# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## Vector-valued invariant means on spaces of bounded linear maps

### Tom 132 / 2013

Colloquium Mathematicum 132 (2013), 1-11 MSC: 46H05, 43A07, 46L10. DOI: 10.4064/cm132-1-1

#### Streszczenie

Let ${\mathcal A}$ be a Banach algebra and let ${\mathcal M}$ be a $W^*$-algebra. For a homomorphism $\varPhi$ from ${\mathcal A}$ into ${\mathcal M}$, we introduce and study ${\mathcal M}$-valued invariant $\varPhi$-means on the space of bounded linear maps from ${\mathcal A}$ into ${\mathcal M}$. We establish several characterizations of existence of an ${\mathcal M}$-valued invariant $\varPhi$-mean on $B({\mathcal A},{\mathcal M})$. We also study the relation between existence of an ${\mathcal M}$-valued invariant $\varPhi$-mean on $B({\mathcal A},{\mathcal M})$ and amenability of ${\mathcal A}$. Finally, for a character $\phi$ of ${\mathcal A}$, we give some descriptions for $\phi$-amenability of $\mathcal A$ in terms of ${\mathcal M}$-valued invariant $\varPhi$-means.

#### Autorzy

• Mahshid DashtiDepartment of Mathematical Sciences
Isfahan University of Technology
84156-83111 Isfahan, Iran
and
Department of Mathematics
Malayer University
65719-95863 Malayer, Hamedan, Iran
e-mail
• Rasoul Nasr-IsfahaniDepartment of Mathematical Sciences
Isfahan University of Technology
84156-83111 Isfahan, Iran
and
School of Mathematics
Institute for Research
in Fundamental Science (IPM)
19395-5746 Tehran, Iran
e-mail
• Sima Soltani RenaniDepartment of Mathematical Sciences
Isfahan University of Technology
84156-83111 Isfahan, Iran
e-mail

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