A Characterization of Pick Bodies

7/28/2019 A Characterization of Pick Bodies

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A CHARACTERIZATION OF PICK BODIES

B. COLE, K. LEWIS AND J. WERMER

ABSTRACTLet 2 = (zv..., zn) be an /i-tuple of distinct points in the open unit disk. We define the Pick body @(z)as the totality of points w = (wv...,wn) in Cn such that there exists feH00 with H/l^ < 1 andJ(zf) = wjtfor 1 - (J[M X),... ,J(Mn)) forfeA, where [f\ denotes

the coset of/ in A/1. The image of the closed unit ball of A/1 under this map isn | 3feA with/(M,) = wp 1


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A CHARACTERIZATION OF PICK BODIES 3 1 7We may obtain an example of a hyperconvex set as follows: we select an n-tuple

of distinct points z l 5 . . . ,zn in the open unit disk in C and put(z) = {v v e C " 1 3 /e / / 0 0 with/(* , ) = " 1 ^j ^ n, and \\f\\n ^ 1},

where z represents the point (z 1 5 . . . ,z n ) in C n .We call the set 3>{z) a Pick body. It follows at once from the definition that 2)(z)is hyperconvex.

Let A(A) denote the disk algebra and let / be the ideal in A(A) consisting of

CLAIM. For z e Cn, put9 = {weCn\lfeA(A) withf[Zj) = wp 1 ^j^n, and M I L < 1},

where \\[F]\\ denotes the norm of[F] in A(A)/I. Then (z) = .Proof. Fix w e (z). Choose

(zj) for each/. PutA ( 0 = 0 ( ( 1 - 1 A ) O - T h e n / f c e ^ (A) a n d \\[fk]\\ 1. Hence

It follows from the definition that 9) is closed. Therefore w = l i m ^ ^ wConversely, let WE Si . F o r k any positive integer, there exists geA(A),| | g | | ^ l + l/fc , wi th g(Zj) = Wj for all / Passing to a pointw ise conv ergen tsubsequence, we obta in gsH with HgH^ ^ 1 an d g(zj) = w } for ally. Hence we3)(z).LEMMA 1. Fix = (( 1 9 . . . ,) in C n wi7/i |Cy| < 1 for each] and ( , # Cy ' / ' # / Letd denote the boundary of 3){Q. Then the points C = (Cf.---.CS) fo/ong to 5 ^ fork= 1 , . . . , - 1 .

The following argument is due to Herbert Alexander. Ifthen there exists geA(A) such that g(Q = C* f o r y = 1,...,, an d | |g| | < 1. H ence\g(z)\ < \z k\ for z on the unit circle. By Rouche's theorem, then, the functions zkand zkg have the same number of zeros in \z\ < 1. But zkg vanishes at then points p 1 < / < n, while z* has exactly k zeros. This contradiction yields that>, as desired.

Thus each Pick body (Q has the two properties: (i) @(Q is hyperconvex in C nand (ii) the kth power of C belongs to the boundary of (Q fo r k = 1,..., 1. W eshall show that these two properties characterize Pick bodies.THEOREM 1. Let K be a subset ofC n. Assume that(1) K is hyperconvex,(2) dK contains a point z = (z 1 5 . . . , zn) with |z^| < 1 for each j and z< ^ z, if i # y ,such that z, . . . , z n - 1 all belong to dK.

Then K is a Pick body.1. Lemmas concerning operators

Let Jf be a finite dimensional Hilbert space and B a linear operator on df.


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318 B. COLE, K. LEWIS AND J. WERMERLEMMA 2. Assume that each eigenvalue of B has modulus less than 1 and that

\\B\\ = ll^""1! = 1. Choose a unit vector in tf with ||5n"VII = 1- Then tn e n-tuple, B^,...,^'1^ is linearly independent.

Proof. Note that Wy/f = WBy/f is equivalent to {{I-B*B)\j/, y/) = 0 and sinceIB*B is positive this is equivalent to (IB*B)y/ = 0. Note that

so all the inequalities are equalities. Hence we have that (IB*B)Bi = 0 forj = 0,\,...,n-2.Now suppose that the w-tuple $, i?0 ,. . . , i?""10 is linearly dependent; then thereexists k < n 1 such that Bk is a linear combination of , B(f>,...,Bk~x. The spacespanned by these vectors is invarian t und er B and is contained in the kernel o f / - B*Bby the above. Hence B is unitary on this subspace and thus has an eigenvalue ofmodulus 1. This contradicts our hypothesis and so the n-tuple (f>, B(J>,...,B n~ l(j> islinearly independent.

