Return to Answer

First note that it is sufficient to reconstruct the $*$-ring $M$ of "affiliated locally measurable operators" (defined by Berberian and Saito using "strongly dense domains" in $L$, but algebraically it is the ring of classical quotients of $A$); in fact, $A$ is the subring of $M$ generated by its projections (or also the $*$-subring of bounded elements in the algebraic sense first used by von Neumann).

 

Then $M$, being a direct product of matrix rings of order at least 3, is generated as a ring by its idempotents $e,f,\dots$ using (besides idempotence) the relations given by a restriction of the classical "circle operation" to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent when $fe=0$; moreover, these generators and relations depend only by lattice theory: idempotents are identified with complementary ordered pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of right ideals of $M$, and the partial circle operation becomes $(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$ (moreover, the join is an independent join and dually for the meet). (All this follows from the easy part of von Neumann's coordinatization, in any ring even without regularity conditions).

 

Finally: the above pairs $(K,I)$ and the circle partial operation on them only depends upon the lattice $L$ (which is the same for $A$ and $M$; it is the lattice associated to these Rickart rings): these are exactly the complementary and modular pairs in the lattice (by $O$-symmetry of such ortholattices, all known reasonable modularity conditions for pairs of elements are equivalent), with join and meet computed in $L$; lastly, the projections (as opposed to generic idempotents) are the pairs $(K,I)$ which are orthocomplementary (as opposed to only modular complementary) in $L$; the involution in $A$ is the only one that makes such projections (that ring generate $A$) self-adjoint (and then the involution is also unique on the classical quotient ring $M$).

First note that it is sufficient to reconstruct the $*$-ring $M$ of "affiliated locally measurable operators" (defined by Berberian and Saito using "strongly dense domains" in $L$, but algebraically it is the ring of classical quotients of $A$); in fact, $A$ is the subring of $M$ generated by its projections (or also the $*$-subring of bounded elements in the algebraic sense first used by von Neumann).

Then $M$, being a direct product of matrix rings of order at least 3, is generated as a ring by its idempotents $e,f,\dots$ using (besides idempotence) the relations given by a restriction of the classical "circle operation" to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent when $fe=0$; moreover, these generators and relations depend only by lattice theory: idempotents are identified with complementary ordered pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of right ideals of $M$, and the partial circle operation becomes $(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$ (moreover, the join is an independent join and dually for the meet). (All this follows from the easy part of von Neumann's coordinatization, in any ring even without regularity conditions).

Finally: the above pairs $(K,I)$ and the circle partial operation on them only depends upon the lattice $L$ (which is the same for $A$ and $M$; it is the lattice associated to these Rickart rings): these are exactly the complementary and modular pairs in the lattice (by $O$-symmetry of such ortholattices, all known reasonable modularity conditions for pairs of elements are equivalent), with join and meet computed in $L$; lastly, the projections (as opposed to generic idempotents) are the pairs $(K,I)$ which are orthocomplementary (as opposed to only modular complementary) in $L$; the involution in $A$ is the only one that makes such projections (that ring generate $A$) self-adjoint (and then the involution is also unique on the classical quotient ring $M$).

A characterization of projectioon ortholattices of von Neumann algebras
 (and more generally JBW algebras) with no type I$_2$ component was given
by Bunce and J.D.M. Wright, Comm. mat. Phys; on projecteuclid.org they
are euclid.cmp/1104114067 and euclid.cmp/1103941854

You obtain an answer since "factorial" and "type III" are expressed
in ortholattice terms (see the Loomis - Maeda dimension theory, in
particoular the last version [Maeda, 1961] where equidimensionality is
identified with lattice semi-projectivity).

As expected by professor Handelman, a big role in such a
characterization is played by (the faces of) the convex set of normal
states (exclusion of I$_2$ components is needed to use Gleason's theorem
to identifiy states with completely additive probability measures on the
projection ortholattice. It is also needed since, as well known from
projective geometry, not every (ortho)lattice automorphisms of a
projective (ortho)line (i.e. a arbitray permutaion of the points, or
half of them in the orthocomplemented case) is semilinearly induced. In
particoular, order two matrices over real, complex or quanternion
numbers all give the same projection ortholattice (as it trivially
happens also with order 1 matrices); by artificially restricting to the
complex case one has unicity, but only up to a noncanonical
isomorphism).

