Representations of *-regular rings and their ortholattices of projections

TL;DR

This paper demonstrates that subdirectly irreducible *-regular rings can be represented within inner product spaces if their ortholattice of projections can also be represented, linking algebraic and lattice-theoretic structures.

Contribution

It establishes a connection between the representability of *-regular rings and their ortholattices of projections, providing new insights into their structural relationship.

Findings

Subdirectly irreducible *-regular rings admit inner product space representations.

Representation of the ortholattice of projections is key to representing the ring.

The work bridges algebraic and lattice-theoretic perspectives in *-regular rings.

Abstract

We show that a subdirectly irreducible *-regular ring admits a representation within some inner product space provided so does its ortholattice of projections.

Equations

$$⟨({\pi }_U-\phi )(v)\mid ({\pi }_W+\psi )(w)⟩=⟨{\pi }_U(v)\mid \psi (w)⟩-⟨\phi (v)\mid {\pi }_W(w)⟩$$

Full text

1. Introduction

The motivating examples of $*$-regular rings, due to Murray and von Neumann, were the $*$-rings of unbounded operators affiliated with finite von Neumann algebra factors; to be subsumed, later, as $*$-rings of quotients of finite Rickart $C^{*}$-algebras. All the latter have been shown to be $*$-regular and unit-regular (Handelman [7]). Representations of these as $*$-rings of endomorphisms of suitable inner product spaces have been obtained first, in the von Neumann case, by Luca Giudici (cf. [19]), in general in joint work with Marina Semenova [13].

The principal right ideals of a $*$-regular ring $R$ form a modular ortholattice $\operatorname{{\sf L}}^{\perp}(R)$, also to be viewed as the ortholattice of projections of $R$. As observed by Giudici [4], any representation of $R$ induces one of $\operatorname{{\sf L}}^{\perp}(R)$. Here, a representation of an ortholattice $L$ in an inner product space $V$ means an embedding $\eta$ of $L$ into the lattice of all linear subspaces of $V$ such that, for any $u\in L$, $\eta(u^{\perp})$ is the orthogonal of $\eta(u)$. In his thesis [18], the second author established the converse for subdirectly irreducible $R$ (cf. [11]). This involved a coordinatization of representable ortholattices in terms of a variant, including orthogonality conditions, of Jónsson’s large partial frames [15]. The purpose of the present note is to give a short presentation to the result, relying on the review of Coordinatization Theory given in [8] and the fact that every variety of $*$-regular rings is generated by its simple members [10].

As general refences we use [5, 1, 2] for regular and $*$-regular rings, [3] for lattices, [19, 16, 20] for coordinatization, and [6] for inner product spaces. Thanks are due to the referee for a lot of helpful suggestions, in particular correcting the proof of the crucial Claims 5.4 and 5.6.

2. Regular rings and vector space representations

Unless stated otherwise, rings will be associative, with unit $1$ as constant (constants in the signature have to be preserved under homomorphisms and in substructures). A (von Neumann) regular ring $R$ is such that for each $a\in R$ there is $x\in R$ such that $axa=a$; equivalently, every right (left) principal ideal is generated by an idempotent.

A representation of a ring $R$ within a vector space $V$ is an embedding of $R$ into the endomorphism ring $\operatorname{{\sf End}}(V)$ of $V$. It appears to be well known that every subdirectly irreducible regular ring $R$ admits some representation. Indeed, each maximal left ideal $M_{i}$ of $R$ gives rise to a homorphism $\varphi_{i}:R\to\operatorname{{\sf End}}(V_{i})$, $\varphi_{i}(r)(a+M_{i})=ra+M_{i}$; here $V_{i}$ is the (right) vector space over the division ring of endomorphisms of the simple left $R$-module $R/M_{i}$. These homomorphisms $\varphi_{i}$ yield a subdirect representation of $R$ since $\bigcap_{i}M_{i}=0$ (for $r\neq 0$ and idempotent $e$ with $Rr=Re$ choose $M_{i}$ such that $1-e\in M_{i}$ to obtain $r\not\in M_{i}=\ker\varphi_{i}$). On the other hand, examples of non-representable regular rings are obtained as products of matrix rings over fields of different characteristics.

