A remark on embedding of a cylinder on a real commutative Banach algebra

A remark on embedding of a cylinder on a real commutative Banach algebra Hiroki Yagisita (Kyoto Sangyo University) Abstract: Let Abe a real commutative Banach algebra with unity. Let a0∈A\{0}. Let Za0:= {na0}n∈Z. Then, Za0is a discrete subgroup of A. For any n∈Z, the Frechet derivative of the mapping x∈A7→ x+na0∈A is the identity map on Aand, especially, an A-linear transformation on A. So, the quotient group A/(Za0) is a 1-dimensional A-manifold and the covering projection x∈A7→ x+Za0∈A/(Za0) is an A-map. We call A/(Za0) the 1-dimensional A-cylinder by a0. Let Tbe a compact Hausdorff space. Suppose that there exist t1∈T and t2∈Tsuch that t16=t2holds. Then, the set C(T;R) of all real-valued continuous functions on Tis a real commutative Banach algebra with unity and R ⊊ C(T;R) holds. In this paper, we show that there exists a0∈ C(T;R)\Rsuch that for any k∈N, the 1-dimensional C(T;R)-cylinder (C(T;R))/(Za0) by a0cannot be embedded in the finite direct product space (C(T;R))kas a C(T;R)-submanifold. Keywords: immersion, isotopy, bump function, partition of unity, Gelfand representation, Radon measure, von Neumann algebra, C∗-algebra, SerreSwan theorem, complex vector bundle, locally trivial fiber space, infinitedimensional Lie group, Cartesian product, Euclidean space, Affine space, vector sheaf. 1 In this paper, we give a remark concerned with an A-manifold (a manifold on a commutative topological algebra A). There are already various studies related to A-manifolds or their analogues (e.g., [1], [2], · · · , [9]). In [10], we showed the existence of a C([0,1]; R)-manifold that cannot be embedded in the finite-dimensional Affine space (C([0,1]; R))kas a C([0,1]; R)- submanifold. In this paper, we show the following existence theorem, which is a generalization. Theorem : Let Tbe a compact Hausdorff space. Suppose that there exist t1∈Tand t2∈Tsuch that t16=t2holds. Then, there exists a0∈C(T;R)\R such that for any k∈N, the cylinder (C(T;R))/(Za0) cannot be embedded in the Cartesian product (C(T;R))kas a C(T;R)-submanifold. Proof : Because Tis normal, by Urysohn’s lemma, there exists a0∈ C(T;R)\Rsuch that a0(t1) = −1 and a0(t2) = +1 hold. Let k∈N. Then, by a contradiction, we show that (C(T;R))/(Za0) cannot be embedded in (C(T;R))kas a C(T;R)-submanifold. Suppose that (C(T;R))/(Za0) can be embedded in (C(T;R))kas a C(T;R)- submanifold. Then, there exists a C(T;R)-injection Ψ from (C(T;R))/(Za0) to (C(T;R))k. Let (1) b0(t) := max{0, a0(t)}(t∈T). Then, b0∈C(T;R) and [0] = [a0]6= [b0] hold, where [x] denotes the equivalence class of x∈C(T;R). That is, we put [x] := x+Za0(x∈C(T;R)). So, because Ψ([0]) = Ψ([a0]) 6= Ψ([b0]) holds, there exists l0∈ {1,2,· · · , k} such that (2) Ψl0([0]) = Ψl0([a0]) 6= Ψl0([b0]) holds, where Ψl0denotes the l0-th component of Ψ. Then, because Ψ is a C(T;R)-injection from (C(T;R))/(Za0) to (C(T;R))k, as we put Φ(x) := Ψl0([x]) (x∈C(T;R)), the map x∈C(T;R)7→ Φ(x)∈C(T;R) 2 is a C(T;R)-map and, especially, Frechet derivatives of Φ are C(T;R)-linear transformations on C(T;R). That is, (3) Φ′(x)∈C(T;R) (x∈C(T;R)) holds, because, in general, if Ais a commutative ring with the unity 1Aand Lis an A-linear transformation on A, then L(1A)∈A and L(x) = L(x·1A) = x·(L(1A)) = (L(1A)) ·x(x∈A) hold. Now, from (2), there exists t0∈Tsuch that (4) (Φ(0))(t0) = (Φ(a0))(t0)6= (Φ(b0))(t0) holds. Then, from (1), Case 1 : “0 = b0(t0)” or Case 2 : “a0(t0) = b0(t0)” holds. First, we consider Case 1. Let 0 = b0(t0). Let F1(s) := sb0(s∈[0,1]). Then, the map s∈[0,1] 7→ F1(s)∈C(T;R) is a line from 0 to b0in C(T;R) and (F1)′(s) = b0(s∈[0,1]) holds. Hence, Φ(b0)−Φ(0) = Φ(F1(1)) −Φ(F1(0)) =∫1 0 (Φ′(F1(s))) ((F1)′(s)) ds =∫1 0 (Φ′(F1(s))) b0ds holds. So, in virtue of (3), for any t∈T, 3 (Φ(b0))(t)−(Φ(0))(t) =∫1 0 ( (Φ′(F1(s))) b0) (t)ds =∫1 0 ((Φ′(F1(s)))(t)) ·(b0(t)) ds holds. However, because of 0 = b0(t0), (Φ(b0))(t0)−(Φ(0))(t0) = 0 holds. It contradicts (4). Next, we consider Case 2. Let a0(t0) = b0(t0). Let F2(s) := a0+s(b0−a0) (s∈[0,1]). Then, similarly, because Φ(b0)−Φ(a0) =∫1 0 (Φ′(F2(s))) (b0−a0)ds holds, in virtue of (3) and a0(t0) = b0(t0), (Φ(b0))(t0)−(Φ(a0))(t0) =∫1 0 ((Φ′(F2(s)))(t0)) ·(b0(t0)−a0(t0)) ds = 0 holds. It contradicts (4). ■ Comment : Kasuya suggested that an Rn-manifold and a Cn-manifold might be related to a differential web and a holomorphic web, respectively. He proposed a candidate for a compact C2-manifold Nsuch that for any C-manifolds M1and M2,Ncan not be embedded in M1×M2as a C2submanifold. The construction of our example was inspired by this proposal. — Acknowledgment: As in Comment, Professor Naohiko Kasuya suggested it. This work was supported by JSPS KAKENHI Grant Number JP16K05245. 4 References [1] B. W. Glickfeld, The Riemann sphere of a commutative Banach algebra, Trans. Amer. Math. Soc., 134 (1968), 1-28. [2] S. Kobayashi, Manifolds over function algebras and mapping spaces, Tohoku Math. J., 41 (1989), 263-282. [3] A. Mallios and E. E. Rosinger, Space-time foam dense singularities and de Rham cohomology, Acta Appl. Math., 67 (2001), 59-89. [4] P. Manoharan, A characterization for spaces of sections, Proc. Amer. Math. Soc., 126 (1998), 1205-1210. [5] M. H. Papatriantafillou, Partitions of unity on A-manifolds, Internat. J. Math., 9 (1998), 877-883. [6] H. Yagisita, Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds, Complex Manifolds, 6 (2019), 228-264. [7] H. Yagisita, Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems, preprint. [8] H. Yagisita, Cartan-Thullen theorem for a Cn-holomorphic function and a related problem, preprint. [9] H. Yagisita, A remark on locally direct product subsets in a topological Cartesian space, preprint. [10] H. Yagisita, A manifold on the real commutative Banach algebra C([0,1]; R) that cannot be embedded in the finite-dimensional Euclidean space C([0,1]; Rn), preprint. 5