A separable type I factor is a multiple of an irreducible representation

By [F3] and the definition of a type-I factor there, fix a nonzero minimal projection $p\in M$. Consider the set of all sets $S$ of pairwise orthogonal minimal projections of $M$, each equivalent to $p$ and with $p\in S$, ordered by inclusion. The set $\{p\}$ is a member, so the poset is nonempty; the union of a chain of members is again a set of pairwise orthogonal minimal projections equivalent to $p$ and containing $p$, hence an upper bound. By Zorn's lemma [F2] there is a maximal member, written $(p_i{)}_{i\in I}$ with $p_{i_0}=p$.

For every $\xi \in H$ and finite $F\subseteq I$ we have ${\sum }_{i\in F}\mathrm{\parallel }{p}_i\xi {\mathrm{\parallel }}^2=\mathrm{\parallel }({\sum }_{i\in F}{p}_i)\xi {\mathrm{\parallel }}^2\le \mathrm{\parallel }\xi {\mathrm{\parallel }}^2$. The supremum of these finite square sums is finite; choosing a finite set within any positive tolerance of the supremum bounds every remaining tail by that tolerance. Orthogonality therefore makes the finite sums ${\sum }_{i\in F}{p}_i\xi$ Cauchy. Completeness gives their limit, which defines the orthogonal projection $s$ onto the closed span of the ranges. Thus the sums converge strongly, [F4] gives $s\in M$, and $r=I-s\in M$ is orthogonal to every $p_i$. If $r\mathrm{\ne }0$, then [F3] applied to the nonzero projections $r$ and $p$ in the factor $M$ supplies nonzero subprojections $r^{\mathrm{\prime }}\le r$ and $p^{\mathrm{\prime }}\le p$ with $r^{\mathrm{\prime }}$ equivalent to $p^{\mathrm{\prime }}$; minimality of $p$ forces $p^{\mathrm{\prime }}=p$, and conjugation by the partial isometry identifies $r^{\mathrm{\prime }}M{r}^{\mathrm{\prime }}$ with $pMp=\mathbb{C}p$, so $r^{\mathrm{\prime }}$ is a minimal projection equivalent to $p$ and orthogonal to every $p_i$, contradicting maximality in step 1.1. Hence $r=0$, the family is exhaustive, and $H$ is the orthogonal direct sum of the nonzero subspaces $p_{i}H$.

For each $i$ choose a unit vector ${\xi }_i\in p_{i}H$ (possible since $p_i\mathrm{\ne }0$) and a point of a fixed countable dense subset $D\subseteq H$ in the ball around ${\xi }_i$ of radius $1\mathrm{/}2$. Distinct $i$ give orthogonal unit vectors, hence centres at distance $\sqrt{2}>1$, so the radius-$1\mathrm{/}2$ balls are pairwise disjoint, and distinct balls contain distinct points of $D$; therefore $I$ is at most countable. For each $i$ choose a partial isometry $t_i\in M$ with $t_i^{\ast }{t}_i=p$ and $t_i{t}_i^{\ast }=p_i$, possible by the equivalence in step 1.1, and set $t_{i_0}:=p$.

For all $i,j,k,l\in I$ the operators $e_{ij}:=t_i{t}_j^{\ast }\in M$ satisfy $e_{ij}^{\ast }=e_{ji}$ and $e_{ij}{e}_{kl}={\delta }_{jk}{e}_{il}$: indeed $t_j^{\ast }{t}_k=t_j^{\ast }(p_j{p}_k)t_k$ vanishes for $j\mathrm{\ne }k$ because $t_j^{\ast }=t_j^{\ast }{p}_j$ and $t_k=p_k{t}_k$, and equals $p$ for $j=k$; in particular the $e_{ii}=p_i$ are orthogonal projections and $e_{i_0{i}_0}=p$.