We consider lattices $L$ with bounds $0,1$ as constants. We use $+$ and $\cap$ to denote joins and meets and write $a\oplus b=c$ if $a+b=c$ and $a\cap b=c$. $L$ is complemented if for any $a$ there is $b$ such that $a\oplus b=1$.

The principal right ideals of a regular ring $R$ form a complemented modular lattice $\operatorname{{\sf L}}(R)$, a sublattice of the lattice of all right ideals. A representation of a lattice $L$ within a vector space $V$ is an embedding of $L$ into the lattice $\operatorname{{\sf L}}(V)$ of linear subspaces of $V$. The following is due to Luca Giudici, proof of (1) in [4, Theorem. 4.2.1], cf. [8, Proposition 10.1].

Fact 2.1**.**

If $\iota$ is a representation of the regular ring $R$ in the vector space $V$, then $\eta(aR)=\operatorname{{\sf im}}\iota(a)$, $a\in R$, defines a representation of $\operatorname{{\sf L}}(R)$ in $V$.

The purpose of this section is to relate representations the other way round making use of coordinatization results due von Neumann and Jónsson, cf. [8]. A coordinatization of a lattice $L$ is an isomorphism onto $\operatorname{{\sf L}}(R)$, $R$ a regular ring. Such are based on ”frames”: suitable coordinate systems. We write $a\sim_{c}b$ if $a+b=a\oplus c=b\oplus c$ and $a\sim b$ if $a\sim_{c}b$ for some $c$. Recall that, for modular $L$, $a\sim b$ and $a^{\prime}\leq a$ implies $a^{\prime}\sim b^{\prime}$ for some $b^{\prime}\leq b$. Following Jónsson [15] a large partial $n$-frame $\Phi$ of $L$ is given by elements of $L$: $a_{i}=a_{ii}$ $(0\leq i<m)$, and $a_{0i}$, $0<i<m$, where $m\geq n$, such that $1=\sum_{i=0}^{m-1}a_{i}$, $a_{0}\neq 0$, $\sum_{i=0}^{n-1}a_{i}=\bigoplus_{i=0}^{n-1}a_{i}$, and $a_{i}\sim_{a_{0i}}b_{i}$ for some $b_{i}\leq a_{0}$ for $0<i<m$. Moreover, for $0<i<n$ one requires $b_{i}=a_{0}$. $\Phi$ is a skew $n$-$m$-frame if, in addition, $1=\bigoplus_{i=0}^{m-1}a_{i}$. Observe that, given such $\Phi$, $m^{\prime}\leq m$, $n^{\prime}\leq n$, and $n^{\prime}\leq m^{\prime}$, the $a_{i},a_{0i}$ with $i<m^{\prime}$ form a skew $n^{\prime}$-$m^{\prime}$-frame in the interval $[0,\sum_{i=0}^{m^{\prime}-1}a_{i}]$. $\Phi$ is a skew $n$-frame if it is a skew $n$-$m$-frame for some $m$. From [15, Theorem 1.7] and [8, Proposition 6.2] one obtains the following

Fact 2.2**.**

Every simple complemented modular lattice of height at least $n$ admits some large partial $n$-frame. Every complemented modular lattice admitting a large partial $n$-frame also admits a skew $n$-frame.

In particular this applies to $\operatorname{{\sf L}}(R)$, $R$ a simple regular ring, due to the following result of Fred Wehrung [21, Theorem 4.3].

Fact 2.3**.**

For a regular ring $R$, the lattice of all congruence relations of $\operatorname{{\sf L}}(R)$ is ismorphic to the lattice of ideals of $R$.

In presence of a skew $n$-frame, coordinatization, if possible, is unique due to the following result of Jónsson, cf. [8, Theorem 11.2].

Fact 2.4**.**

For regular rings $R,R^{\prime}$, if $\operatorname{{\sf L}}(R)$ admits a skew $n$-frame, $n\geq 3$, then for any isomorphism $\theta:\operatorname{{\sf L}}(R)\to\operatorname{{\sf L}}(R^{\prime})$ there is an isomorphism $\iota:R\to R^{\prime}$ such that $\theta(aR)=\iota(a)R^{\prime}$ for all $a\in R$.

