Bulletin of the Belgian Mathematical Society - Simon Stevin

Achievement of continuity of $(\varphi,\psi)$-derivations without linearity

S. Hejazian, A. R. Janfada, M. Mirzavaziri, and M. S. Moslehian

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Suppose that $\frak A$ is a $C^*$-algebra acting on a Hilbert space $\frak K$, and $\varphi, \psi$ are mappings from $\frak A$ into $B(\frak K)$ which are not assumed to be necessarily linear or continuous. A $(\varphi, \psi)$-derivation is a linear mapping $d: \frak A \to B(\frak K)$ such that $$d(ab)=\varphi(a)d(b)+d(a)\psi(b)\quad (a,b\in \frak A).$$ We prove that if $\varphi$ is a multiplicative (not necessarily linear)\ $*$-mapping, then every $*$-$(\varphi,\varphi)$-derivation is automatically continuous. Using this fact, we show that every $*$-$(\varphi,\psi)$-derivation $d$ from $\frak A$ into $B(\frak K)$ is continuous if and only if the $*$-mappings $\varphi$ and $\psi$ are left and right $d$-continuous, respectively.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 4 (2007), 641-652.

First available in Project Euclid: 15 November 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L57: Derivations, dissipations and positive semigroups in C-algebras
Secondary: 46L05: General theory of $C^*$-algebras 47B47: Commutators, derivations, elementary operators, etc.

Automatic continuity $d$-continuous $(\varphi,\psi)$-derivation $*$-mapping derivation $C^*$-algebra


Hejazian, S.; Janfada, A. R.; Mirzavaziri, M.; Moslehian, M. S. Achievement of continuity of $(\varphi,\psi)$-derivations without linearity. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 4, 641--652. doi:10.36045/bbms/1195157133. https://projecteuclid.org/euclid.bbms/1195157133

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