LEMMA 3. Let 3ft'bean n-dimensional Hilbert space, and let B be a linear operator onJtf with eigenvalues zx,...,zn. Assume that(3) \Zj\


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A CHARACTERIZATION OF PICK BODIES 319An elementary calculation gives

C>4 = ^ V l~ie dO,and so (, \ is positive definite and makes Cn into an n-dimensional Hilbert space whichwe designate by C".For each w = (w1,...,wn), we denote by Pw the operator on C" defined byPw(t1,...,tn) = (w1t1,...,wntn). We call Pw a Pick operator on C?. We also setS = P,Let ev..., en denote the standard basis on Cn. Then, suppressing the subscript z,we have

(eit e}) = for 1 ^ ij ^ n. (7)\Zi2jBy the definition of S, we also have

Sei = ziei for 1 ^y ^ n. (8)LEMMA 4. Let B be a linear operator on JiC satisfying (3), (4), (5), and hence (6).Then there exists a unitary map U from Jtf to C? such that S = UBU'1.

Proof Choose vx,...,vn as in Lemma 3. From (6) and (7), it follows that(e(, et) = (v


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320 B. COLE, K. LEWIS AND J. WERMERLet "W be the subspace of M spanned by the vectors , T,..., Tn~x(f>. Since st is n-dimensional, Tn is a linear combination of /, T,..., Tn~l and so if is invariant underT. We shall show that iV is ^-dimensional and that the restriction of T to if isunitarily equivalent to the operator 5 on C2.

Define f to be the restriction of T to HT. By (11),so \\f1| = IIT71"1!! = 1. Let x be an eigenvector of f with corresponding eigenvalue X.Choose jQ such that E} x ^0. Then

Hence X = z} and so |A| < 1. Hence each eigenvalue of f has modulus less than 1. SoLemma 2 applies to f and gives that the w-tuple of vectors , f ,..., fn"10 is linearlyindependent and hence is a basis of #". Since for each k, Tk = ^z*,, the vectorsE1 (j>, E2 0 , . . . , En $ span itr. Also, for each ;, TE i - ( zfc fc) Erf = z, E i . So theset JE*! 0, "2 (j>,..., En (f> is a basis of eigenvectors for T.Thus Lemma 4 applies to T and so there exists a unitary operator U from T^" toCzn with S = UTU*.Fix a in ^ . Then || 2 J < 1. Let Q be a polynomial with Q(z}) = a} fory = 1,...,. If Pa denotes the Pick operator introduced above, then

Pa = Q(S) = UQ(f)U* = UQ(TZ)U* = UTa U*,and so HPJ < | | | | < 1. Hence for all /eC z

n, ||Paf||

2< ||r||

2, the norms being taken inC". Thus we have

t afa^IJLl-ZjZj-1^ t hUl-z^y. (12)By Pick's theorem, (12) implies that there exists feH with H/H^ ^ 1 a n d / ^ ) = a,for ally, and so ae^(z). Thus 2> c Q)(z).On the other hand, fix a in Q)(z). Then, for each e > 0, there exists a function/e,4(A), which we may take to be a polynomial, with \\f\\m ^ 1 +e and^z^) = a} forall j by the Claim before Lemma 1. Then

./TO =and so by von Neumann's inequality, | |./(^)|| ^ 1+e. Hence ||7^|| < 1+e for alle > 0, implying that || 7^|| ^ 1, and so a e Si. Therefore 2>{z) a S>, and we have shownthat Q)(z) = 2.

2. Proof of Theorem 1In the proof of Theorem 1, we shall use Cole's representation theorem, [2,Theorem 7, p. 272] which states that the quotient of a uniform algebra by a closedideal is isometrically isomorphic to an algebra of operators on a Hilbert space.Furthermore, every operator in the algebra achieves its norm on some unit vector.Proof of Theorem 1. By hypothesis K is a hyperconvex subset of C n, and henceis the closed unit ball of a Banach algebra 3& which is algebraically isomorphic to C nand which satisfies the following condition: whenever x',x",...,x{k) are in the closedunit ball of ^ , then P(x',x",...,x lk)) is also in the closed unit ball of 38 provided thatP is a polynomial in k variables with ||P||A* < 1.


