[proofplan] We choose one minimal projection $p_0\in M$ and build a maximal orthogonal family of projections Murray-von Neumann equivalent to $p_0$. Since $M$ is a factor, comparison of projections forces the supremum of this family to be the identity. The resulting partial isometries give a complete system of matrix units, and the scalarity of the minimal corner $p_0Mp_0$ reduces the matrix-unit reconstruction theorem to $\mathcal{L}(\ell^2(I))$. Finally, the cardinality of the index set $I$ is recovered as the dimension of any maximal orthogonal family of equivalent minimal projections, so it is an invariant of $M$. [/proofplan] [step:Choose a maximal orthogonal family equivalent to one minimal projection] Since $M$ is a type I factor, by definition there exists a nonzero minimal projection in $M$. Fix such a projection and denote it by $p_0\in M$. For projections $p,q\in M$, write $p\sim q$ if there exists a partial isometry $v\in M$ such that $v^*v=p$ and $vv^*=q$. Let $\mathcal{F}$ be the set of all families $(p_i)_{i\in I}$ such that $I$ is a set, each $p_i\in M$ is a projection, the projections are pairwise orthogonal, and $p_i\sim p_0$ for every $i\in I$. We partially order $\mathcal{F}$ by extension of families. The family indexed by the one-point set $\{0\}$ with projection $p_0$ belongs to $\mathcal{F}$, so $\mathcal{F}$ is nonempty. If $\mathcal{C}\subseteq\mathcal{F}$ is a chain, the union of the families in $\mathcal{C}$ is again a family of pairwise orthogonal projections, each equivalent to $p_0$. Therefore every chain has an upper bound in $\mathcal{F}$. By [Zorn's lemma](/theorems/1226), there is a maximal family $(p_i)_{i\in I}$ in $\mathcal{F}$. [guided] We first need enough copies of one minimal projection to fill the identity. The definition of a type I factor supplies a nonzero minimal projection; fix it and call it $p_0\in M$. We now organize all possible orthogonal collections of copies of $p_0$. Define $\mathcal{F}$ to be the collection of all families $(p_i)_{i\in I}$ with the following properties: $I$ is a set, each $p_i$ is a projection in $M$, the projections satisfy $p_ip_j=0$ whenever $i\ne j$, and $p_i\sim p_0$ for every $i\in I$. The relation $p_i\sim p_0$ means that there exists a partial isometry $w_i\in M$ such that \begin{align*} w_i^*w_i=p_0 \end{align*} and \begin{align*} w_iw_i^*=p_i. \end{align*} The one-element family consisting of $p_0$ itself is in $\mathcal{F}$, so $\mathcal{F}$ is nonempty. We order $\mathcal{F}$ by extension: one family is below another if the latter contains all projections of the former and possibly more. If $\mathcal{C}$ is a chain in this order, then the union of all families in $\mathcal{C}$ is still pairwise orthogonal, because any two projections in the union already occur together in some member of the chain. Each projection in the union is still equivalent to $p_0$ by construction. Thus the union is an upper bound for $\mathcal{C}$ in $\mathcal{F}$. Zorn's lemma applies, and we obtain a maximal family $(p_i)_{i\in I}$ of pairwise orthogonal projections in $M$, all Murray-von Neumann equivalent to $p_0$. [/guided] [/step] [step:Show the supremum of the maximal family is the identity] Let $\mathcal{P}(M)=\{e\in M : e=e^*=e^2\}$ denote the projection lattice of $M$, ordered by $e\le f$ when $ef=e$. Define \begin{align*} p=\bigvee_{i\in I}p_i \end{align*} to be the supremum of the family in the complete projection lattice $\mathcal{P}(M)$, whose existence is guaranteed by [citetheorem:9267]. Set \begin{align*} r=1-p. \end{align*} Then $r$ is a projection in $M$ and $rp_i=0$ for every $i\in I$. For projections $a,b\in M$, write $a\precsim b$ to mean that $a$ is Murray-von Neumann subequivalent to $b$: there is a projection $b_0\le b$ such that $a\sim b_0$. We prove that $r=0$. Suppose instead that $r\ne 0$. Since $M$ is a factor and $p_0,r$ are nonzero projections in $M$, the [Comparison Theorem for Projections in Factors][citetheorem:9281] applies to this pair and gives comparability in the Murray-von Neumann preorder: either $p_0\precsim r$ or $r\precsim p_0$. If $p_0\precsim r$, then there exists a nonzero projection $q\le r$ such that $q\sim p_0$. This projection is orthogonal to every $p_i$, so adjoining $q$ to the family contradicts maximality. If $r\precsim p_0$, then there is a nonzero projection $q\le r$ and a projection $e\le p_0$ such that $q\sim e$. Since $q\ne0$, also $e\ne0$. The projection $p_0$ is minimal, so $e=p_0$. Hence $q\sim p_0$, again contradicting maximality of $(p_i)_{i\in I}$. Therefore $r=0$, and so \begin{align*} \bigvee_{i\in I}p_i=1. \end{align*} [/step] [step:Build a complete system of matrix units] For each $i\in I$, choose a partial isometry $v_i\in M$ such that \begin{align*} v_i^*v_i=p_0 \end{align*} and \begin{align*} v_iv_i^*=p_i. \end{align*} Choose any index $i_0\in