The Calkin algebra has outer automorphisms. Phillips, N. C. & Weaver, N. arXiv preprint math/0606594, 2006.
bibtex   
@article{phillips_calkin_2006,
	title = {The {Calkin} algebra has outer automorphisms},
	journal = {arXiv preprint math/0606594},
	author = {Phillips, N. Christopher and Weaver, Nik},
	year = {2006},
}

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