Geodesic
Poisson kernel is the only approximative identity asymptotically multiplicative on $H^\infty$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 82-89

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Every approximative identity asymptotically multiplicative with respect to $H^\infty(\mathbb{R})$ (or to $H^\infty(\mathbb{T})$) is necessarily a shift and a contraction of the Poisson kernel. 
@article{ZNSL_1989_170_a4,
     author = {H. Wolf and V. P. Havin},
     title = {Poisson kernel is the only approximative identity asymptotically multiplicative on $H^\infty$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {82--89},
     publisher = {mathdoc},
     volume = {170},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_170_a4/}
}
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H. Wolf; V. P. Havin. Poisson kernel is the only approximative identity asymptotically multiplicative on $H^\infty$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 82-89. http://geodesic.mathdoc.fr/item/ZNSL_1989_170_a4/