.2, the relation 0)1oq = w sets up a bijective correspondence eeoc) 1 between the b o u n d a r y representations for A a n d those for q(A) (note t h a t the equation 0)1oq = w on A entails its validity on B = C*(A), since w, o)1, a n d q are all .-homomorphisms). Thus, for x 6 B, q(x) E N bet 0) 1 implies 0)(x) = 0)loq(x) = 0 for all w Ebd A, hence xE N o E ~ K e r 0)=K, a n d so q(x)=0. 2.2.3 now shows t h a t {0} is the Silov b o u n d a r y for q(A).
I t is natural to ask the e x t e n t to which a subalgebra or subspaee, A, of a C*-algebra B determines the structure of B. E v e n when B is c o m m u t a t i v e a n d is generated b y A, there can be quite a variation of structure. F o r example, let D be the closed unit disc, a n d
172 w r r . r . T A ~ B . ARVES01~"
let A be the closed subalgebra of C(D) consisting of all sup-norm limits of polynomials.
A separates points of D and thus C*(A)= C(D). On the other hand, the unit circle T is a subset of D, and the restriction map /EC(D)~/[TE C(T) is a .-homomorphism of C(D) on C(T) which is completely isometric on A by the maximum modulus principle (cf. 1.2.11).
Thus, A I = A I T is the "same" as A, whereas C*(A1)=C(T) is quite different from C(D).
The following result implies, among other things, t h a t an admissible subspace com- pletely determines the structure of its generated C*-algebra, once one has factored b y the Silov boundary ideal.
THEOREM 2.2.5. Let A (resp. A1) be an admissible subspace o / a C*-algebra B (resp. B1) such that B = C*(A ) (resp. B 1 = C*(A1) ). Assume that both A and A 1 have trivial Silov boundary ideals. Then every completely isometric linear map o / A on A 1, which takes e to e, is imple- mented by a .-isomorphism o] B on B 1.
Proo/. Let S be the set of all equivalence classes of b o u n d a r y representations of B (for A). For each n E S choose a representative w~ for n. Let ~ be a completely isometric linear map of A on A 1 such t h a t ~(e)=e. B y 2.1.2, there exists, for each nES, a b o u n d a r y
9 9 t
representation w~ of B 1 (for A1) such t h a t wno~ =wn on A. As n runs over S, ~o~ runs over all (classes) of boundary representations of B 1. So b y hypothesis, and 2.2.3, we have n n Ker o) n = A n Ker o)n = {0}. Thus the representations 7~ = Qn ~0~ and :~' = (~ neon are, respectively, faithful representations of B and B 1. Moreover, ~ ' o T = ~ on A, by con- struction. I t follows t h a t ~'(B1) = C*(~'o~(A)) = C*(ze(A))=Te(B), and since both 7~ and ~' are injective, the mapping G=(y~')-log is in fact a .-isomorphism of B on B 1. The rela- tion 7~'ocf(a)= g(a), a EA, implies a I A = % and the proof is complete.
COROLLARY 2.2.6. Let A be an admissible subalgebra o / a C*-algebra B, such that B = C*(A ). Then every completely isometric linear mapping o / A onto itsel/ which leaves the identity fixed is an algebra automorphism.
Proo/. Let K be the Silov b o u n d a r y ideal for A and let q be the quotient map of B on B/K. Let ~ be a completely isometric map of A on A such t h a t ~(e)=e, and put ~1=
q o ~ o q - l : q(A)~q(A). B y 2.2.5, ~1 is implemented b y a .-automorphism of B / K , and in particular ~1 is multiplicative on q(A). Since q is an algebra isomorphism of A on q(A), it follows t h a t ~ =q-lO~lO q is multiplicative on A. That completes the proof.
We conclude this section with a discussion of one sufficient condition for a subspaee to be admissible. This condition is not always satisfied, but it is effective in dealing with a variety of examples. Also, we point out t h a t some related questions will be t a k e n up in section 2.4.
L e t A be a linear subspace of a C*-algebra B a n d let to b e an irreducible representa- tion of B. Define the set M~ to be all b in the closure of A +A* for which co(b) has the form lib ]] U, where U is a u n i t a r y operator. ~//~ consitsts of those elements of (A + A*)- which, in a sense, t a k e on their " m a x i m u m modulus" in to. Note, too, t h a t M* = M ~ .
