Let $A$ be a concrete $C^*$-algebra, i.e, embedded in $B(H)$, the set of all bounded operators on some Hilbert space $H$. In the sense of Woronowicz, a closed operator $a$ is said to be affiliated with $A$ if $z_a=a(1+a^*a)^{-\frac{1}{2}}$ is in $M(A)$, the multiplier algebra of $A$, and $(1+a^*a)^{-\frac{1}{2}}A$ is linearly dense in $A$. This affiliation relation is expressed as $a\eta A$. Define $$ A^{\eta}:=\{a\mid a\eta A\} $$
Is $A^{\eta}$ an algebra?
My intuition says that it is not. It is known that if $a_1,a_2\eta A$ then $a_1+a_2\eta A$ but I'm not sure about the product structure as the products of unbounded operators does not behave well. Any hint, direction or a counter example will be highly appreciated. Thanks in advance.