## Taiwanese Journal of Mathematics

### COPIES OF $c_0$ AND $\ell_\infty$ INTO A REGULAR OPERATOR SPACE

#### Abstract

For an Orlicz function $\varphi$ and a Banach lattice $X$, let $\ell_\varphi$ denote the Orlicz sequence space associated to $\varphi$, ${\mathcal L}^r(\ell_\varphi, X)$ denote the space of regular operators from $\ell_\varphi$ to $X$, and ${\mathcal K}^r(\ell_\varphi, X)$ denote the linear span of positive compact operators from $\ell_\varphi$ to $X$. In this paper, we show that if $\varphi$ and its complementary function $\varphi^\ast$ satisfy the $\Delta_2$-condition, then (a) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $\ell_\infty$; and (b) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $c_0$ if and only if ${\mathcal L}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $c_0$ and each positive linear operator from $\ell_\varphi$ to $X$ is compact.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 1 (2012), 207-215.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406537

Digital Object Identifier
doi:10.11650/twjm/1500406537

Mathematical Reviews number (MathSciNet)
MR2887861

Zentralblatt MATH identifier
1247.46018

#### Citation

Li, Yongjin; Ji, Donghai; Bu, Qingying. COPIES OF $c_0$ AND $\ell_\infty$ INTO A REGULAR OPERATOR SPACE. Taiwanese J. Math. 16 (2012), no. 1, 207--215. doi:10.11650/twjm/1500406537. https://projecteuclid.org/euclid.twjm/1500406537

#### References

• Q. Bu, G. Buskes and W. K. Lai, The Radon-Nikodym property for tensor products of Banach lattices II, Positivity, 12 (2008), 45-54.
• Q. Bu, M. Craddock and D. Ji, Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-II, Quaest. Math., 32 (2009), 339-350.
• P. Cembranos and J. Mendoza, Banach Spaces of Vector-Valued Functions, Springer-Verlag, 1997.
• S. Chen, Geometry of Orlicz Spaces, Dissertaions Math., 356, Warszawa, 1996.
• J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, 1984.
• L. Drewnowski, Copies of $\ell_\infty$ in an operator space, Math. Proc. Camb. Phil. Soc., 108 (1990), 523-526.
• G. Emmanuele, A remark on the containment of $c_0$ in spaces of compact operators, Math. Proc. Cambridge Philos. Soc., 111 (1992), 331-335.
• I. Ghenciu and P. Lewis, The embeddability of $c_0$ in spaces of operators, Bull. Pol. Acad. Sci. Math., 56 (2008), 239-256.
• N. Kalton, Spaces of compact operators, Math. Ann., 208 (1974), 267-278.
• J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, 1977.
• P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, 1991.
• H. P. Rosenthal, On relatively disjoint families of measures with some applications to Banach space theory, Studia Math., 37 (1970), 13-36.