THEOREM 2.2.7. Let A be a linear subspace o / a C*-algebra B and let to be an irreducible representation o/ B such that M~ generates B as a C*-algebra. Then ~o is a boundary repre- sentation/or A.
Proo/. L e t ~ be a n y completely positive extension of ~oiA, say Q = V*xeV, where 7~ is a representation of B on a Hilbert space ~ and V is an operator from the Hilbert space on which w acts to ~, such t h a t [~(B) V~] = ~ . As in 2.1.2, it suffices to show t h a t r is a representation, or w h a t is the same, t h a t V is unitary. Now V*V=~(e)=og(e)=I, so V is isometric, and we need only prove t h a t [V~] = ~ .
N o t e t h a t Q m u s t equal ~o on the closure of A + A*, since both are bounded self-adjoint linear maps. Take z E (A +A*)-. Then for every ~ E,~, we have
I1~(~) v ~ - v ~ ( z ) ~ l l ~ = II~(z) v~ll ~ - 2 R e ( v * ~ ( ~ ) v ~ , ~ ( z ) ~ ) + II v ~ ( z ) ~ l l ~
= I1~(~) v~ll ~ - II~(z)~ll :,
since V'st(z)V=eo(z) and V is isometric. So if zEM+, t h e n I1~(~)~11~=11~11:11~11 ~, hence I1~(~) v~ll 2 - I1~(~)~ll 2 ~< 0, and it follows t h a t :re(z) V~ = Vw(z)~ E [V~]. Thus, [V~] is invariant under the self-adjoint family of operators u(M~), which generates u(B) as a C*-algebra.
We conclude t h a t ~ = [ s t ( B ) V ~ ] _ ~ [V~], as required.
Some examples are noteworthy. Suppose A is a linear subspaee of B such t h a t the u n i t a r y elements of (A + A * ) - generate B as a C*-algebra; for example, A + A * could be dense in B, or A could be the algebra generated b y a semigroup of u n i t a r y operators which contains e and generates B as a C*-algebra. Then b y 2.2.7, every irreducible representation of B is a b o u n d a r y representation for A; hence A is admissible and, in fact, the Silov b o u n d a r y of A is the trivial ideal {0}. Thus, making use also of 2.2.5, we can state the following.
COROLLARY 2.2.8. Let A be a subspace o / a C*-algebra B such that the unitary elements in (A + A*)- generate B as a C*-algebra. Then every irreducible representation o / B is a bound- ary representation/or A , the Silov boundary ideal/or A is trivial, and every completely iso- metric linear map o/ A onto itsel/ which leaves the identity fixed is implemented by a *-auto- morphism o/ B.
We r e m a r k t h a t the same conclusion can be drawn from weaker, though less easily verified, hypotheses. For convenience, let us call an irreducible representation o~ of B
174 WttJJTAM B. 2kR~?'~SON
peakin9 for A if M~(_~ (A + A * ) - ) generates B as a C*-algebra. T h e n the conclusion of 2.2.8 is valid provided only t h a t the intersection of the kernels of all peaking representa- tions is trivial.
2.3. Finite representations o/ operator algebras. This section serves two purposes; it contains material which is p r e p a r a t o r y for the characterization of b o u n d a r y representa- tions in 2.4, a n d we introduce here certain notions a n d terminology which will be used in the entire sequel.
I n this section a n d the next, we shall be interested in subalgebras (rather t h a n sub- spaces) of C*-algebras. Recall first the definition of semi-invariant subspaces. A closed subspace ~j~ of a t t i l b e r t space ~ is said to be semi-invariant under a subalgebra t4 of L ( ~ ) (• is assumed to contain the identity) if the m a p r = P ~ T I ~ is multiplicative on M.
The definition is due to Sarason [20], who pointed out the following characterization. I f is semi-invariant for A, t h e n ~rJ~ o = [M~J~] @ ~)~ is ~4-invariant, so t h a t ~j~ = [A~J~] G ~J~o is a nested difference of M-invariant subspaces; conversely if ~ = ~ 1 @ ~J~0 where ~0~0~ ~J~
are A-invariant, t h e n ~ is semi-invariant for M. Thus when M is a self-adjoint algebra the semi-invariant subspaces are reducing subspaces. I n general, of course, a semi-invariant subspace need not even be invariant.
L e t A be a subalgebra of a C*-algebra B; A is always assumed to contain the identity of B. A representation of A is a h o m o m o r p h i s m ~ of A into the algebra of operators on some Hilbert space, such t h a t
(i) ? ( e ) = I , a n d