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On Algebras of Harmonic Quaternion Fields in R

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On Algebras of Harmonic Quaternion Fields in R On algebras of harmonic quaternion fields in R3 M.I.Belishev∗and A.F.Vakulenko† Abstract Let A (D) be an algebra of functions continuous in the disk D = {z ∈ C | |z| 6 1} and holomorphic into D. The well-known fact is that the set M of its characters (homomorphisms A (D) → C) is exhausted M ∼ by the Dirac measures {δz0 | z0 ∈ D} and a homeomorphism = D holds. We present a 3d analog of this classical result as follows. Let B = {x ∈ R3 | |x| 6 1}. A quaternion field is a pair p = {α, u} of a function α and vector field u in the ball B. A field p is harmonic if α, u are continuous in B and ∇α = rot u, div u = 0 holds into B. The space Q(B) of such fields is not an algebra w.r.t. the relevant (point- wise quaternion) multiplication. However, it contains the commuta- 2 tive algebras Aω(B) = {p ∈ Q(B) | ∇ωα = 0, ∇ωu = 0} (ω ∈ S ), each Aω(B) being isometrically isomorphic to A (D). This enables one to introduce a set M H of the H-valued linear functionals on Q(B) (H- characters), which are multiplicative on each Aω(B), and prove that M H H ∼ H = {δx0 | x0 ∈ B} = B, where δx0 (p)= p(x0). Key words: 3d quaternion harmonic fields, real uniform Banach algebras, arXiv:1710.00577v3 [math.FA] 11 Oct 2017 characters. MSC: 30F15, 35Qxx, 46Jxx. ∗Saint-Petersburg Department of the Steklov Mathematical Institute, RAS, beli- [email protected]; Saint-Petersburg State University, 7/9 Universitetskaya nab., St. Pe- tersburg, 199034, Russia, [email protected]. Supported by the RFBR grant 17-01- 00529- and Volks-Wagen Foundation. †Saint-Petersburg Department of the Steklov Mathematical Institute, [email protected] 1 0 Introduction • The result, which our paper is devoted to, is announced in [6]1. In a sense, it is a ‘by-product’ of activity in the framework of the algebraic approach to tomography problems on manifolds [2]–[9]. However, we hope that this result is of certain independent interest for the real uniform Banach algebras theory [1, 10]. Namely, we propose a 3d generalization of the well-known theorem on the characters of the disk-algebra of holomorphic functions. Perhaps, the most curious point is that such a generalization does exist although the relevant 3d analog of the disc-algebra is not an algebra. • A subject to be generalized is the following well-known result. Let A (D) be a commutative Banach algebra of functions continuous in the disk D = {z ∈ C | |z| 6 1} and holomorphic into D. The well-known fact (see, e.g., [12]) is that the set M of its characters (homeomorphisms A (D) → C) endowed with the Gelfand topology, is exhausted by the Dirac M ∼ measures {δz0 | z0 ∈ D} and the homeomorphism = D holds. Shortly, a 3d generalization, which is our main result, looks as follows. Let B = {x ∈ R3 | |x| 6 1}. A quaternion field is a pair p = {α,u} of a function α and vector field u in the ball B. Such fields are identified with the H-valued functions and, hence, can be multiplied point-wise as quaternions. We say a field p to be harmonic if α,u are continuous in B and ∇α = rot u, div u = 0 holds into B. A space Q(B) of such fields is not an algebra w.r.t. the above mentioned multiplication. However, it contains the 2 commutative algebras Aω(B) = {p ∈ Q(B) | ∇ωα =0, ∇ωu = 0} (ω ∈ S ), which we call the axial algebras (ω is an axis). Each Aω(B) is isometrically isomorphic to A (D). This enables one to introduce the set M H of the H- valued linear functionals on Q(B)(H-characters), which are multiplicative on A M H H ∼ H all ω(B), and prove that = {δx0 | x0 ∈ B} = B, where δx0 (p)= p(x0). Sections 1,2 contain the definitions, statements of the results and com- ments on them. The basic proofs are placed in 3. • We’d like to thank Dr C.Shonkwiler for helpful remarks and very useful references. 