I$. For $i,j\in I$, define \begin{align*} e_{ij}=v_iv_j^*. \end{align*} Then $e_{ij}\in M$ for all $i,j\in I$. For $i,j,k,l\in I$, the orthogonality of the ranges gives \begin{align*} v_j^*v_k=0 \end{align*} when $j\ne k$, because $v_j=p_jv_j$, $v_k=p_kv_k$, and $p_jp_k=0$. Also $v_j^*v_j=p_0$. Let $\delta_{jk}=1$ if $j=k$ and $\delta_{jk}=0$ if $j\ne k$. Hence \begin{align*} e_{ij}e_{kl}=v_iv_j^*v_kv_l^*=\delta_{jk}v_ip_0v_l^*=\delta_{jk}e_{il}. \end{align*} Also \begin{align*} e_{ij}^*=v_jv_i^*=e_{ji}. \end{align*} Thus $(e_{ij})_{i,j\in I}$ is a system of matrix units in $M$, and its diagonal projections are $e_{ii}=p_i$. From the previous step, \begin{align*} \bigvee_{i\in I}e_{ii}=1. \end{align*} [/step] [step:Apply matrix unit reconstruction with scalar corner] The projection $p_{i_0}$ is minimal and nonzero because $p_{i_0}\sim p_0$ and minimality is preserved under Murray-von Neumann equivalence. By the [[Minimal Projection Corner Criterion](/theorems/9290)][citetheorem:9290], \begin{align*} p_{i_0}Mp_{i_0}=\mathbb{C}p_{i_0}. \end{align*} Let $\ell^2(I)$ denote the [Hilbert space](/page/Hilbert%20Space) of square-summable scalar families indexed by $I$. Define \begin{align*} \ell^2(I)=\{a:I\to\mathbb{C} : \sum_{i\in I}|a_i|^2<\infty\} \end{align*} with its usual Hilbert-space [inner product](/page/Inner%20Product), and let $\overline{\otimes}$ denote the von Neumann algebra [tensor product](/page/Tensor%20Product). The system $(e_{ij})_{i,j\in I}$ satisfies the hypotheses of the [[Matrix Unit Reconstruction Theorem](/theorems/9292)][citetheorem:9292], with \begin{align*} N=e_{i_0i_0}Me_{i_0i_0}=p_{i_0}Mp_{i_0}=\mathbb{C}p_{i_0}. \end{align*} Therefore there is a normal $*$-isomorphism \begin{align*} M\cong (\mathbb{C}p_{i_0})\overline{\otimes}\mathcal{L}(\ell^2(I)). \end{align*} Identifying $\mathbb{C}p_{i_0}$ with $\mathbb{C}$ as von Neumann algebras gives \begin{align*} M\cong \mathcal{L}(\ell^2(I)). \end{align*} Concretely, the identification sends $(\lambda p_{i_0})\otimes T$ to $\lambda T$ for $\lambda\in\mathbb{C}$ and $T\in\mathcal{L}(\ell^2(I))$. Thus the theorem holds with $K=\ell^2(I)$. [guided] The reconstruction theorem needs a distinguished diagonal corner, but it does not need that corner to be the original projection $p_0$. We have chosen an index $i_0\in I$, and the corresponding diagonal projection is \begin{align*} e_{i_0i_0}=p_{i_0}. \end{align*} Since $p_{i_0}\sim p_0$ and $p_0$ is minimal, the projection $p_{i_0}$ is also minimal. Indeed, equivalence transports subprojections of $p_{i_0}$ to subprojections of $p_0$, so a nonzero proper subprojection of $p_{i_0}$ would give a nonzero proper subprojection of $p_0$, contradicting minimality. Hence the [Minimal Projection Corner Criterion][citetheorem:9290] gives \begin{align*} p_{i_0}Mp_{i_0}=\mathbb{C}p_{i_0}. \end{align*} Now define $\ell^2(I)$ to be the Hilbert space of square-summable scalar families indexed by $I$, and let $\overline{\otimes}$ denote the von Neumann algebra tensor product. The family $(e_{ij})_{i,j\in I}$ is a system of matrix units and satisfies \begin{align*} \bigvee_{i\in I}e_{ii}=1. \end{align*} Therefore the [Matrix Unit Reconstruction Theorem][citetheorem:9292] applies with \begin{align*} N=e_{i_0i_0}Me_{i_0i_0}=p_{i_0}Mp_{i_0}=\mathbb{C}p_{i_0}. \end{align*} It yields a normal $*$-isomorphism \begin{align*} M\cong (\mathbb{C}p_{i_0})\overline{\otimes}\mathcal{L}(\ell^2(I)). \end{align*} The one-dimensional von Neumann algebra $\mathbb{C}p_{i_0}$ is identified with $\mathbb{C}$ by the map $\lambda p_{i_0}\mapsto \lambda$. Under this identification, tensoring with $\mathbb{C}p_{i_0}$ does not change the operator algebra factor, and we obtain \begin{align*} M\cong \mathcal{L}(\ell^2(I)). \end{align*} Thus the required Hilbert space may be taken to be $K=\ell^2(I)$. [/guided] [/step] [step:Recover the Hilbert-space dimension from the factor] It remains to prove uniqueness of the Hilbert-space dimension. In $\mathcal{L}(K)$, the minimal projections are precisely the rank-one orthogonal projections, and a maximal family of pairwise orthogonal equivalent minimal projections is exactly the family of rank-one projections onto the vectors of an [orthonormal basis](/page/Orthonormal%20Basis) of $K$. Hence the cardinality of such a family is $\dim K$. A normal $*$-isomorphism of von Neumann algebras preserves projections, orthogonality, suprema of projection families, minimality, and Murray-von Neumann equivalence. Therefore it preserves the cardinality of maximal orthogonal families of equivalent minimal projections whose supremum is the identity. Consequently, if \begin{align*} M\cong \mathcal{L}(K_1) \end{align*} and \begin{align*} M\cong \mathcal{L}(K_2), \end{align*} then $\dim K_1=\dim K_2$. This proves that the Hilbert-space dimension of $K$ is uniquely determined by $M$. [/step]