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Semicrossed Products of the Disk Algebra: Contractive Representations and Maximal Ideals

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Semicrossed Products of the Disk Algebra: Contractive Representations and Maximal Ideals pacific journal of mathematics Vol. 185, No. 1, 1998 SEMICROSSED PRODUCTS OF THE DISK ALGEBRA: CONTRACTIVE REPRESENTATIONS AND MAXIMAL IDEALS Dale R. Buske and Justin R. Peters Given the disk algebra A(D) and an automorphism α, there + is associated a non-self-adjoint operator algebra Z ×α A(D) called the semicrossed product of A(D) with α. We consider those algebras where the automorphism arises via composi- tion with parabolic, hyperbolic, and elliptic conformal maps ϕ of D onto itself. To characterize the contractive representa- + tions of Z ×α A(D), a noncommutative dilation result is ob- tained. The result states that given a pair of contractions S, T on some Hilbert space H which satisfy TS = Sϕ(T), there exist unitaries U, V on some Hilbert space K⊃Hwhich dilate S and T respectively and satisfy VU =Uϕ(V). It is then shown that there is a one-to-one correspondence between the contractive + (and completely contractive) representations of Z ×α A(D) on a Hilbert space H and pairs of contractions S and T on H satis- fying TS = Sϕ(T). The characters, maximal ideals, and strong + radical of Z ×α A(D) are then computed. In the last section, we compare the strong radical to the Jacobson radical. I. Introduction. A semicrossed product of the disk algebra is an operator algebra associated to the pair (A(D),α), where A(D) is the disk algebra and α an automor- phism of A(D). Any such α has the form α(f)=f◦ϕ(f∈A(D)) for a linear fractional transformation ϕ. It is well-known there is a one-to-one correspondence between contractions (i.e., bounded linear operators T on some Hilbert space with kT k≤1) and contractive representations of A(D). Here, analogously, there is a one-to-one correspondence between pairs S, T of contractions satisfying the relation TS = Sϕ(T) and contractive repre- + sentations of the semicrossed product, denoted Z ×α A(D), associated to (A(D),α). This is meaningful since linear fractional transformations map contractions to contractions (cf. [Sz-NF]). The question of whether con- + tractive representations of Z ×α A(D) can be dilated to representations of ∗ the C -crossed product Z ×α C(T) is equivalent to the following: given a pair of contractions S, T on some Hilbert space H satisfying TS = Sϕ(T)do 97 98 DALE R. BUSKE AND JUSTIN R. PETERS there exist unitaries U, V on some Hilbert space K⊃Hwhich are dilations of S, T respectively, and satisfy VU = Uϕ(V)? This question, which is of interest in its own right, has an affirmative solution [Theorem II.4]. Fur- thermore, it marks the starting point in our study of semicrossed products + of the disk algebra, by giving faithful representations of Z ×α A(D). Section V deals with the characterization of the maximal ideal space. The character space, or equivalently the space of maximal ideals of codimension one, was easily obtained from the representation theory [Corollary III.11]. It turns out there are no maximal ideals of codimension greater than one – unless the automorphism (i.e., the linear fractional transformation) is elliptic of finite period, say K, in which case the maximal ideals have codimension either 1 or K2. Finally, these results, together with [HPW] are used to com- pare the strong radical with the Jacobson radical: the two radicals coincide except when α is elliptic and nonperiodic (that is, an irrational rotation). (Theorem VI.1.) II. Dilating Noncommuting Contractions. It is well-known that each contraction T on a Hilbert space H can be dilated to a unitary U on a Hilbert space K⊆H[Sz-NF]. That is, n n for n ≥ 1,T h = PHU h ∀ h ∈H. This result then yields the fact that the contractive representations of the disk algebra A(D)onHare in a one-to-one correspondence with contractions on H [DP], [Sz-NF]. Moreover, it shows that each contractive representation is completely con- tractive [DP]. Andˆo then generalized this result by showing that every commuting pair of contractions S and T on H have a unitary