On finite-support families $({\eta }_j)$ in ${\widehat{⨁}}_{I}L$, define $\rho (A)({\eta }_j{)}_i={\sum }_j{A}_{ij}{\eta }_j$ for $A\in B(E)$. Choose an orthonormal basis $b_1,\dots ,b_d$ of the finite-dimensional span of these input vectors by [F9], and write ${\eta }_j={\sum }_k{a}_{jk}{b}_k$. Then ${\sum }_i\mathrm{\parallel }{\sum }_j{A}_{ij}{\eta }_j{\mathrm{\parallel }}^2={\sum }_k\mathrm{\parallel }A(a_{jk}{)}_j{\mathrm{\parallel }}_E^2\le \mathrm{\parallel }A{\mathrm{\parallel }}^2{\sum }_j\mathrm{\parallel }{\eta }_j{\mathrm{\parallel }}^2$. Thus $\rho (A)$ extends boundedly to the direct sum with norm at most $\mathrm{\parallel }A\mathrm{\parallel }$; testing families $(z_{j}b)$ for one fixed unit $b\in L$ gives equality. The identity $\rho (A)(z_{j}b{)}_j=((Az{)}_{i}b{)}_i$ holds first for finite-support $z$ and then for all $z\in E$ by continuity. Finite linear combinations of these separated families are dense, so this identity gives the product and adjoint laws for $\rho$; the norm equality makes it injective. The formula $W({\eta }_i)={\sum }_i{t}_i{\eta }_i$ is unitary by orthogonality and exhaustion. For $a\in M$, $t_i^{\ast }a{t}_j={\lambda }_{ij}(a)p$ since $pMp=\mathbb{C}p$; testing $W(z_{i}b)$ shows that the scalar matrix $\mathrm{\Lambda }(a)$ defines an operator on $E$ with norm at most $\mathrm{\parallel }a\mathrm{\parallel }$, and its blocks give $W^{\ast }aW=\rho (\mathrm{\Lambda }(a))$.

The unique scalar blocks and injectivity of $\rho$ show that $\mathrm{\Lambda }$ is a unital injective $\ast$-homomorphism: the identities follow by conjugating sums, products and adjoints with $W$. The matrix units satisfy $W^{\ast }{e}_{ij}W=\rho (E_{ij})$. For finite-coordinate projections $P_F$ on $E$, put $R_F=\rho (P_F)$; direct-sum tails give $R_F\to I$ strongly. For every $A\in B(E)$, $\rho (P_{F}A{P}_F)=R_F\rho (A)R_F\to \rho (A)$ strongly, since $\mathrm{\parallel }{R}_F\mathrm{\parallel }\le 1$ and $R_F\to I$. Each compression is a finite linear combination of the represented matrix units and lies in $W^{\ast }MW$.

Strong closedness [F4] now gives $\rho (A)\in W^{\ast }MW$ for every $A\in B(E)$, while step 5.1 gives the reverse inclusion. Hence $W^{\ast }MW=\rho (B(E))$, and $\mathrm{\Lambda }$ is onto. To compute its commutant, let $T$ commute with all $\rho (E_{ij})$. Commuting with $\rho (E_{ii})$ makes $T$ block diagonal with blocks $S_i\in B(L)$; commuting with $\rho (E_{ij})$ makes $S_i=S_j$ for every $i,j$. Thus $T=I\star S$ for one bounded $S\in B(L)$, and conversely every such operator commutes with all $\rho (A)$. Consequently $W^{\ast }{M}^{\mathrm{\prime }}W=I_E\otimes B(L)$.

Suppose now that $\pi (G{)}^{\mathrm{\prime }\mathrm{\prime }}=M$ and put $\sigma (g):=\mathrm{\Lambda }(\pi (g))\in B(E)$. Then $\sigma$ is a group homomorphism into the unitary group of $E$ by step 6.1, and $W^{\ast }\pi (g)W=\rho (\sigma (g))=\sigma (g)\otimes I_L$ in the notation of the Statement. For $\eta \in E$ fix a unit vector $b\in L$. The identity $\mathrm{\parallel }\sigma (g)\eta -\sigma (g_0)\eta {\mathrm{\parallel }}_E=\mathrm{\parallel }\pi (g)W(\eta \star b)-\pi (g_0)W(\eta \star b){\mathrm{\parallel }}_H$ makes this orbit continuous at each $g_0$ directly by strong continuity of $\pi$; hence $\sigma$ is strongly continuous.

The commutant of $\sigma (G)$ in $B(E)$ is computed by transporting along $\rho$: an operator $A\in B(E)$ commutes with every $\sigma (g)$ exactly when $\rho (A)$ commutes with every $\rho (\sigma (g))=W^{\ast }\pi (g)W$, that is, when $\rho (A)\in W^{\ast }\pi (G{)}^{\mathrm{\prime }}W=W^{\ast }{M}^{\mathrm{\prime }}W$; intersecting with $\rho (B(E))=W^{\ast }MW$ gives $W^{\ast }(M\cap M^{\mathrm{\prime }})W=\mathbb{C}{I}_H$, because $M$ is a factor. Hence $\sigma (G{)}^{\mathrm{\prime }}=\mathbb{C}{I}_E$, and by the double commutant theorem [F4] and the irreducibility criterion of [F6], $\sigma$ is irreducible with $\sigma (G{)}^{\mathrm{\prime }\mathrm{\prime }}=B(E)$.

