Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a *-representation $\pi\colon A\to B(K)$ on some Hilbert space $K$. Does it follow that $\|\pi(a)\|\leqslant \|a\|_{K(H)}$?
I was trying to play with the uniqueness of the *-irreducible representation of $K(H)$ but with no success so far.
Note that $$\tag{1} \|\pi(a)\|^2=\|\pi(a)^*\pi(a)\|=\|\pi(a^*a)\|. $$ So we only care about the norm of positive elements in $A$. For $a\in A$, the operator $a^*a$ is compact and so it is of the form $a^*a=\sum_{j=1}^\infty\lambda_j\,p_j$, for rank-one projections $p_1,p_2,\ldots\in A$ (since $A$ contains all finite-rank projections). Then $$\tag{2} \|\pi(a^*a)\|=\left\|\sum_{j=1}^\infty\lambda_j\,\pi(p_j)\right\|\leq\max\{\lambda_j:\ j=1,2,\ldots\}=\|a^*a\|=\|a\|^2 $$ (note that since $\pi$ is a representation, $\pi(p_1),\pi(p_2),\ldots$ are pairwise orthogonal projections, with some of them possibly zero).
Combining $(1)$ and $(2)$, we obtain that $\pi$ is a contraction.