\[Prp6.2.Tom06.b\] ${{\operatorname{rc}}}(M_n \otimes A)= \frac{1}{n} \cdot {{\operatorname{rc}}}(A)$ for $n \in {{\mathbb{Z}}_{> 0}}$.

The following proposition, which is a special case of results in [@BRTTW12] and is Theorem 12.4.4 of [@GKPT18], is an algebraic reformulation of the radius of comparison for simple C\*-algebras.

\[rc\_alg\_def\] Let $A$ be a stably finite simple unital C\*-algebra. Then ${{\operatorname{rc}}}(A)$ is the least number $s \in [0, \infty]$ such that whenever $m,n \in {{\mathbb{Z}}_{> 0}}$ satisfy $m/n > s$, and $a,b \in M_{\infty} (A)_+$ satisfy $$(n + 1)\langle a \rangle_ A + m\langle 1 \rangle_ A \leq \langle b \rangle_ A$$ in $W(A)$, then $a \precsim_A b$.

The following theorem, providing bounds on the radius of comparison of a full corner in a [C\*-algebra]{}, is a special case of Theorem 2.18 in [@AGP19].

\[Ourcornertheorem\] Let $A$ be a stably finite unital [C\*-algebra]{} and let $q$ be a full projection in $A$. Define $$\lambda
 = \inf \bigl( \bigl\{ \rho (q) \colon
     \rho \in {{\operatorname{QT}}}(A) \bigr\} \bigr)
{\qquad {\mbox{and}} \qquad}\eta
 = \sup \bigl( \bigl\{ \rho (q) \colon
     \rho \in {{\operatorname{QT}}}(A) \bigr\} \bigr).$$ Then $0 < {\lambda}\leq {\eta}\leq 1$ and $
\frac{1}{\eta} \cdot {{\operatorname{rc}}}(A)
 \leq {{\operatorname{rc}}}( q A q )
 \leq \frac{1}{\lambda} \cdot {{\operatorname{rc}}}(A).
$

The radius of comparison 