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The Cone and Cylinder Algebra THE CONE AND CYLINDER ALGEBRA RAYMOND MORTINI1∗ AND RUDOLF RUPP2 Abstract. In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder alge- bras. These include a determination of the maximal ideals, the solution of the B´ezoutequation and a computation of the stable ranks by elementary methods. Let D = fz 2 C : jzj < 1g be the open unit disk in the complex plane C and D its closure. As usual, C(D; C) denotes the space of continuous, complex-valued functions on D and A(D) the disk-algebra, that is the algebra of all functions in C(D; C) which are holomorphic in D. By the Stone-Weierstrass Theorem, C(D; C) = [ z; z ]alg and A(D) = [ z]alg, the uniformly closed subalgebras gener- ated by z and z, respectively z on D. In this expositional note we study the uniformly closed subalgebra Aco = [ z; jzj ]alg ⊆ C(D; C) of C(D; C) which is generated by z and jzj as well as the algebra Cyl(D) = f 2 C D × [0; 1]; C : f(·; t) 2 A(D) for all t 2 [0; 1] : We will dub the algebra Aco the cone algebra and the algebra Cyl(D) the cylinder algebra. The reason for chosing these names will become clear later in Theorem 1.3 and Proposition 3.1. The cone-algebra appeared first in [15]; the cylinder algebra was around already at the beginning of the development of Gelfand's theory. Their role was mostly reduced to the category \Examples" to illustrate the general Gelfand theory; no detailed proofs appeared though. We think that these algebras deserve a thorough analysis of their interesting algebraic properties and that is the aim of this note. The main new results will be the determination of the stable ranks for the cone-algebra, the absence of the corona property (Cn) when (and only when) n ≥ 2, explicit examples of peak functions and a simple approach to the calculation of the Bass stable rank for the cylinder algebra. arXiv:1511.04567v1 [math.RA] 14 Nov 2015 This paper forms part of an ongoing textbook project of the authors on stable ranks of function algebras, due to be finished only in a couple of years from now (now = 2015). Therefore we decided to make this chapter already available to the mathematical community (mainly for readers of this special issue of Annals of Functional Analysis (AFA) dedicated to Professor Anthony To-Ming Lau and for master students interested in function theory and function algebras). Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 46J10, Secondary 46J15; 46J20; 30H50. Key words and phrases. cone algebra; cylinder algebra; B´ezoutequation; maximal ideals; stable ranks. 1 2 R. MORTINI, R. RUPP 1. The cone algebra We begin with some examples of non-trivial elements in Aco. The general situation will be dealt with in Theorem 1.4. Example 1.1. q p −1 (1) For every δ > 0, q > 0 and p 2 R, jzj + δ 2 (Aco) . 8 z <p if 0 < jzj ≤ 1 (2) f(z) := jzj belongs to Aco. :0 if z = 0 Proof. (1) Since by Weierstrass' Theorem (xq + δ)±p is a uniform limit of poly- q p −1 nomials in x = jzj 2 [0; 1], we have that (jzj + δ) 2 (Aco) . (2) First we note that f is continuous on D. For δ > 0, let z fδ(z) := : pjzj + δ Then, by (1), fδ 2 Aco. Since jjf − fδjj1 ≤ δ=(1 + δ) (see below), we conclude that f is a uniform limit of functions in Aco; hence f itself belongs to Aco. Now we prove our estimate: z z δ pjzj p p − p = jzj p p = p δ =: h jzj : jzj jzj + δ jzj ( jzj + δ) jzj + δ δx 0 δ2 Since the derivative of h(x) = x+δ , namely h (x) = (x+δ)2 > 0 on [0; 1], we have h(x) ≤ h(1) = δ=(δ + 1). Next, we derive several Banach algebraic properties for Aco. Definition 1.2. Let X be a topological space and A a subalgebra of C(X; C). (1) The set of invertible n-tuples in A is denoted by Un(A); that is n n n X Un(A) = f(f1; : : : ; fn) 2 A : 9(r1; : : : ; rn) 2 A : rjfj = 1g: j=1 (2) A is said to be inverse-closed (on X), if f 2 A and jfj ≥ δ > 0 on X imply that f is invertible in A. (3) A is said to satisfy condition (Cn) 1 if n n n \ o Un(A) = (f1; : : : ; fn) 2 A : ZX (fj) = ; ; j=1 where ZX (f) = fx 2 X : f(x) = 0g denotes the zero set of f. (4) If A is a commutative unital Banach algebra over C, then its spectrum (or set of multiplicative linear functionals on A endowed with the weak- ∗-topology) is denoted by M(A). Moreover, Ab is the set of Gelfand trans- forms fb of elements in A. 1 Here (Cn) stands for \Corona-condition for n-tuples". 