The approach of [8] to coordinatization relied on the following, combining Theorem 7.1 and Corollary 9.2 in [8].

Fact 2.5**.**

For any vector space $V$, complemented sublattice $L$ of $\operatorname{{\sf L}}(V)$, and skew $n$-frame $\Phi$ in $L$, $n\geq 3$, there is a regular subring $R_{0}$ of $\operatorname{{\sf End}}(V)$ and an isomorphism $\omega:\operatorname{{\sf L}}(R_{0})\to L$ such that $\omega(\varphi R_{0})=\operatorname{{\sf im}}\varphi$ for all $\varphi\in R_{0}$.

Now, we are in position to derive a representation of $R$ from a representation of $\operatorname{{\sf L}}(R)$.

Proposition 2.6**.**

Given a regular ring $R$, a skew $n$-frame $\Phi$, $n\geq 3$, in $\operatorname{{\sf L}}(R)$, a vector space $V$, and an embedding $\eta:\operatorname{{\sf L}}(R)\to\operatorname{{\sf L}}(V)$, there is an embedding $\iota:R\to\operatorname{{\sf End}}(V)$ such that $\eta(aR)=\operatorname{{\sf im}}\iota(a)$ for all $a\in R$.

Proof.

Let $L$ denote the sublattice $\eta(\operatorname{{\sf L}}(R))$ of $\operatorname{{\sf L}}(V)$. With $R_{0}$ and $\omega$ according to Fact 2.5 one obtains an isomorphism $\omega^{-1}\circ\eta:\operatorname{{\sf L}}(R)\to\operatorname{{\sf L}}(R_{0})$. By Fact 2.4 there is an isomorphism $\iota:R\to R_{0}$ such that $(\omega^{-1}\circ\eta)(aR)=\iota(a)R_{0}$ for all $a\in R$. It follows that $\eta(aR)=\omega(\iota(a)R)=\operatorname{{\sf im}}\iota(a)$ for all $a\in R$. ∎

3. $*$-regular rings and inner product spaces

A $*$-ring is a ring $R$ endowed with an involution $r\mapsto r^{*}$. Such $R$ is $*$-regular if it is regular and $rr^{*}=0$ only for $r=0$. A projection is an idempotent $e$ such that $e=e^{*}$; we write $e\in P(R)$. A $*$-ring is $*$-regular if and only if for any $a\in R$ there is $e\in P(R)$ with $aR=eR$; such $e$ is unique. In particular, for $*$-regular $R$, each ideal is closed under the involution. It follows

Fact 3.1**.**

A $*$-regular ring is simple (subdirectly irreducible) if and only if so is its ring reduct.

For a $*$-ring $R$ and projection $e\in R$. the corner $eRe$ is the $*$-ring consisting of all $eae$, $a\in R$, with unit $e$ and operations inherited from $R$, otherwise. The following is Lemma 2 together with Theorem 3 in [10].

Fact 3.2**.**

Given a subdirectly irreducible $*$-regular ring $R$ with minimal ideal $I$ and a projection $e$ in $I$, the $*$-ring $eRe$ is $*$-regular. Moreover, $R$ is a homomorphic image of a $*$-regular sub-$*$-ring of an ultraproduct of $*$-rings $eRe$, where $e$ ranges over the set of projections in $I$.