1unfortunately, with some inaccuracies in the formulations 2 1 The 2d classics We begin with the result, which is planned to be generalized. Algebras • Let D = {z ∈ C | |z| 6 1} be the disc on the complex plane. The continuous function algebra CC(D) := {f = ϕ + iψ | ϕ, ψ ∈ CR(D)}; kfk = sup |f| , D kfgk 6 kfkkgk , fg = gf , kf 2k = kfk2 is a Banach commutative uniform algebra. • A function f = ϕ + iψ is holomorphic if the Cauchy-Riemann conditions dψ = ⋆dϕ (δψ = δϕ =0) (1) hold in the inner points of Dom f ⊂ C. Here ϕ and ψ are regarded as the 0-forms, ⋆ is the Hodge operator corresponding to the standard orientation of C, d and δ are the differential and codifferential respectively. The condi- tions in the brackets are fulfilled just by the well-known definitions. It is a form of writing the CR-conditions, which is most relevant to the forthcoming generalization. • The (sub)algebra A (D) := {f ∈ CC(D) | f is holomorphic into D} is also a commutative uniform Banach algebra. Characters By L(F,G) we denote the normed space of the linear continuous operators from a Banach space F to a Banach space G. Let A ′(D) := L(A (D), C) be the dual space. Characters (multiplicative functionals) are defined as elements of the set M := {µ ∈ A ′(D) | µ(fg)= µ(f)µ(g), f,g ∈ A (D)} 3 endowed with the Gelfand (∗-weak) topology, which is determined by the convergence C {µj → µ}⇔{µj(f) → µ(f) , f ∈ A (D)} . The set M is also called a spectrum of the algebra A (D). An example of characters, which turns out to be universal, is provided by M the Dirac measures δz0 ∈ : A δz0 (f) := f(z0) , f ∈ (D) (z0 ∈ D) . Basic fact For topological spaces, we wright S ∼= T if S and T are homeomorphic. For algebras, A ∼= B means that A and B are isometrically isomorphic. Our goal is to provide a 3d analog of the following classical result (see, e.g., [12]: Chapter III, paragraph 11, item 3). µ Theorem 1. For any µ ∈ M there is a point z ∈ D such that µ = δzµ . The map γ : M → D, µ 7→ zµ is a homeomorphism, so that M =∼ D holds. The Gelfand transform Γ: A (D) → CC(M ), (Γf)(µ)= µ(f), µ ∈ M is an isometric isomorphism onto its image, so that A (D) ∼= Γ(A (D)) holds. Thus, the spectrum of A (D) is exhausted by Dirac measures: M = µ {δz0 | z0 ∈ D}. Also, since (Γf)(µ) = µ(f) = δzµ (f) = f(z ), the Gelfand transform just transfers functions from D to M along the map γ. Notice in addition that the same set of the Dirac measures exhausts the spectrum of the ‘big’ algebra CC(D) [12]. 2 The 3d analogs Quaternions • Recall that H is a real algebra of the collections (quaternions) h = α + u1i + u2j + u3k, α,ui ∈ R 4 endowed with the component-wise linear operations and the norm (module) 1 2 2 2 2 2 |h| = [α + u1 + u2 + u3] . A multiplication is determined by the table ii = jj = kk = −1; ij = k, jk = i, ki = j and extended to H by linearity and distributivity. The multiplication is associative but not commutative. The module obeys |gh| = |g||h|. • A geometric quaternion is a pair of a scalar and 3d vector. The set of pairs 3 Hg = p = {α,u}| α ∈ R, u ∈ R is a real algebra w.r.t. the component-wise linear operations, the module 1 |p| = [α2 + |u|2] 2 , and multiplication pq := {αβ − u · v, αv + βu + u ∧ v} for p = {α,u}, q = {β, v} , where · and ∧ are the standard inner and vector products in R3. The multi- plication is noncommutative. The module obeys |pq| = |p||q|. The correspondence u1 H ∋ h = α + u i + u j + u k ↔ h = {α, u } ∈ H (2) 1 2 3 2 g u3 ∼ determines an isometric isomorphism of algebras: H = Hg. As a consequence, Hg is the left H-module with the action hp = hp , h ∈ H, p ∈ Hg . (3) 3 • A quaternion field is an Hg-valued function on a domain in R , i.e., a pair p = {α,u} , where α is a function and u is a vector field defined on a domain in R3. The set of such fields is a real algebra and left H-module w.r.t. the relevant point-wise operations. Algebras and spaces • Let B = {x ∈ R3 | |x| 6 1} be a ball. The space of continuous fields CH(B) := { p = {α,u}| α ∈ CR(B), u ∈ C(B; R3)} 5 with the norm kpk = sup |p| obeying kpqk 6 kpkkqk and kp2k = kpk2 is a B (noncommutative) Banach uniform algebra and a left H-module. • A field p = {α,u} is said to be harmonic if the Cauchy-Riemann conditions dα = ⋆du′, δu′ =0 (δα =0) (4) hold in the inner points of Dom p ⊂ R3. Here α is regarded as a 0-form, ⋆ is the Hodge operator corresponding to the standard orientation of R3, d and δ are a differential and codifferential respectively, u′ is a 1-form dual to u, i.e., u′(b)= b · u on vector-fields b. The condition in the brackets is fulfilled automatically, whereas the ‘gauge condition’ δu′ = 0 is now not trivial.
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