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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.10654 |
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| _version_ | 1866908845652049920 |
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| author | Pisier, Gilles |
| author_facet | Pisier, Gilles |
| contents | We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(F_\infty)$ the full $C^*$-algebra of the free group $F_\infty$ with infinitely many generators and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $F_\infty$. Incidentally we observe an analogue for strong similarity of Haagerup's (and Paulsen's) similarity formula for the cb-norm : if $C$ is any unital $C^*$-algebra and $N$ any von Neumann algebra then for any bounded unital homomorphism $u: C \to N$ we have $$\|u\|_{mb}= \inf\{ \|S\|\|S^{-1}\| \}$$ where the inf (which is attained) runs over all invertible $S\in N$ such that $S u(.) S^{-1}$ is a $*$-homomorphism. We end the note by a quick proof of the main point using the mb-norm and the space $R_n\cap C_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10654 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on strong similarity and the Connes embedding problem Pisier, Gilles Operator Algebras We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(F_\infty)$ the full $C^*$-algebra of the free group $F_\infty$ with infinitely many generators and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $F_\infty$. Incidentally we observe an analogue for strong similarity of Haagerup's (and Paulsen's) similarity formula for the cb-norm : if $C$ is any unital $C^*$-algebra and $N$ any von Neumann algebra then for any bounded unital homomorphism $u: C \to N$ we have $$\|u\|_{mb}= \inf\{ \|S\|\|S^{-1}\| \}$$ where the inf (which is attained) runs over all invertible $S\in N$ such that $S u(.) S^{-1}$ is a $*$-homomorphism. We end the note by a quick proof of the main point using the mb-norm and the space $R_n\cap C_n$. |
| title | A note on strong similarity and the Connes embedding problem |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2601.10654 |