By an inner product space $V$ we will mean a vector space (also denoted by $V$) over a division $*$-ring $F$, endowed with a sesqui-linear form $\langle.\mid.\rangle$ which is anisotropic ($\langle v\mid v\rangle=0$ only for $v=0$) and orthosymmetric ( $\langle v\mid w\rangle=0$ if and only if $\langle w\mid v\rangle=0$). The orthogonal of a subset $X$ is the subspace $X^{\perp}=\{y\in V\mid\forall x\in X.\,\langle x\mid y\rangle=0\}$. For subspaces $U,W$ of $V$ we write $U\perp W$ if $W\subseteq U^{\perp}$; in this case we write $U+W=U\oplus^{\perp}W$. A subspace $U$ is closed if $U^{\perp\perp}=U$; equivalently, $V=U\oplus^{\perp}W$ for some $W$. Here, $W=U^{\perp}$ and one has the orthogonal projection $\pi_{U}$ where $\pi_{U}(x+y)=x$ for $x\in U$ and $y\in U^{\perp}$. Let $\operatorname{{\sf End}}^{*}(V)$ denote the $*$-ring consisting of those endomorphisms $\varphi$ of the vector space $V$ which have an adjoint $\varphi^{*}$ w.r.t. $\langle.\mid.\rangle$. If $\varphi$ is a projection in $\operatorname{{\sf End}}^{*}(V)$ then $V=\operatorname{{\sf im}}\varphi\oplus^{\perp}\operatorname{{\sf im}}({\sf id}_{V}-\varphi)$. It follows

Fact 3.3**.**

An endomorphism $\varphi$ of $V$ is a projection in $\operatorname{{\sf End}}^{*}(V)$ if and only if $\varphi=\pi_{U}$, where $U=\operatorname{{\sf im}}\varphi$.

A representation of a $*$-ring $R$ within $V$ is an embedding of $R$ into $\operatorname{{\sf End}}^{*}(V)$. Of course, any representation $\iota$ of a $*$-ring $R$ within $V$ gives rise to representations of corners $eRe$ within $\operatorname{{\sf im}}\iota(e)$.

Inner product spaces will be considered as $2$-sorted structures with sorts $V$ and $F$. In particular, the class of inner product spaces is closed under formation of ultraproducts. In this setting, representations of $*$-rings $R$ can be viewed as $3$-sorted structures (with third sort $R$), again forming a class closed under ultraproducts [13, Proposition 13]. On the other hand, a representation of $R$ in $V$ gives rise to representations of homomorphic images of $R$ in closed subspaces of certain ultrapowers of $V$ [13, Proposition 25]. It follows

Fact 3.4**.**

In the context of Fact 3.2, if each $*$-ring $eRe$ admits a representation within some inner product space $V_{e}$ for each projection $e\in I$, then the $*$-ring $R$ admits a representation within a closed subspace of an ultraproduct of spaces $V_{e}$, where $e$ ranges over the set of projections in $I$.

4. Modular ortholattices

An ortholattice is a lattice $L$ together with an order reversing involution $a\mapsto a^{\perp}$ such that $1=a\oplus a^{\perp}$. Elements $a,b$ are orthogonal to each other, $a\perp b$, if $b\leq a^{\perp}$; this then implies $a\cap b=0$ and we write $c=a\oplus^{\perp}b$ if $c=a+b$. If $L$ is modular and $u\in L$, then the section $[0,u]$ is again an ortholattice under $a\mapsto u\cap a^{\perp}$; that is, $a,b\leq u$ are orthogonal in $[0,u]$ if and only if they are so in $L$. Also, if $L$ is modular and $a\leq b$ then each of the quotients $b/a$, $(b\cap a^{\perp})/0$, and $a^{\perp}/b^{\perp}$ generate the same lattice congruence. It follows

Fact 4.1**.**

In a modular ortholattice, any lattice congruence is also a congruence w.r.t. the operation $a\mapsto a^{\perp}$.

The notion of skew frame can be adapted to the ortholattice setting requiring the $a_{i}$ to be pairwise orthogonal, see Niemann [18]. A weaker version will suffice, here. We write $a\sim^{\perp}b$ if $a\perp b$ and $a\sim b$. An orthogonal semiframe in an ortholattice $L$ consists of elements $a_{0},\ldots,a_{k-1}$ such that $1=\bigoplus_{i=0}^{k-1}a_{i}$ and for each $a_{i}$ there is $b_{i}\sim^{\perp}a_{i}$.

Lemma 4.2**.**

Every modular ortholattice $L$ admitting some skew $2$-$m$-frame also admits an orthogonal semiframe. In particular, any simple $L$ of height at least $2$ admits an orthogonal semiframe.

Proof.

We first observe that the following hold in any modular ortholattice.

• (1)

If $v\oplus b=1$ and $v^{\perp}\cap b=0$ then $v^{\perp}\sim^{\perp}v^{\prime}$ for some $v^{\prime}\leq v$.