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A CHARACTERIZATION OF PICK BODIES 3 2 1Proposition 5 in [2, 50] implies that J 1 is isometrically isomorphic tothe quotient A/I of a uniform algebra A by a closed ideal /. Also there exist

Mx,...,MnzMA with / = {feA\j{M^ = 0, j = 1,...,}. By Cole's representationtheorem stated above, there exists a further isometric isomorphism of A/1 on analgebra of operators, d, on a Hilbert space, Jf . Combining these two maps,we have an isometric isomorphism x'M -> s/ and

for all Test, there exists e3? with ||0|| = 1 and \\T\\ = || r| |. (13)Let |/J e A/1 for 7 = 1,..., be such that /, (M t) = (z), and so the closed unit ball of 38 is Q)(z). Thus K = Q){z) andTheorem 1 is proved.3. Operator algebras

In this section we give an alternative approach to some of the questions consideredin the previous sections of this paper. These results are independent of those obtainedearlier and are somewhat more general. Theorem 2 implies Theorem 1, andother results generalize Theorem 1 in various ways. The price to be paid is thatthese calculations are less explicit and elementary than those used in the proof ofTheorem 1.The main objects of interest here are finite dimensional commutative Banachalgebras, principally operator algebras on Hilbert space (which are not assumed to beself-adjoint). When $t is any semisimple Banach algebra of dimension n, the Gelfandtheory identifies the closed unit ball of stf with a subset Q)^ of Cn. Specifically, we put

% = {{x{M1),...,x{M n))\xesf,\\x\\ ^ 1},where the maximal ideal space of $0 consists of the n distinct po ints M l s . . . , M n. Sincethe Gelfand representation is essentially the map

Cn with O(x) = (x(M x),..., x(M n)) ,we see that O is an isomorphism when C n is viewed as an algebra undercoordinatewise operations. Furthermore, under this map, the closed unit ball of s#corresponds to ^ .The hyperconvex set Si, associated with the Q-algebra s$ = All in theIntroduction, is precisely the set S>^ defined above. Moreover, as discussed in theproof of Theorem 1, every hyperconvex set arises in this way. Thus, from the Banachalgebra po int of view, Theorem 1 can be interpreted as a statement about the unit ballof a finite dimensional semisimple Q-algebra.We now turn our a ttention to a broader class of Banach algebras: finitedimensional commutative algebras of operators on Hilbert space. Since we know thatevery Q-algebra is in fact an operator algebra, results about operator algebrasautomatically apply to Q-algebras. And consequently, general results about the unitball of an operator algebra provide information about hyperconvex sets.

11 JLM 48


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3 2 2 B. COLE, K. LEWIS AND J. WERMERFor the remainder of this section, we let A denote the disk algebra A(A) and letn be a positive integer. All Banach algebras considered are assumed to contain anidentity.DEFINITION. A Pick algebra is a finite dimensional Banach algebra J / which isisometrically isomorphic to A/J where / is a closed ideal in A with hull(/) int(A).Furthermore, an element xesf is said to be a Pick generator if

{p(x)\p is a polynomial, \\p\\A ^ 1}constitutes a dense subset of the closed unit ball of $0.

Note that J in the above definition is of the form bA where b is a finite Blaschkeproduct and that the Pick algebra A/J has the Pick generator [idj where idA is theidentity map on A (the complex coordinate function). Moreover, s/ = A/J issemisimple if and only if b has simple zeros, and this is exactly the situationconsidered in the Introduction; so we see that Pick bodies arise as S>^ where si asemisimple Pick algebra.