A much better (from the quantum logic point of view) characterization of
the projection ortholattices of (real or complex, always excluding type
I$_2$ cases) finite factors as "continuous geometries with transition
probability" is due to von Neumann (and then unfortunately well
forgotten by modern quantum logicians). Form this, two generalizations
are obtainable with standard methods: (1) to the decomposable case,
using Boolean valued analysis (a decomposable case is the same as
indecomposable object of a boolean valued universe); (2) to the
semifinite (instead of finite) case, using the fact that in the
semifinite case the join-dense ideal of finite elements completely
determines $L$ (a standard method "to adjoin 1 to a generalized
orthomodular lattice", due to Janowitz, produces the lattice of all
finite and cofinite elements; then the Dedekind completion produces
$L$).

[Digression. In particoular, this gives a characteriazion of Hilbertian
logics (of type I factors) that is physically much better than the
characterization that modern quantum logicians deduce from Soler's
theorem (which is however mathematically wonderful). The modern theorem
must exclude all finite dimensional factors (why a finite dimensional
irreducibly quantum logic should be automatically embeddable in a
infinite dimensional one? von Neumann's method instead excludes only the
"spin factors", which are not really quantum since they are the only
factors with nonclassical logic but with "noncontextual hidden variables",
and the nonarguesian planes, which cannot be embedded in any larger
logic except by direct product, which means that these exceptional
components can have only classical, not quantum, interactions with the
other components) and must presuppose together a complete lattice and
orthomodularity without physical reasons (orthomodularity is justified
by restricting only to certain propositions, and the "complete
lattice" property is justified by enlarging using completions, like
Dedekind completions. Unfortunately this only produces two possibly
different structures, a restricted orthomodular one and a a larger
complete lattice; almost no known mathematical theorem produces
automatically a orthomodular completion. The only exception is precisely
von Neumann's method when applied to type I cases (and, analogoulsy, the metric completion of pre-Hilbert spaces): the only completeness
axiom which is not trivially satisfied in the finite dimensional case is
used only in the last step, to show that an already constructed
Hilbertian representation is surjective; so, were this last completeness
axiom not satisfied, one can always take as completion the bicommutant
of the algebra in the Hilbertian representation: one has proved that a
completion exists, a conceptual case analogous to the well known proof
that, assuming the archimedean axiom for the measures of physical
quantities, then one can assume that the measures are real numbers: the
archimedean axiom, involving two magnitudes and a simple arithmetic
progression, is experimentally falsifiable at least ideally, but the
completeness axions for real numbers, with arbitrary infinite sets, is
physically hopeless). End digression.]

I know no attempts to concretize the details of a last, third step in
the extensions of von Neumann's characterization: using Tomita -
Takesaki modular theory to obtain a generic type III factor starting
from a type II infinite factor with a suitable automorphism, one has
that, in principle, the projection ortholattice $L$ of a type III
factor, being equivalent to the factor itself, is somehow obtained from
a type II factor with a given automorphism, which is equivalent to a
projection ortholattice, with fixed automorphism, of a type II factor. I
hope that someone one day will write down the deatails of this method.

The possibility of characterization of complex AW$^*$-algebras with no
type I$_2$ components by their projection otholattices follows from's
Dye's theorem: each projection ortholattice isomorphism among them
extends to one and only one (necessarilly real linear) $*$-ring
isomorphism (or equivalently a unique complex linear Jordan isomorphism;
however, since there are type II finite factors not anti-isomorphic to
themselves, there are cases where a complex linear $*$-ring isomorphism
is impossible).

Dye proved his theorem in 1955 (on jstor.org it is number 1969620) for
von Neumann algebras, but Yen in 1957 (proc. ams. S0002-9939-1957-0084123-X ) and Berberian in 1982 (on projecteuclid.org
see euclid.rmjm/1250128413 ) remarked that the proof works also for
AW$^*$-algebras. [Recent interest in Dye's theorem appares in C.
Heunen, M. L. Reyes pdf/1212.5778 on arxiv.org; these authors seem
unaware of the concept of orthosymetric otholattices introduced by
Mayet]

Really, the theorem (but not Dye's own proof) also holds for real AW$^*$
algebras with no abelian or type I$_2$ component (or even more generally
for Rickart real $C^*$-algebras of matrix order at least 3 and
 $C^*$-direct sums of such algebras). An explicit reconstruction of the
Rickart $C^*$-algebra $A$ from its projection ortholattice $L$ is the
following:

first note that it is sufficient to recontruct the $*$-ring
$M$ of "affiliated locally measurable operators" (defined by Berberian
and Saito using "strongly dense domains" in $L$, but algebraically it
is the ring of classical quotients of $A$); infact, $A$ is the subring
of $M$ generated by its projections (or also the $*$-subring of bounded
elements in the algebraic sense first used by von Neumann).

