Let $A$ and $B$ are two $C^*$-algebras such that $A\subset B$. If an operator $T$ is affiliated with $A$, then does it imply that $T$ is also affiliated with $B$?
As per my intuition, I think it is true but i can't verify using the definition.
Definition: An operator $T$ is said to be affiliated with $A$ if $z_T=T(1 + T^*T)^{-\frac{1}{2}}\in M(A)$, where $M(A)$ denotes the multiplier algebra of $A$, and $(1 + T^*T)^{-\frac{1}{2}}A$ is linearly dense in $A$.
This affiliation relation is denoted by $T\eta A$.
Now, if $T\eta A$, then by definition, $z_T\in M(A)\subset M(B)$ and $(1 + T^*T)^{-\frac{1}{2}}A$ is linearly dense in $A$. Does it follow that $(1 + T^*T)^{-\frac{1}{2}}B$ is linearly dense in $B$?
Help in any direction will be highly appreciated!