• (2)

Assume $u\oplus a=1$ and $a\sim^{\perp}a^{\prime}$ for some $a^{\prime}\leq u$. Then there are $d,f$ such that $1=u\oplus^{\perp}d\oplus^{\perp}f$, $d\sim^{\perp}d^{\prime}$ and $e\sim^{\perp}e^{\prime}$ for some $d^{\prime}\leq u$ and $e^{\prime}\leq u+d$.

(1) follows from $v^{\perp}\sim_{b}v\cap(v^{\perp}+b)$. To prove (2), put $d:=a\cap u^{\perp}$ and $v:=u+d$. Then $v=u\oplus^{\perp}d$ and $d\sim^{\perp}d^{\prime}$ for some $d^{\prime}\leq a^{\prime}\leq u$. Moreover, $a\cap v^{\perp}=a\cap d^{\perp}\cap u^{\perp}=d\cap d^{\perp}=0$. Now, put $b:=a\cap d^{\perp}$, the orthocomplement of $d$ in the ortholattice $[0,a]$; thus, $b\oplus d=a$ and $v\oplus b=1$. On the other hand, from $b\leq a$ it follows $b\cap v^{\perp}=0$. Now, $1=u\oplus d\oplus v^{\perp}$ and (2) follows applying (1).

Finally, observe that (2) deals with the case $m=2$ as well with the inductive step from $m-1$ to $m$. The second claim follows from Facts 2.2 and 4.1 ∎

A representation of an ortholattice $L$ in an inner product space $V$ is an embedding $\eta$ of the lattice $L$ into $\operatorname{{\sf L}}(V)$ such that $\eta(a^{\perp})=\eta(a)^{\perp}$ for all $a\in L$. In particular, $\eta(L)$ is a modular sub-ortholattice of the (in general, non-modular) lattice of all closed subspaces of $V$.

Fact 4.3**.**

Given a $*$-regular ring $R$, the lattice $\operatorname{{\sf L}}(R)$ expands to an ortholattice $\operatorname{{\sf L}}^{\perp}(R)$ defining $(aR)^{\perp}=(1-e)R$ where $e\in P(R)$ such that $aR=eR$. In particular, for $e,f\in P(R)$ one has $eR\subseteq fR$ if and only if $fe=e$.

For $e,f\in P(R)$, we write $e\perp f$ if $eR\perp fR$; that is, $fe=0=ef$. Now, in view of Fact 3.3, Fact 2.1 transfers as follows.

Fact 4.4**.**

If $\iota$ is a representation of the $*$-regular ring $R$ in the inner product space $V$ then $\eta(aR)=\operatorname{{\sf im}}\iota(a)$, defines a representation of the ortholattice $\operatorname{{\sf L}}^{\perp}(R)$ in $V$.

In the presence of orthogonal semiframes, we will relate such representations the other way round.

5. Main Lemma

Lemma 5.1**.**

Given a $*$-regular ring $R$, an orthogonal semiframe $\Phi$ in $\operatorname{{\sf L}}^{\perp}(R)$, an inner product space $V$, and representation $\iota$ of the ring $R$in the vector space $V$, then $\iota$ is a representation of the $*$-ring $R$ within $V$, provided that $\eta:\operatorname{{\sf L}}^{\perp}(R)\to\operatorname{{\sf L}}(V)$, $\eta(aR):=\operatorname{{\sf im}}\iota(a)$, $a\in R$, defines an ortholattice representation in the inner product space $V$.

Recall that $\eta(aR)=\eta(bR)$ if $aR=bR$ (and we may write $\eta(a):=\eta(aR)$) and that $\eta$ is a lattice representation in view of Fact 2.1. Thus, the point is to show $\iota(a^{*})=\iota(a)^{*}$ for all $a\in R$ using the fact that $\eta$ preserves orthogonality. For the remainder of this section we assume the hypotheses of the Lemma.

