There is a quotation of a book "C*-algebras Finite-Dimensional Approximations" below:
Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero compact operators. ($A$ is a C*-algebra.)
Essential representations are easy to construct: if $\pi: A\rightarrow B(H)$ is any representation, then its infinite inflation (i.e., the direct sum of infinitely many copies of $\pi$) will be essential.
I can not understand why the construction above is essential?
The assertion is that for any $x\in B(H)$, the operator $\bigoplus x$ is not compact when the sum is infinite. This new operator acts on an infinite direct sum of Hilbert spaces. The image of the unit ball $H_1$ will be the direct sum of $xH_1$ (you can play with the vectors that have a single nonzero direct summand), which is not compact.