We shall prove a slight modification of a theorem due to D. Sarason which givesa representation of Pick algebras as singly generated algebras of operators on aHilbert space. Sarason's theorem uses if00 in place of A and an arbitrary innerfunction in place of b. Our proof will also work in this situation; however, we needthe disk algebra version in what follows.We review some basic facts concerning Hardy spaces, denoted by Hp. Let mdenote Lebesgue measure on the unit circle. Let S be the shift operator ofmultiplication by z on H2: Sf{z) = zj{z) for feH2. A function beH2 is called inner if\b\ = 1 a.e. [m] and a function geH2 is called outer if g is a cyclic vector for S, thatis, the set of all pg, p a polynomial, is dense in H2. A theorem of Beurling states thatevery closed subspace of H2 invariant under S has the form bH2, where b in an innerfunction, and every fu nc tion /ei/ 2 can be factored a s / = bg with b inner and g outer.Also, for every heH1, there exists keH2 with \h\ = \k\2 a.e. [m].Following Sarason [8], wefix an inner function b, let Jfb = H2Q bH 2, denote byPjrb the orthogonal projection from H2 to J^, and put Tb = PjCbS\:t-b, where S ismultiplication by z on H2. The orthogonal projection from L2 to H2 is denoted by P.ForfeA, define A S) on H2 by AS)g =fg for geH2 and define b(p). This shows that O6 is multiplicative on polynomials, and so bynorm continuity of multiplication,


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A CHARACTERIZATION OF PICK BODIES 3 2 3PROPOSITION 1 (Sarason). Let b be a finite Blaschke product and letfeA. Then= ||/TO || In particular, the algebra st generated by Tb is a Pick algebraand has Pick generator Tb.Proof. By the Hahn-Banach Theorem, there exists a complex measure ji with\\f+bA\\ = jfdfi, ||//|| < 1, and jbgd/i = 0 for all geA. We shall show that thereexist x,yeki,, \\x\\, \\y\\ ^ 1, with ffdfi = ffxydm. This will prove the theoremsince we shall have

\\f+bA\\ = jfdM = (fx,y) = (P Kbfx,y) =


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3 2 4 B. COLE, K. LEWIS AND J. WERMERThe next result provides the key link between operator algebras and quotients of

the disk algebra.PROPOSITION 2. Let JF be n-dimensional Hilbert space, and let s&bea comm utative

subalgebra of @{tf). Let Test so that \\T\\ = 1, r a n k ( / - T * r ) = 1, and T hasno eigenvalue of modulus 1. Then, $0 is an n-dimensional Pick algebra with Pickgenerator T.Proof Since r a n k ( / - T*T) = r a n k ( 7 - TT*) and since the spectral radius of T,

and hence T*, is less than 1, the hypotheses of Lemma 6 are satisfied by T*. So T*is unitarily equivalent to S* restricted to an invariant subspace which, by Beurling'stheorem, has the form Jfb for some inner function b. Consequently, $0 is unitarilyequivalent to a commutative subalgebra si of 88{Xb) containing Tb, and, under thisequivalence, Tb corresponds to T. Also, since tf has finite dimension, b is a finiteBlaschke product.

F o r the vector = Px 1, {, Tb $,..., Tb~x0} is a basis for Xb, and thereforeR s j / w i t h R = 0 implies tha t R = 0. It follows that dim (j/ ) = n and j / i s the a lgebragenerated by Tb. So, by Proposition 1, s/ is a Pick algebra with Pick generator T.

A Banach algebra s& satisfies the von Neumann Inequality of order 1 if, wheneverxestf with ||x|| < 1, ||/?(x)|| < ma x j / j l for every polynomial/?. Von Neumann showedthat this condition is satisfied if $$ is a subalgebra of &(&) for a Hilbert space 3f?\see [4] for more details. The next result is used in this pape r only when sdx is such analgebra.

LEMMA 7. Let s^ be a commutative Banach algebra satisfying the von NeumannInequality of order 1, and let s^be a Pick algebra. Let Q> be a norm 1 homomorphismfrom s% into #, and let xes^ with \\x\\ ^ 1 so that O(x) is a Pick generator for s^.Then, induces an isometric isomorphism between j ^ /k e rC O) and s^. In particular, ifO is one-to-one, then O establishes an isometric isomorphism between s^ an

Proof. Let (*) = y. For every polynomial p, maps p(x) to p(y). Restrictingattention to polynomials in the unit ball of the disk algebra, the hypotheses guaranteethat O maps the unit ball of s^, onto a dense subset of the unit ball of s%. Theassertion now follows.LEMMA 8. Let n ^ 2. If T is an operator on a Hilbert space #? with dim(jf) ^ n,II Tk\\ = \ for 0 < k < n, and T has no eigenvalue of modulus 1, then I T*T has rank1 and d i m ( ^ f ) = n.Proof. Choose a unit vector

,..., Tn~x 0 is a basis for 2tf. Repeating the argument of the first paragraphof the proof of Lemma 3, we conclude that rank(7 T*T) = 1.

