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H(x))2 dx. 0 The upshot of this ansatz is this: Let H be the Hilbert space of functions 2π g such that g(x + 2π) ≡ −g(x) with norm ||g||2 = 0 g(x)2 dx. Let G be the Grassmannian of 2-planes in H and let G0 be the open subset of 2-planes such that there is no x where all functions in the 2-plane vanish. Then using orthonormal bases of these 2-planes as g and h, we ﬁnd that G0 is isomorphic to the space of parameterized immersed plane curves of odd index mod translations, rotations and scaling. Not only that but the natural metric on this Grassmannian corresponds to a very natural metric on this space of curves.
S. H ERZ, Remarques sur la note pr´ec´edente de M. Varopoulos, C. R. Acad. Sci. Paris 260 (1965), 6001–6004.  C. S. H ERZ, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23(3) (1973), 91–123.  B. H OST, Le th´eor`eme des idempotents dans B(G), Bull. Soc. Math. France 114 (1986), 215–221.  J. R. H UBBUCK and R. M. K ANE, The homotopy types of compact Lie groups, Israel J. Math. 51 (1985), 20–26.  M. I LIE and N. S PRONK, Completely bounded homomorphisms of the Fourier algebras, J.
It would sufﬁce to know that the conjecture was true for the “ax + b group” to know it in general. References  M. BARONTI, Algebre di Banach A p di gruppi localmente compatti, Riv. Mat. Univ. Parma 11 (1985), 399–407. ˇ AP, M. G. C OWLING , F. D E M ARI , M. G. E ASTWOOD and  A. C R. G. M C C ALLUM, The Heisenberg group, SL(3, R) and rigidity, pages 41–52, In: “Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory”, edited by Jian-Shu Li, Eng-Chye Tan, Nolan Wallach and Chen-Bo Zhu.