[b_i\eta,1]_i \big).$$ ]{}

In general, a unital homomorphism $\phi: C[0,1]\rightarrow B$ with finite dimensional image is always of the form : $$\phi(f)=\sum f(x_k)p_k,\,\,\forall f\in C[0,1]$$ where $\{x_k\}$ is a finite subset of \[0,1\] and $\{p_k\}$ is a set of mutually orthogonal projections with $\sum p_k=1_B$. A homomorphism $\phi: A=M_n(C([0,1]))\rightarrow B$ with finite dimensional image is of the form: $$\phi(f)=\sum f(x_k)\otimes p_k,\,\,\forall f\in A$$ for a certain identification of $\phi(1_A)B\phi(1_A)\cong M_n(\mathbb{C})\otimes (\phi(e_{11})B\phi(e_{11}))$, where $\{p_k\}$ is a set of mutually orthogonal projections in $\phi(e_{11})B\phi(e_{11})$.

Pairing results
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A unital ${\mathrm{C}}^*$-algebra is said to have $real\ rank\ zero$, written $RR(A)=0$, if the set of invertible self-adjoint elements is dense in $A_{sa}$.

[ Suppose $A$ is a ${\mathrm{C}}^*$-algebra, $B\subset A$ is a subalgebra, $F\subset A$ is a finite subset and let $\varepsilon>0$. If for each $f\in F$, there exists an element $g\in B$ such that $\|f-g\|<\varepsilon$, then we shall say that $F$ is approximately contained in $B$ to within $\varepsilon$, and denote this by $F\subset_{\varepsilon} B$. ]{}

The following is clear by the standard techniques of spectral theory [@BBEK].

Let $A=\underrightarrow{lim}(A_n,\phi_{m,n})$ be a ${\mathrm{C}}^*$-algebra which is the inductive limit of ${\mathrm{C}}^*$-algebras $A_n$ with morphisms $\phi_{m,n}:A_m\rightarrow A_n$. Then $A$ has $RR(A)=0$ if and only if 