Then $M$,
being a direct product of matrix rings of order at least 3, is generated
as ring by its idempotents $e,f,\dots$ using (besides idempotence) the
relations given by a restriction of the classical ``circle operation''
to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent
when $fe=0$; moreover, these generators and relations depend only by
lattice theory: idempotents are identified with complementary ordered
pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of
right ideals of $M$, and the partial circle operation becomes
$(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$
(moreover, the join is a independent join and dually for the meet). (All
this follows from the easy part of von Neumann's coordinatization, in
any ring even without regularity conditions).

Finally: the above pairs
$(K,I)$ and the circle partial operation on them only depends upon the
lattice $L$ (which is the same for $A$ and $M$; it is the lattice
associated to these Rickart rings): these are exactly the complementary
and modular pairs in the lattice (by $O$-symmetry of such ortholattices,
all known reasonable modularity conditions for pairs of elements are
equivalent), with join and meet computed in $L$; lastly, the projections
(as opposed to generic idempotents) are the pairs $(K,I)$ which are
orthocomplementary (as opposed to only modular complementary) in $L$;
the involution in $A$ is the only one that makes such projections (that
ring generate $A$) self-adjoint (and then the involution is also unique
on the classical quotient ring $M$).

Note that $A$ contains only some of
the idempotents of $M$; precisely, the idempotents corresponding to
pairs $(K,I)$ which are "nonasymptotic" (for this classical concept
see Topping, Bures [with improvements by S. Maeda in the interaction
with lattice theory], and more recently M. Anoussis, A. Katavolos, I. G.
Todorov math/0601003v2 on arxiv.org). In the von Neumann algebra case, a
 (external) lattice description of "nonasimptoticy" is "absolute
modularity": for one normal embedding of $L$ in a Hilbert lattice (as
its own bicommutant; note that the commuting of projections is
ortholattice definable), the pair is modular in the larger lattice (then
the same happens for each normal embedding of $L$ in any projection
ortholattice of a von Neumann algebra).

[Since the ortholattice $L$
determines $A$, it also determines the orthosymmetric structure that $A$
defines on $L$; really, Mayet's othosymmetric structure on $L$ is unique
since for each element $e$ of $L$ there is only one involutory
automorphisms of $L$ i.e. of $A$ that fixes exactly the projections that
commute with $e$ (see for example lemma 2.4 in euclid.cmp/1103859692 on projecteuclid.org ; it is sufficient the even more folklore case of
factors: then apply a subdirect decomposition into factors for the
general case)]

A characterization of projection ortholattices of von Neumann algebras  (and more generally JBW algebras) with no type I$_2$ component was given by Bunce and J.D.M. Wright in two papers: [1] and [2] (full references at end).

You obtain an answer since "factorial" and "type III" are expressed in ortholattice terms (see the Loomis - Maeda dimension theory, in particular the last version [3] where equidimensionality is identified with lattice semi-projectivity).

As expected by professor Handelman, a big role in such a characterization is played by (the faces of) the convex set of normal states (exclusion of I$_2$ components is needed to use Gleason's theorem to identify states with completely additive probability measures on the projection ortholattice. It is also needed since, as well known from projective geometry, not every (ortho)lattice automorphism of a projective (ortho)line (i.e. a arbitrary permutation of the points, or half of them in the orthocomplemented case) is semilinearly induced. In particular, order two matrices over real, complex or quaternion numbers all give the same projection ortholattice (as it trivially happens also with order 1 matrices); by artificially restricting to the complex case one has unicity, but only up to a noncanonical isomorphism).

