Forms of Isometries Between Function Spaces Kathleen Roberts1 Iowa State University, Ames, IA Introduction Takeshi's Challenge Results The classic Banach-Stone Theorem establishes a form for surjective, Consider the unit circle in the complex plane In the Banach-Stone proof we referred to, we were able to complex-linear isometries (distance preserving functions) between function relate the functions we sought, and ', to the inverse A = faz + b : a, b 2 , jzj = 1g spaces. Mathematician Takeshi Miura from Niigata University questioned C image of a set of functions, Q such that for w, α 2 T, what could be said about surjective, real-linear isometries after finding a and isometry Q(w, α) = ff 2 A: f (w) = α, kf k = 1g. counter-example that demonstrated the shortcomings of the Banach-Stone T (az + b) = az + b. T Theorem. Through a careful examination of the Banach-Stone Theorem We found that proof, we hope to gain insight into why the theorem does not hold in general T -1[Q(w, 1)] = Q( (w), '(w)). and to find a form for real-linear isometries between function spaces. This along with the complex-linearity of T allowed us to access any of the complex-numbers. However, since we Objectives sought to create such a generalization for a real-linear function, we needed to use two different such sets, one to I We seek to determine why the Banach-Stone Theorem fails to classify allow us to access any real part and one to allow us to forms of isometries for general subspaces. access any imaginary part: We seek to understand the failure of the classification for the I T -1[Q(w, 1)] = Q( (w), ' (w)) counterexample. 1 1 and I We seek to prove a conjecture as to the correct form of such isometries of -1 the unit circle in the complex plane. T [Q(w, i)] = Q( i(w), 'i(w)). Methods Conclusion I Understand problem background and definitions needed to understand the As stated by the original Banach-Stone Theorem, Banach-Stone Theorem. complex-linear isometries T between function spaces take I Begin by examining a proof of the Banach-Stone Theorem and see where this form: the failure occurs when we have only real-linearity instead of T (f ) = T (1) · (f ◦ ). complex-linearity. However, real-linear isometries T between function spaces I Seek to prove the conjecture for forms of real-linear isometries. take one of the following forms: T (f )(z) = Re[ 1(z)f ( 1(z))] + i Im[ 1(z)f ( i(z))] Classic Banach-Stone Result or The Banach-Stone Theorem says if T is a surjective, complex-linear isometry T (f )(z) = Re[ 1(z)f ( 1(z))] - i Im[ 1(z)f ( i(z))]. between function spaces, then there is a weighting fuction ' and a We found that this was indeed possible to prove as we continuous bijection such that had hoped. So we now have a "nice" formula to tell us T (f ) = ' · (f ◦ ) the forms of such isometries. for all f in the function space. References Note: This function is in fact a surjective isometry and is real-linear, but it does not follow the form set by the S. Banach, \Theorie des operations linearse," Chelsea, Banach-Stone Theorem for complex-linear isometries. Warsaw, 1932. Also note that the isometry can be written in the following way: T. Miura \Surjective isometries between function T (az + b) = Re(az + b)- i Im(a(-z) + b), spaces," Contemp. Math., Vol. 645 (2015), pp. 231{240. which matches one of the conjectured forms. M. Stone, \Applications of the theory of boolean rings in topology," Trans. Amer. Math. Soc., Vol. 41 (1937), pp. 375{481. Project Advisor: Dr. Kristopher Lee 1Department of Mathematics, Iowa State University, Ames, IA 50011 ([email protected]).