Claim 5.2**.**

Consider closed subspaces $U,W$ of $V$ such that $U\perp W$ and $\varphi,\psi\in\operatorname{{\sf End}}(V)$ such that $\varphi=\pi_{W}\varphi\pi_{U}$ and $\psi=\pi_{U}\psi\pi_{W}$. Then $\psi=\varphi^{*}$ if and only if $\operatorname{{\sf im}}(\pi_{U}-\varphi)\perp\operatorname{{\sf im}}(\pi_{W}+\psi)$.

Proof.

This follows immediately since for all $v,w\in V$ one has

[TABLE]

∎

Claim 5.3**.**

If $e\perp f$ in $P(R)$ and $a\in fRe$ then $\iota(a^{*})=\iota(a)^{*}$.

Proof.

Put $b=a^{*}$. Then $b\in eRf$ and $(e-a)^{*}(f+b)=0$, that is $(e-a)R\perp(f+b)R$. It follows $\eta(e-a)\perp\eta(f+b)$. Now, $\eta(e-a)=\operatorname{{\sf im}}\iota(e-a)=\operatorname{{\sf im}}(\iota(e)-\iota(a))$ and $\eta(f+b)=\operatorname{{\sf im}}(\iota(f)+\iota(b))$ and Claim 5.2 applies with $\varphi=\iota(a)$, $U=\operatorname{{\sf im}}\iota(e)$, $\psi=\iota(b)$, $W=\operatorname{{\sf im}}\iota(f)$. ∎

Claim 5.4**.**

If $eR\sim fR$ in $\operatorname{{\sf L}}^{\perp}(R)$ for idempotents $e,f\in R$ then there is $c\in fRe$ such that $cx=cy$ implies $x=y$ for all $x,y\in eRe$.

Proof.

Assume $eR\sim_{gR}fR$; then $\omega(x)=y\Leftrightarrow x-y\in gR$ defines an isomorphism $\omega:eR\to fR$ of right $R$-modules. Put $c=\omega(e)e$ and observe that $\omega(x)=\beta(ex)=\beta(e)x=cx$ for all $x\in eRe$. Thus, assuming $cx=cy$ for given $x,y\in eRe$ it follows $\omega(x)=cx=cy=\omega(y)$ whence $x=y$. ∎

Claim 5.5**.**

Consider closed subspaces $U\perp W$ of $V$ and $\varepsilon\in\operatorname{{\sf End}}^{*}(V)$ such that $\varepsilon\circ\xi=\varepsilon\circ\chi$ implies $\xi=\chi$ for all $\xi,\chi\in\pi_{U}\circ\operatorname{{\sf End}}^{*}(V)\circ\pi_{U}$. Then $\varphi^{*}=\psi$ provided that $\varphi,\psi\in\pi_{U}\circ\operatorname{{\sf End}}^{*}(V)\circ\pi_{U}$ and $(\varepsilon\circ\varphi)^{*}=\psi\circ\varepsilon^{*}$.

Proof.

From $\varphi^{*}\circ\varepsilon^{*}=(\varepsilon\circ\varphi)^{*}=\psi\circ\varepsilon^{*}$ it follows $\varepsilon\circ\varphi=\varepsilon\circ\psi^{*}$, whence $\varphi=\psi^{*}$ and $\varphi^{*}=\psi$. ∎

Claim 5.6**.**

Given $e,f$ as in Claim 5.4 such that $e\perp f$ one has $\iota(a^{*})=\iota(a)^{*}$ for all $a\in eRe$.

Proof.

We put $b=a^{*}$ and have to show $\iota(b)=\iota(a)^{*}$. Choose $c$ according to Claim 5.4. By Claim 5.3 one has $\iota(c^{*})=\iota(c)^{*}$ and $\iota((ca)^{*})=(\iota(ca))^{*}$ since $c,ca\in fRe$. It follows $\iota(b)\iota(c)^{*}=\iota(b)\iota(c^{*})=\iota(bc^{*})=\iota((ca)^{*})=(\iota(ca))^{*}=(\iota(c)\iota(a))^{*}$ whence $\iota(b)=\iota(a)^{*}$ applying Claim 5.5 with $U=\operatorname{{\sf im}}e$, $W=\operatorname{{\sf im}}f$, $\varphi=\iota(a)$, $\psi=\iota(b)$, and $\varepsilon=\iota(c)$. ∎

