Irreducible class and multiplicity of a type I factor representation are well defined

Let $T:K^{\oplus m}\to (K^{\mathrm{\prime }}{)}^{\oplus m^{\mathrm{\prime }}}$ be a unitary equivalence. Its coordinate blocks $T_{ji}:K\to K^{\mathrm{\prime }}$ intertwine $\sigma$ and ${\sigma }^{\mathrm{\prime }}$. Some block is nonzero: otherwise $T$ vanishes on every coordinate inclusion and hence on the dense finite-coordinate vectors, contradicting unitarity on the nonzero domain. Fix such a block $A$. By [F2], $A^{\ast }A=a{I}_K$ and $A{A}^{\ast }=b{I}_{K^{\mathrm{\prime }}}$, where $a,b>0$; $A{A}^{\ast }A=bA=aA$ gives $a=b$, so $S=a^{-1\mathrm{/}2}A$ is a unitary equivalence $K\to K^{\mathrm{\prime }}$.

Apply $S^{-1}$ in each target coordinate to obtain a unitary $R:K^{\oplus m}\to K^{\oplus m^{\mathrm{\prime }}}$ intertwining the two amplifications of $\sigma$. Every block of $R$ is $v_{ji}{I}_K$ by [F2]. Fix a unit $\eta \in K$. On finite-coordinate scalar vectors $z$, the norm identity for $R(z_i\eta {)}_i$ gives ${\sum }_j\mathrm{\mid }{\sum }_i{v}_{ji}{z}_i{\mathrm{\mid }}^2={\sum }_i\mathrm{\mid }{z}_i{\mathrm{\mid }}^2$. Thus the scalar matrix defines an isometry $v:{\mathrm{\ell }}^2(m)\to {\mathrm{\ell }}^2(m^{\mathrm{\prime }})$. The same block argument for $R^{\ast }$ gives its adjoint matrix, and $R^{\ast }R=I$, $R{R}^{\ast }=I$ imply $v^{\ast }v=I$, $v{v}^{\ast }=I$ by testing these vectors; hence $v$ is onto. A unitary preserves dimension: finite dimensions agree by linear independence of bases; finite versus infinite is impossible because the infinite space has arbitrarily large independent coordinate sets. The only remaining case is both countably infinite. Therefore $m=m^{\mathrm{\prime }}$. Existence from [F1] and this uniqueness prove the consequence.