Phase-isometries on the unit sphere of C ( K )
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00099-1 ORIGINAL PAPER
Phase‑isometries on the unit sphere of C(K) Dongni Tan1 · Yueli Gao1 Received: 21 August 2020 / Accepted: 16 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We say that a map T ∶ SX → SY between the unit spheres of two real normed-spaces X and Y is a phase-isometry if it satisfies
{‖T(x) + T(y)‖, ‖T(x) − T(y)‖} = {‖x + y‖, ‖x − y‖} for all x, y ∈ SX . In the present paper, we show that there is a phase function 𝜀 ∶ SX → {−1, 1} such that 𝜀 ⋅ T is an isometry which can be extended a real linear isometry on X whenever T is surjective, X = C(K) and Y is an arbitrary Banach space. Additionally, if T is a surjective phase-isometry between the unit spheres of C(K) and C(𝛺) , where K and 𝛺 are compact Hausdorff spaces, we prove that there are a homeomorphism 𝜑 ∶ 𝛺 → K and a continuous unimodular function h on 𝛺 such that
T(f ) ∈ {h ⋅ f ◦𝜑, −h ⋅ f ◦𝜑} for all f ∈ SC(K) . This also can be seen as a Banach-Stone type representation for phase-isometries in C(K) spaces. Keywords Wigner’s theorem · Tingley’s problem · Banach-Stone · Phase-isometry · Phase-equivalent Mathematics Subject Classification 46B04 · 46B20
1 Introduction Let X and Y be real normed spaces, and let A and B be subsets of X and Y, respectively. A map T ∶ A → B is called an isometry if
Communicated by Jesús Castillo. * Dongni Tan [email protected] 1
Tianjin University of Technology, Tianjin, China Vol.:(0123456789)
D. Tan and Y. Gao
‖T(x) − T(y)‖ = ‖x − y‖,
(x, y ∈ A)
and a phase-isometry if
{‖T(x) + T(y)‖, ‖T(x) − T(y)‖} = {‖x + y‖, ‖x − y‖},
(x, y ∈ A).
(1)
Two maps S, T ∶ A → B are said to be phase-equivalent if there is a phase function 𝜀 ∶ A → {−1, 1} such that 𝜀 ⋅ T = S . It should be noted that 𝜀 does not need to be continuous. In the present paper, it is the connection between phase-isometries and isometries restrained on the unit spheres that will concern us. We explore the following Question 1 Is a phase-isometry between the unit spheres of two real normed spaces X and Y necessarily phase-equivalent to an isometry which can be extended to a linear isometry from X onto Y? This question is motivated by Tingley’s problem and Wigner’s theorem. Tingley’s problem asked whether every isometry between the unit spheres of real normed spaces can be extended to a linear isometry between the corresponding entire spaces. This problem raised by Tingley [23] in 1987 has attracted many people’s attention. Although it is still open in general case, there is a large number of papers dealing with this topic (Zentralblatt Math. shows 61 related papers published from 2002 to 2020). Please see [20] and the survey [5] for classical Banach spaces, and see the survey [16] which contains a good description of non-commutative operator algebras. For some of the most recent papers, one can see [1, 3, 4, 7, 12, 15, 17]. Wigner’s theorem plays a fundamental role in mathematical foundations of quantum mechanics, and there are many equivalent
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