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Some weakly inner automorphisms of the Cuntz algebras


Author: Sze-Kai Tsui
Journal: Proc. Amer. Math. Soc. 123 (1995), 1719-1725
MSC: Primary 46L40; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1995-1239808-X
MathSciNet review: 1239808
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Abstract: Let V be an $n \times n$ unitary matrix. Then V induces an automorphism ${O_V}$ of ${O_\infty }$ and ${O_n}$. It is shown in this paper that ${O_V}$ is weakly inner. Let ${F_n}$ be the UHF ${C^ \ast }$-subalgebra of ${O_n}$ and U be a unitary operator in the diagonal maximal abelian $\ast$-subalgebra of ${F_n}$. Then the automorphism ${\lambda _U}$ of On defined by ${\lambda _U}({S_i}) = {U^ \ast }{S_i}$, for each generator ${S_i}$ of ${O_n}$, is weakly inner. Any automorphism conjugate to either one mentioned above is also weakly inner.


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Keywords: Cuntz algebras, weakly inner automorphisms, FOCK representation, separable simple <!– MATH ${C^ \ast }$ –> <IMG WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img23.gif" ALT="${C^ \ast }$"> algebras, universally weakly inner automorphisms, outer automorphisms, permutations
Article copyright: © Copyright 1995 American Mathematical Society