Communications of the
Korean Mathematical Society CKMS

We consider the $C^*$-algebra $O_\infty$ generated by infinite isometries $s_1$, $s_2$, $\cdots$ on Hilbert spaces with the property $\sum_{i=1}^ns_i s_i^*\leqq 1$ for every $n\in \Bbb N$. We present certain type of representations of $C^*$-algebra $O_\infty$ on a separable Hilbert space and study the conditions for irreducibility of these representations.

Keywords: $C^*$-algebra, irreducible representation, decomposition

MSC numbers: Primary 47D30; Secondary 46L45, 47D45, 47S50