dilation on some K⊇H[A]. That is, there exist unitaries U and V on K such that m n m n ∀ m ≥ 1,n≥1,S Th=PHU Vh. Hence the contractive represen- tations of the bidisk algebra A(D2)onHare in one-to-one correspondence with commuting pairs of contractions on H. Furthermore, each representa- tion of A(D2) is completely contractive [DP]. Recently, Sebesty´en showed that every anti-commuting pair of contractions have such a dilation [S]. In this section we show that when ϕ is a conformal automorphism of D and S and T are contractions on H satisfying TS = Sϕ(T) then a unitary dilation exists. This result is then used to characterize the contractive representa- tions of a semicrossed product. Proofs in this section closely resemble those in [S]. Lemma II.1 is directly lifted from [S]. Lemma II.1. Let K and K0 be Hilbert spaces, H⊂Kand H0 ⊂K0 be subspaces and X : H→K0 and X0 : H0 →Kbe given bounded linear transformations. Then, there exists an operator Y : K→K0 extending X so that Y ∗ extends X0 if and only if hXh,h0i = hh, X0h0i∀h∈H,h0 ∈H0. SEMICROSSED PRODUCTS OF THE DISK ALGEBRA 99 Moreover, kY k≤max{kXk, kX0k}. Suppose now that S and T are contractions on H which satisfy TS = Sϕ(T) for some linear fractional transformation ϕ of D. Note that ϕ(T )isa well-defined contraction by the functional calculus found in [Sz-NF]. Let U be the minimal isometric dilation of S acting on a Hilbert space K containing W W ∗ ∗ ∞ n k H. Then, U extends S , where K = n=0 Kn and Kn = k=0 U (H). Lemma II.2. Let S and T be contractions on H such that TS = Sϕ(T). If U is the minimal isometric dilation of S acting on K, then there exists Tϕ ∗ ∗ an operator on K such that Tϕ extends T , kTϕk≤1, and TϕU = Uϕ(Tϕ). Proof. The proof is similar to [S]. At the n-th step of induction, Lemma II.1 is applied to the maps Uϕ(T )U∗| : H ≡ U(K ) →K and n−1 Hn−1 n−1 n−1 n ∗ Tn−1 : Kn−1 →Kn and kTnk≤1 since kϕ(Tn−1)k≤1. Supposing that ϕ = µϕ where |µ| = 1 and ϕ (z)= z−a, the conditions of Lemma II.1 a a 1−az ∗ ∗ ∗ are satisfied since hx, [µϕa(Tn)] yi = hx, µϕa(Tn )yi = hx, µϕa(Tn−1)yi = ∗ hx, [µϕa(Tn−1)] yi for x ∈Hn−1 and y ∈Kn−1 [Sz-NF, I.4]. Lemma II.3. Let S and T be contractions on H such that TS = Sϕ(T).If Uis the minimal unitary dilation of S acting on a Hilbert space K, then there exists Tϕ on K which is a dilation of T with kTϕk≤1and TϕU = Uϕ(Tϕ). Proof. Let U+ be the minimal isometric dilation of S acting on K+; H⊆K+ ⊆ K. By Lemma II.2, there exists a dilation T of T to K with kT k≤1 and ϕ+ + ϕ+ −1 Tϕ+U+ = U+ϕ(Tϕ+). By considering a sequence of polynomials pn → ϕ , −1 it follows from Tϕ+U+ = U+ϕ(Tϕ+) that ϕ (Tϕ+)U+ = U+Tϕ+. Taking ∗ ∗ ∗ −1 ∗ ∗ adjoints yields Tϕ+U+ = U+ϕ (Tϕ+) . Since U is the unique minimal ∗ isometric dilation of U+ it follows by Lemma II.2 and [Sz-NF, I.4] that ∗ ∗ there exists an operator Tϕ on K such that Tϕ extends Tϕ+, kTϕk≤1, and ∗ ∗ ∗ −1 ∗ TϕU = U ϕ (Tϕ) . Reasoning as above, TϕU = Uϕ(Tϕ). Theorem II.4. Let S and T be contractions on H such that TS = Sϕ(T). Then there exists a pair of unitaries U and V such that VU =Uϕ(V) and m n m n S T = PHU V |H for every m, n ∈ N. Proof. As in [S], let U0 be the minimal unitary dilation of S and T0 a con- tractive dilation of T with T0U0 = U0ϕ(T0). Then let V be the minimal unitary dilation of T0 and proceed to extend U0 to a unitary such that VU = Uϕ(V). The proof follows [S] after it is shown that U defined on W∞ n n −1 n K = V (K0)byU(V k0)=ϕ (V)U0k0 is isometric. However, since −∞ −1 V is the minimal unitary dilation of T0, it follows that ϕ (V ) is the mini- −1 m n mal unitary dilation of ϕ (T0)[Sz-NF, I.4.3] and so hU(V h0),U(V k0)i= 100 DALE R. BUSKE AND JUSTIN R. PETERS −1 m −1 n −1 m −1 n hϕ (V ) U0h0,ϕ (V) U0k0i = hϕ (T0) U0h0,ϕ (T0) U0k0i = m n m n hU0T0 h0,U0T0 k0i=hV h0,V k0i. III. The Semicrossed Product. Example III.1. Let D be an operator algebra and α an automorphism of D. Let (ρ, K) be a contractive representation of D and let H = H 2(K), the space L∞ of square summable elements in 0 K. Define a contractive representation 2 π of D on H by π(f )(ξ0,ξ1,ξ2,...)=(ρ(f)ξ0,ρ(α(f))ξ1,ρ(α (f))ξ2,...). If U+ is the unilateral shift on H, then U+π(α(f)) = π(f)U+∀ f ∈D.
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