Proof.

of the Lemma. We fix an orthogonal semiframe $\Phi$ of $\operatorname{{\sf L}}^{\perp}(R)$, that is pairwise orthogonal projections $e_{i}$, $0\leq i<k$, such that $\bigoplus_{i=0}^{k-1}e_{i}R=R$ and for each $i<k$ there are $f_{i},g_{i}\in P(R)$ with $e_{i}R\sim^{\perp}f_{i}R$. By Claims 5.3 and 5.6 one has $\iota(a^{*})=\iota(a)^{*}$ for all $a\in e_{j}Re_{i}$, $i,j<k$.

Now, $e_{i}e_{j}=0$ for $i\neq j$ since $e_{i}\perp e_{j}$. Thus $e=\sum_{i0}^{k-1}e_{i}$ is a projection and $eR=R$ whence $e=1$ by uniqueness. It follows, for each $a\in R$, that $a=\sum_{i,j=0}^{k-1}e_{j}ae_{i}$ and $a^{*}=\sum_{i,j=0}^{k-1}e_{j}a^{*}e_{i}$. Thus $\iota(a)^{*}=(\sum_{i,j=0}^{k-1}\iota(e_{j}ae_{i}))^{*}=\sum_{i,j=0}^{k-1}(\iota(e_{j}ae_{i}))^{*}=\sum_{i,j=0}^{k-1}\iota((e_{j}ae_{i})^{*})=\iota(\sum_{i,j=0}^{k-1}(e_{j}ae_{i})^{*})=\iota(\sum_{i,j=0}^{k-1}e_{i}a^{*}e_{j})=\iota(a^{*})$. ∎

6. Results

Facts 3.1, 2.3, and 2.2 yield the following.

Fact 6.1**.**

A simple $*$-regular ring admits a large partial $n$-frame if $\operatorname{{\sf L}}(R)$ is of height at least $n$.

Theorem 6.2**.**

Given a $*$-regular ring $R$ such that its ortholattice $\operatorname{{\sf L}}^{\perp}(R)$ of projections admits a large partial $n$-frame, $n\geq 3$, and a representation $\eta$ within some inner product space $V$. Then there is a representation $\iota$ of the $*$-ring $R$ within $V$ such that $\eta(a)=\operatorname{{\sf im}}\iota(a)$ for all $a\in R$.

Proof.

By Fact 2.2 one has a skew $n$-frame, $n\geq 3$ and so Proposition 2.6 provides the ring embedding $\iota:R\to\operatorname{{\sf End}}(V)$ such that $\eta(a)=\operatorname{{\sf im}}\iota(a)$. Now, by Lemma 4.2 there is an orthogonal semiframe and Lemma 5.1 shows that $\iota$ is a representation of the $*$-ring $R$. ∎

Corollary 6.3**.**

Consider a subdirectly irreducible $*$-regular ring $R$ such that $\operatorname{{\sf L}}^{\perp}(R)$ is of height at least $3$ and has a representation in the inner product space $V$. Then the $*$-ring $R$ has a representation within a closed subspace of some ultrapower of $V$.

Proof.

Let $P$ denote the set of projections $e$ in the minimal ideal $I$ of $R$ such that $\operatorname{{\sf L}}^{\perp}(eRe)$ has height at least $3$. Observe that for any projection $f\in I$, $fRf$ embeds into $eRe$ for some $e\in P$. Also, for $e\in P$, $\operatorname{{\sf L}}^{\perp}(eRe)$ is a section of $\operatorname{{\sf L}}^{\perp}(R)$ and any representation $\eta$ of $\operatorname{{\sf L}}^{\perp}(R)$ in some inner product space $V$ restricts to a representation of $\operatorname{{\sf L}}^{\perp}(eRe)$ in a closed subspace $V_{e}$ of $V$. By Lemma 5.1 one obtains a representation of the $*$-ring $eRe$ within $V_{e}$ for each $e\in P$. By Fact 3.4 this gives rise to a representation of $R$ in an ultraproduct of the $V_{e}$, that is a closed subspace of an ultrapower of $V$. ∎

Let $\mathcal{MOL}$ and $\mathcal{MOL}_{art}$ denote the ortholattice varieties generated by all respectively all finite height modular ortholattices.