Observe that, if T belongs to a finite-dimensional subalgebra of @}(3ff), then thespectrum of T coincides with the set of eigenvalues. For such a T with || T\\ = 1, Thasno eigenvalue of m odu lus 1 if and only if its spectral radiu s is less tha n 1 which occursif and only if ||!T*|| < 1 for some k.


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A CHARACTERIZATION OF PICK BODIES 3 2 5THEOREM 2. Let n^2, and let si be a commutative Banach algebra of operatorson a Hilbert space tf with dim(j^) < . If there exists Tesi with \\P\\ = 1 for0 #(JT) with WniT^y/W = WT^W = 1. Define

by T(R) = R n(R) for Resi. Then T is an isometric algebra homomorphism, x{T)satisfies all of the original hypotheses, and \\x{T)n~\Q ^)| | = ||0 y/\\ = 1.The subspace X = {R(j) \ R e jaf} is invariant under d and has dimension at most n.The algebra J / = st\x and the operator f = T\^ satisfy the hypotheses of Lemma 8and Proposition 2; so si is an n-dimensional Pick algebra with Pick generator f.Since the restriction homomorphism Q>:si -^ si satisfies the conditions of Lemma 7with T in the role of x, is an isometric isomorphism, and the proof is complete.When si is semisimple, the above theorem shows that 2^ is a Pick body. So, weare led to a strengthened form of Theorem 1. Note that z( ^ z if/ # j:is a consequenceof the theorem, not an assum ption. This follows since, by the conclusion of Theorem2, T must have n distinct eigenvalues.THEOREM V. Let Kbea hyperconvex subset ofC n. Assume that K contains a pointz = (z15... ,zn) with \zt\ < 1 for each j and with zn~xedK. Then K is a Pick body.We now identify all Pick generators for a Pick algebra. Let Aut(A) denote the setof conformal automorphisms of the unit disk.LEMMA 9. Let si be a Pick algebra A/Jfor which dim(j^) ^ 2. Then, x is a Pickgenerator for si if and only if x = [y/]for ^eAut(A). In particular, if xx, x2 are both

Pick generators for si, then x2 = y/(x^) for someProof Let x = [a] be a Pick generator for A/J; so ||[fl]|| < 1. Observe that since2, the coset x cannot contain a constant function. We write / = bA for afinite Blaschke product b.Let y = [y/0] where y/0 = idA, and no te that ||_y|| ^ 1. Select polynomials gn with| |g j | ^ 1 so that gn(x)-*y in A/J, and select fne[a] with ||/J| < 1 + 1/n. Using anormal families argument, we obtain /, g, h, keH^ so that || /| | < 1, ||g|| < 1,f=a + bh, and gof= y/0 + bk. Clearly / cannot be constant; so the compositionT = g o / i s well defined as a holomorphic map on int(A) with values in A. Since T leavesfixed the zeros of b and has derivative 1 at each non-simple zero, and since the sumof the multiplicities of the zeros of b is at least 2, Schwarz's Lemma implies that

T = y/ 0. Hence, / e Aut(A) and x = [f], as desired.Since, for ^e A ut(A ), {po y/\p is a polynomial, \\p\\ < 1} is a dense subset of theclosed unit ball of A, the converse is clear.To prove the last statement of the lemma, suppose that xlfx2 are both Pickgenerators. Then, xx = [y/^ and x2 = [^2] for y/x, ^ 2eAut(A). So, x2 = ^(xx) fory/ = y/ 2o y/^1 e Aut(A).


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3 2 6 B. COLE, K. LEWIS AND J. WERMERThe next result extends Lemma 8 by replacing Tn~l with a product Tx'~ Tn_x ofcommuting operators .LEMMA 10. Let n^2, and let 2tf be a Hilbert space with d im(^ f ) < n. Let

Tx,...,Tn_x belong to a commutative subalgebra of &(3f) so that ||7J|| = 1 andTt has no eigenvalue of modulus 1 for i= 1 ,. .. , w 1, while \\TX - - Tn_x\\ = 1. Then,= n an drank(/= T* 1$ = 1 for i = 1 , . . . , n - 1 .Proof Selective xeJif so that ||x|| = ||>>|| = 1 for y = Tx Tn_xx. Let