A much better (from the quantum logic point of view) characterization of the projection ortholattices of (real or complex, always excluding type I$_2$ cases) finite factors as "continuous geometries with transition probability" is due to von Neumann [4] (and then unfortunately well forgotten by modern quantum logicians). From this, two generalizations are obtainable with standard methods:

1. to the decomposable case, using Boolean valued analysis (a decomposable case is the same as indecomposable object of a boolean valued universe);

2. to the semifinite (instead of finite) case, using the fact that in the semifinite case the join-dense ideal of finite elements completely determines $L$ (a standard method "to adjoin 1 to a generalized orthomodular lattice", due to Janowitz, produces the lattice of all finite and cofinite elements; then the Dedekind completion produces $L$).

[Digression. In particular, this gives a characteriazion of Hilbertian logics (of type I factors) that is physically much better than the characterization that modern quantum logicians deduce from Soler's theorem (which is however mathematically wonderful). The modern theorem must exclude all finite dimensional factors (why a finite dimensional irreducibly quantum logic should be automatically embeddable in a infinite dimensional one? von Neumann's method instead excludes only the "spin factors", which are not really quantum since they are the only factors with nonclassical logic but with "noncontextual hidden variables", and the nonarguesian planes, which cannot be embedded in any larger logic except by direct product, which means that these exceptional components can have only classical, not quantum, interactions with the other components) and must presuppose together a complete lattice and orthomodularity without physical reasons (orthomodularity is justified by restricting only to certain propositions, and the "complete lattice" property is justified by enlarging using completions, like Dedekind completions. Unfortunately this only produces two possibly different structures, a restricted orthomodular one and a a larger complete lattice; almost no known mathematical theorem produces automatically a orthomodular completion. The only exception is precisely von Neumann's method when applied to type I cases (and, analogoulsy, the metric completion of pre-Hilbert spaces): the only completeness axiom which is not trivially satisfied in the finite dimensional case is used only in the last step, to show that an already constructed Hilbertian representation is surjective; so, were this last completeness axiom not satisfied, one can always take as completion the bicommutant of the algebra in the Hilbertian representation: one has proved that a completion exists, a conceptual case analogous to the well known proof that, assuming the archimedean axiom for the measures of physical quantities, then one can assume that the measures are real numbers: the archimedean axiom, involving two magnitudes and a simple arithmetic progression, is experimentally falsifiable at least ideally, but the completeness axioms for real numbers, with arbitrary infinite sets, is physically hopeless). End digression.]

I know no attempts to concretize the details of a last, third step in the extensions of von Neumann's characterization: using Tomita - Takesaki modular theory to obtain a generic type III factor starting from a type II infinite factor with a suitable automorphism, one has that, in principle, the projection ortholattice $L$ of a type III factor, being equivalent to the factor itself, is somehow obtained from a type II factor with a given automorphism, which is equivalent to a projection ortholattice, with fixed automorphism, of a type II factor. I hope that someone one day will write down the details of this method.

The possibility of characterization of complex AW$^*$-algebras with no type I$_2$ components by their projection otholattices follows from Dye's theorem [5]: each projection ortholattice isomorphism among them extends to one and only one (necessarilly real linear) $*$-ring isomorphism (or equivalently a unique complex linear Jordan isomorphism; however, since there are type II finite factors not anti-isomorphic to themselves, there are cases where a complex linear $*$-ring isomorphism is impossible).

Dye proved his theorem in 1955 for von Neumann algebras, but Yen in 1957 [6] and Berberian in 1982 [7] remarked that the proof works also for AW$^*$-algebras. (Recent interest in Dye's theorem appears in C. Heunen, M. L. Reyes [8]; these authors seem unaware of the concept of orthosymmetric ortholattices introduced by Mayet [9].)

Really, the theorem (but not Dye's own proof) also holds for real AW$^*$ algebras with no abelian or type I$_2$ component (or even more generally for Rickart real $C^*$-algebras of matrix order at least 3 and  $C^*$-direct sums of such algebras). An explicit reconstruction of the Rickart $C^*$-algebra $A$ from its projection ortholattice $L$ is the following:

First note that it is sufficient to reconstruct the $*$-ring $M$ of "affiliated locally measurable operators" (defined by Berberian and Saito using "strongly dense domains" in $L$, but algebraically it is the ring of classical quotients of $A$); in fact, $A$ is the subring of $M$ generated by its projections (or also the $*$-subring of bounded elements in the algebraic sense first used by von Neumann).