Corollary 6.4**.**

A $*$-regular ring $R$ is a subdirect product of representables if and only if $\operatorname{{\sf L}}^{\perp}(R)\in\mathcal{MOL}_{art}$.

Proof.

Consider a homomorphism $\iota_{k}$ of $R$ onto $S_{k}$. Then $S_{k}$ is also $*$-regular and $\iota_{k}$ induces a homomorphism $\eta_{k}$ of $\operatorname{{\sf L}}^{\perp}(R)$ onto $\operatorname{{\sf L}}^{\perp}(S_{k})$ given by $\eta_{k}(eR)=\iota_{k}(e)S_{k}$, see Proposition 5.4(iv) [14]. Moreover, the $\iota_{k}$ yield a subdirect decomposition if and only if so do the $\eta_{k}$.

Thus, it suffices to consider subdirectly irreducible $R$; that is, subdirectly irreducible $\operatorname{{\sf L}}^{\perp}(R)$. If $R$ is representable then, by Fact 2.3, $\operatorname{{\sf L}}^{\perp}(R)$ is representable, too, and so in the variety generated by subspace ortholattices of finite dimensional inner product spaces by [14, Theorem 10.1], whence in $\mathcal{MOL}_{art}$. For the converse, we may assume that $\operatorname{{\sf L}}^{\perp}(R)$ is of height at least $4$, since otherwise $R$ is simple artinian whence representable. Now any finite height modular ortholattice is a direct product of simple ones and, by Jónsson’s Lemma, $\operatorname{{\sf L}}^{\perp}(R)$ is is a homomorphic image of a sub-ortholattice of an ultraproduct of such $L_{i}$. Since $\operatorname{{\sf L}}^{\perp}(R)$ contains a 5-element chain, the ultraproduct may be restricted to be formed from the $L_{i}$ of height at least $4$. Such are representable whence, by Lemma 8.3 and Corollary 8.6 in [14], so is $\operatorname{{\sf L}}^{\perp}(R)$. The claim follows by Corollary 6.3. ∎

Corollary 6.5**.**

$\mathcal{MOL}=\mathcal{MOL}_{art}$* if and only if every subdirectly irreducible $*$-regular ring is representable.*

Proof.

Assume that every subdirectly irreducible $*$-regular ring is representable. According to [12] it suffices to show $L\in\mathcal{MOL}_{art}$ for each simple $L\in\mathcal{MOL}$. Of course, we have to consider $L$ of infinite height, only. Such $L$ is coordinatizable, that is $L\cong\operatorname{{\sf L}}^{\perp}(R)$ for some $*$-regular ring $R$ (cf. [17, Section 4.3] respectively Fact 2.2 and Theorems 11.2 and 13.2 in [8]). Thus, by Fact 2.3, $L$ is representable, whence in $\mathcal{MOL}_{art}$ cf. [14, Theorem 10.1]. The other direction is immediate by Corollary 6.4. ∎

In case of $*$-regular rings $R$ without unit, $R$ is the directed union of the $eRe$, $e$ a projection in $R$, and $\operatorname{{\sf L}}^{\perp}(R)$ the directed union of the $\operatorname{{\sf L}}^{\perp}(eRe)$. The latter is a modular sectional ortholattice $L$, a modular lattice with [math] and a binary operation $(a,u)\mapsto a^{\perp_{u}}$ such that $a\mapsto a^{\perp_{u}}$ is an orthocomplementation on $[0,u]$ and $a^{\perp_{v}}=a^{\perp_{u}}\cap v$ if $v\leq u$. A representation of such $L$ is given by an inner product space $V$ and an embedding $\eta$ of the lattice $L$ with [math] into $\operatorname{{\sf L}}(V)$ such that, for each $u\in L$, $\eta(u)$ is closed in $V$ and the restriction of $\eta$ is a representation of the ortholattice $[0,u]$ within $\eta(u)$.

Corollary 6.6**.**

Corollaries 6.3 and 6.4 hold for $*$-regular rings without unit, analogously.