Er = {T ti - T i t x \ 0 ^ k , l ^ i x < < ik ^ r)(interpret Eo to mean {*}), and let Vr = span(2sr) for 0 r ^ n 1. For eE r,y = Rfor some R of the form T} 7J ; so 1 = \\y\\ < |||| ^ 1 and hen ce ||f || = 1 for all

N ow let l ^ r ^ n - l . For eEr_x,Z' = Tr(Z)eEr. Since K\\ = Kl = h wededuce that T?Tr = for all eE r_ lt and therefore / - T * Tr = 0on Vr_x. It is clearthat Vr_x e V r and Tr(Vr_x) V r. If V r_ x = V r, then V r is ^-invariant. Consequently,is unitary on Vr_x, implying that Tr has an eigenvalue of modulus 1, contrary toassumption. So 1 = d i m ( ^ ) < < &\m(Vn_x) ^ d i m ( J f ) ^ n, and in particular,= n and dim(Fn_2) = n-1. Since / - r *^ 7 ; ^ = 0 on Vn_2,

and equahty holds, as claimed, because Tn_x is not unitary on Jf.After re-labeling the operators, the above argument can be applied to any T { inplace of Tn_x.The preceding lemmas lead to a generalization of Theorem 2.THEOREM 3. Let Jf be a Hilbert space, and let n^2. Suppose that $fis a commutative subalgebra of fflffl) with dim(j^) < n. Let Tx,...,Tn_x belong tos# so that \\Tf\\ = 1 and Tt has no eigenvalue of modulus 1 for i= \,...,n 1, while\\Tx-Tn_x\\ = \

(i) s/ is an n-dimensional Pick algebra,(ii) each Tt is a Pick generator for s/,(hi) Tx = i//i(Tx)for 2 ^ i ^ n-1 where y/te Aut(A).Proof We repea t the proof of Theo rem 2, with the following changes. En largingif necessary, we can assume that there exists 0 e 3f? so that

Then, Lemma 10 is used in place of Lemma 8. Assertion (iii) is a consequence of thelast sentence of Lemma 9.

As discussed at the beginning of this section, operator-theoretic results can beapplied to finite-dimensional, semisimple Q-algebras to yield information abouthyperconvex sets, and those hyperconvex sets associated with semisimple Pickalgebras are just the Pick bodies. So, we have the following extension of Theorem 1.


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A CHARACTERIZATION OF PICK BODIES 3 2 7COROLLARY 1. Let K be a hyperconvex subset of C n, and let xa),x{2) ,...,x{n-X)edKbesuch that

(i) xwx{2) JC(B~1) e dK (multiplying coordinatewise),(ii) \xf\ < 1 for 1 ^ i ^ n-\ and 1 ^j < n.Then, K is a P ick body such that

(iii) K= (x) = = 9(x (n~X)),(iv) for 2 < / < n 1, there exists ^ (eAut(A) so that x{i) = ^(x (1 )) (applying \f/tcoordinatewise).4. Additional comments

1. The converse to Theorem 2 is true also. In particu lar, if s/ is an n-dimensionalPick algebra with Pick generator T, then 1 = || T\\ = || T"**"11| > | | r n || and st isisometrically isomorphic to a subalgebra of ^ ( J f ) where ?f is an w-dimensionalHilbert space. This can be seen using Proposition 1 and Rouche's theorem as in theproof of Lemma 1.2. Let A be the disk algebra, and let J = bA where b is a finite Blaschke product.With an argument similar to that in the proof to Proposition 1, it can be shown thateach coset in A/J contains an element of minimum norm. Specifically, ifxeA/J, thenthere exists a unique gex with ||g|| = ||JC|| ; moreover, g is a constant multiple of afinite Blaschke product of order less than the order of b.3. A recurring assumption used in Section 3 is that T is a norm 1 operator withno eigenvalue of modulus 1, which clearly corresponds to the requirement in Theorem1 that zedK with \z}\ < 1 for e a c h / We now analyse this assumption, showing howour main results take on a simpler form after applying an appropriate reduction.Proofs are omitted.In view of the comments before Theorem 2, for each operator T in ourinvestigation, we can deal with its spectrum, a(T), instead of its eigenvalues. We beginwith a result about Banach algebras.LEMMA. Let $4 be a finite dimensional Banach algebra with identity denoted by 1.There exists x erf so that \\x\\ = 1, Xea(x) with \X\ = 1, and x # X if and only if thereexists yerf so that y2 = y, \\y\\ = 1, andy ^ 1.The proof of this lemma is a straightforward exercise infinite-dimensional inearalgebra. For example, y is calculated from x by the formula y = limJfc_00(|(l +Xx))k.Consequently, we call s4 decomposable if there exists yerf with y2 y, || j>|| = 1,and y # 1; otherwise, s$ is called non-decomposable. If s/ is a finite-dimensionalsubalgebra of ^(3f) where Jf is a Hilbert space, then s$ is decomposable exactlywhen st contains a non-trivial orthogonal projection. Theorems 2 and 3 can be re-stated for non-decomposable algebras in a slightly simpler form. For instance, wehave the following theorem.THEOREM 2''. Let n ^ 2, and let s/ be a non-decomposable, commutative Banachalgebra of operators on a Hilbert space #? with dim(j^) ^ n. If there exists Test with|| T}\\ = I/or 0