3 As usual, for a compact set X in C, P (X) is the uniform closure in C(X; C) of the set C[z] of polynomials. Theorem 1.3. Let Aco = [ z; jzj ]alg ⊆ C(D; C) be the cone algebra. Then (1) A(D) ⊆ Aco ⊆ C(D; C). (2) AcojT = A(D)jT = P (T). (3) For every 0 < r < 1, AcojrT = P (rT). (4) M(Aco) is homeomorphic to the cone 3 p K := f(x; y; t) 2 R : x2 + y2 ≤ t; 0 ≤ t ≤ 1g; a three dimensional set. Figure 1. The cone as spectrum (5) For f 2 C(D; C) and 0 < r < 1, let fr be the dilation of f given by fr(z) = f(rz). Moreover, let Z 2π 2 2 1 R − jzj it PR[f](z) := it 2 f(Re ) dt (1.1) 2π 0 jRe − zj be the Poisson integral and P [f] := P1[f]. Then M(Aco) = δ0 [ r;a : 0 < r ≤ 1; a 2 D with jaj ≤ r ; where δ0 is point evaluation at 0, r;a = δa if jaj = r and ( Aco ! C r;a : f 7! P [(fr)jT](a=r) = Pr[fjrT](a); if jaj < r. (6) Abco is the uniform closure of the polynomials p(z; r) on the cone K = f(z; t) 2 D × [0; 1] : jzj ≤ tg 4 R. MORTINI, R. RUPP and coincides with A := f 2 C(K; C): f(·; r) 2 A(rD) 8 r 2 ]0; 1] : (7) Aco is inverse-closed; that is, it has property (C1), but Aco does not have property (Cn) for any n ≥ 2. (8) The Shilov boundary of Aco coincides with the outer surface S := f(z; r) 2 C × R : 0 ≤ r ≤ 1; jzj = rg of the cone K (this is the boundary of K without the upper disk f(z; 1) 2 2 C × R : jzj < 1g). The Bear-Shilov boundary is the closed unit disk . Proof. (1) This is clear since the polynomials in z are dense in A(D). (2) Let f 2 Aco. If we choose a sequence of polynomials pn 2 C[z; w] such that pn(z; jzj) converges uniformly to f on D, then pn(z; 1) converges uniformly on T to fjT. Hence, fjT 2 P (T) = A(D)jT. Together with (1), we conclude that A(D)jT = AcojT. (3) Fix 0 < r < 1 and let f 2 Aco. Then, for jzj = r, f(z) = lim pn(z; r) 2 P (rT). Hence AcojrT ⊆ P (rT). Conversely, given h 2 P (rT), we let H := Pr[h] be the Poisson extension of h to rD. Then H 2 A(rD). Now we define the function f by 8 <H(z) if jzj ≤ r f(z) = z H r if r ≤ jzj ≤ 1: : jzj Note that f is an extension of H to the unit disk that stays constant on every ray seiθ beginning at the radius r. Then f is continuous on D. Now f(z) can be written as f(z) = Hzg(jzj), where g is defined by (1 if 0 ≤ s ≤ r g(s) = r if r ≤ s ≤ 1: s Then g is continuous on [0; 1]. Next, we uniformly approximate on [0; 1] the function g by a sequence (qn) of polynomials in C[s] and H on fjzj ≤ rg by a sequence of polynomials (pn) 2 C[z]. Let rqn(s) Qn(s) := max jsqn(s)j 0≤s≤1 Then also (Qn) converges uniformly to g on [0; 1], because max0≤s≤1 jsg(s)j = r. What we have gained is that jz Qn(jzj)j ≤ r for every z 2 D. Hence H(z Qn(jzj)) 2 Recall that if X is a compact Hausdorff space and L a point-separating K-linear subspace of C(X; K) with K ⊆ L, then L admits a smallest closed boundary, which we will call the Bear-Shilov boundary (see [2]). The Shilov boundary of a uniform algebra A is the smallest closed boundary of A on its spectrum M(A). 5 is well defined on D and H(z Qn(jzj)) converges uniformly on D to f(z). We claim that pn(zQn(jzj)) converges uniformly on D to f(z), too. In fact, jpn(zQn(jzj)) − f(z)j ≤ jpn(zQn(jzj)) − H(zQn(jzj))j + jH(zQn(jzj)) − f(z)j ≤ max jpn(w) − H(w)j + max jH(ξQn(jξj)) − f(ξ)j: jw|≤r jξ|≤1 Now Qn(jzj) 2 Aco implies zQn(jzj) 2 Aco and so pn(zQn(jzj) 2 Aco.
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