If $D$ is a countable dense subset of $H$, the set $pD$ is dense in $L=pH$ since $p$ is a contraction. A dense sequence and [F9] therefore supply a finite or countable orthonormal basis of $L$. Expanding $L$ in that basis, the identity $W^{\ast }\pi (g)W=\sigma (g)\otimes I_L$ exhibits $\pi$ as the Hilbert direct sum of $\dim (L)$ copies of $\sigma$ in the sense of [F6], where $\dim (L)\in \{1,2,\dots ,\mathrm{\infty }\}$ is the cardinality of that basis; the space $L=pH$ is thereby the multiplicity space of this decomposition, while $E$ is the carrier of the irreducible $\sigma$. Moreover $N=M^{\mathrm{\prime }}$ is a factor of type I: by the computation of step 7.1 we have $W^{\ast }NW=\{I\star S:S\in B(L)\}\cong B(L)$; commuting with its rank-one matrix units forces a scalar operator, so its centre is scalar, and $B(L)$ contains a nonzero minimal projection, namely the rank-one projection $q_0$ onto any line $\mathbb{C}b$ with $b\in L$ a unit vector, since $q_{0}B(L)q_0=\mathbb{C}{q}_0$.

Let $q\in N$ be a nonzero minimal projection and let $(q_i{)}_{i\in I^{\mathrm{\prime }}}$ be an exhaustive orthogonal family of minimal projections in $N$ equivalent to $q$, with partial isometries $u_i\in N$ satisfying $u_i^{\ast }{u}_i=q_i$ and $u_i{u}_i^{\ast }=q$; such data exist by the maximal-family argument of steps 1.1-2.1 applied to the type-I factor $N$ of step 10.1. Put $K:=qH$. The formula $V\xi :=(u_i\xi {)}_{i\in I^{\mathrm{\prime }}}$ defines a unitary $V:H\to {\widehat{⨁}}_{i\in I^{\mathrm{\prime }}}K$: it is isometric because ${\sum }_i\mathrm{\parallel }{u}_i\xi {\mathrm{\parallel }}^2={\sum }_i⟨q_i\xi ,\xi ⟩=\mathrm{\parallel }\xi {\mathrm{\parallel }}^2$ by exhaustion; moreover $u_i{u}_j^{\ast }={\delta }_{ij}q$, since $u_i=u_i{q}_i$ and $u_j^{\ast }=q_j{u}_j^{\ast }$. Its image contains every summand, since for $\eta \in K$ the vector $u_i^{\ast }\eta$ is mapped to the vector with $\eta$ in the $i$-th slot, and the image is closed as the isometric image of a complete space.

Each $u_i$ lies in $N=\pi (G{)}^{\mathrm{\prime }}$, so $V$ intertwines: $V\pi (g)=({⨁}_{i\in I^{\mathrm{\prime }}}\pi (g)\mathrm{\mid }K)V$. Finally $\pi \mathrm{\mid }K$ is irreducible: for $T\in \mathcal{B}(K)$ the operator $Tq$ on $H$ commutes with $\pi (G)$ exactly when $T$ commutes with $\pi (G)\mathrm{\mid }K$, so $(\pi \mathrm{\mid }K)(G{)}^{\mathrm{\prime }}=q\pi (G{)}^{\mathrm{\prime }}q=qNq=\mathbb{C}q=\mathbb{C}{I}_K$ by minimality of $q$; hence $V$ exhibits $\pi$ as $m:=\mathrm{\mid }{I}^{\mathrm{\prime }}\mathrm{\mid }$ copies of the irreducible representation $\pi \mathrm{\mid }K$, as claimed.

Conversely, for an irreducible strongly continuous unitary $\sigma$ on $E$, [F6] and [F4] give $\sigma (G{)}^{\mathrm{\prime }\mathrm{\prime }}=B(E)$. The block-commutant calculation in step 7.1 applied to $\sigma (g)\otimes I_L$ gives generated algebra $B(E)\otimes I_L$, which has a nonzero minimal projection $P_b\otimes I_L$ for a unit vector $b\in E$. Thus a nonzero multiple of an irreducible is a type-I factor representation. The same spatial calculation, with $M$ and $M^{\mathrm{\prime }}$ interchanged, proves the equivalence of their type-I property.