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3 28 A CHARACTERIZATION OF PICK BODIESUsing minim al orthog ona l projections, we obtain the following decom position for

operator algebras. A similar result holds for Banach algebras satisfying the vonNeumann Inequality of order 1.

PROPOSITION. Let s& be a finite dimensional, commutative subalgebra ofwhere Jff is a Hilbert space. Then, $4 is isometrically isomorphic to $$ xx x s / r ,where each st{ is a non-decomposable subalgebra of&(3%) and 3% is a Hilbert space.The product algebra is normed so that | | (7 i , . . . , Tr)\\ = m ax1;S(!gr ||7J|| for Ttejtft.When s& is isometrically isomorphic to the ^-algebra AIJ where A is a uniform

algebra and J is a closed ideal in A, this decomposition has the followinginterpretation : the minimal norm 1 idempotents of A/J (and hence the algebras s/ tin the above proposition) are in one-to-one correspondence with those Gleason partsof A that meet hu\\(J). See [5] for information on Gleason parts.As in Section 3, it is easy to translate these results into statements abouthyperconvex sets. For a hyperconvex set K C n , we say that it is decomposable if,after re-ordering the coordinates, K = K x x K2 where K x s C"1, K2 c C"a, n = nx + n2.Otherwise, K is said to be non-decomposable.

PROPOSITION. Let K be a hyperconvex subset of C n. Then,(i) K is non-decompo sable if and only if zsK with max, |zy| = 1 implies thatz = (A,...,X) for some XeC with \X\ = 1,

(ii) after re-ordering the coordinates, K= K x x x K r, where each K { is a non-decomposable hyperconvex subset of Cn< and n = nx + + nr.Thus we obtain another modified form of Theorem 1.THEOREM 1". Let K be a non-decomposable, hyperconvex subset ofC n. Let zeKso that

(i) z ? (A,...,A) for each AeC with \X\ = 1,(ii) z* -xedK.

Then K is a Pick body.References

1. W. ARVESON, An invitation to C*-algebras (Springer-Verlag, New York, 1976).2. F. F. BONSALL and J. D U N C A N , Complete normed algebras (Springer-Verlag, Berlin, Heidelberg, NewYork, 1973).3. L. DE BRANGES and J. ROVNYAK, Perturbation theory and its applications in quantum mechanics (ed.Calvin H. Wilcox; Wiley, New York, 1966).4. B. COLE, K. LEWIS and J. WERMER, 'Pick conditions on a uniform algebra and von Neumanninequalities', J. Func. Anal, 107 (1992) 235-254.5. T. W. GAMELIN, Uniform algebras (Prentice Hall, Englewood Cliffs, 1969).6. K. LEWIS and J. WERMER, 'On the theorems of Pick and von Neumann', Function Spaces (ed. K.Jarosz; Marcel Dekker, 1992).7. G. PICK ' Uber die Beschrankungen analytischer Funktionen, welche durch vorgegebeneFuntionswerte bewirkt werden', Math. Ann. 11 (1916) 7-23.8. D . SARASON, 'Generalized interpolation in /f00', Trans. Amer. Math. Soc. 127 (1967) 179-203.

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A Characterization of Pick Bodies

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