Then $M$, being a direct product of matrix rings of order at least 3, is generated as a ring by its idempotents $e,f,\dots$ using (besides idempotence) the relations given by a restriction of the classical "circle operation" to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent when $fe=0$; moreover, these generators and relations depend only by lattice theory: idempotents are identified with complementary ordered pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of right ideals of $M$, and the partial circle operation becomes $(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$ (moreover, the join is an independent join and dually for the meet). (All this follows from the easy part of von Neumann's coordinatization, in any ring even without regularity conditions).

Finally: the above pairs $(K,I)$ and the circle partial operation on them only depends upon the lattice $L$ (which is the same for $A$ and $M$; it is the lattice associated to these Rickart rings): these are exactly the complementary and modular pairs in the lattice (by $O$-symmetry of such ortholattices, all known reasonable modularity conditions for pairs of elements are equivalent), with join and meet computed in $L$; lastly, the projections (as opposed to generic idempotents) are the pairs $(K,I)$ which are orthocomplementary (as opposed to only modular complementary) in $L$; the involution in $A$ is the only one that makes such projections (that ring generate $A$) self-adjoint (and then the involution is also unique on the classical quotient ring $M$).

Note that $A$ contains only some of the idempotents of $M$; precisely, the idempotents corresponding to pairs $(K,I)$ which are "nonasymptotic" (for this classical concept see Topping, Bures [with improvements by S. Maeda in the interaction with lattice theory], and more recently M. Anoussis, A. Katavolos, I. G. Todorov [10]). In the von Neumann algebra case, a  (external) lattice description of "nonasymptoticy" is "absolute modularity": for one normal embedding of $L$ in a Hilbert lattice (as its own bicommutant; note that the commuting of projections is ortholattice definable), the pair is modular in the larger lattice (then the same happens for each normal embedding of $L$ in any projection ortholattice of a von Neumann algebra).

[Since the ortholattice $L$ determines $A$, it also determines the orthosymmetric structure that $A$ defines on $L$; really, Mayet's orthosymmetric structure on $L$ is unique since for each element $e$ of $L$ there is only one involutory automorphism of $L$ i.e. of $A$ that fixes exactly the projections that commute with $e$ (see for example lemma 2.4 in euclid.cmp/1103859692 on projecteuclid.org ; it is sufficient the even more folklore case of factors: then apply a subdirect decomposition into factors for the general case)]

Full references:

[1] Bunce, L. J.; Wright, J. D. Maitland. Quantum logics and convex geometry. Comm. Math. Phys. 101 (1985), no. 1, 87--96.

[2] Bunce, L. J.; Wright, J. D. Maitland. Quantum logic, state space geometry and operator algebras. Comm. Math. Phys. 96 (1984), no. 3, 345--348.

[3]Maeda, Shûichirô. Dimension theory on relatively semi-orthocomplemented complete lattices. J. Sci. Hiroshima Univ. Ser. A-I Math. 25 (1961), no. 2, 369--404. doi:10.32917/hmj/1206139804.

[4] Von Neumann, John. Continuous geometries with a transition probability. Vol. 252. American Mathematical Soc., 1981.

[5] Dye, H. A. “On the Geometry of Projections in Certain Operator Algebras.” Annals of Mathematics, vol. 61, no. 1, 1955, pp. 73–89. JSTOR

[6] Yen, Ti. Isomorphism of AW∗-algebras. Proc. Amer. Math. Soc. 8 (1957), 345–349.

[7] Berberian, S.K. The maximal ring of quotients of a finite Von Neumann algebra. Rocky Mountain J. Math. 12 (1982), no. 1, 149--164. doi:10.1216/RMJ-1982-12-1-149.

[8] Heunen, Chris, and Manuel L. Reyes. "Active lattices determine AW*-algebras." Journal of Mathematical Analysis and Applications 416.1 (2014): 289-313.

[9] Mayet, R. “Orthosymmetric Ortholattices.” Proceedings of the American Mathematical Society, vol. 114, no. 2, 1992, pp. 295–306. JSTOR

[10] Anoussis, M., Aristides Katavolos, and Ivan G. Todorov. "Angles in C*-algebras." arXiv preprint math/0601003 (2005).

You obtain an answer since factorial'' and type III'' are expressed
in ortholattice terms (see the Loomis - Maeda dimension theory, in
particoular the last version [Maeda, 1961] where equidimensionality is
identified with lattice semi-projectivity).

A much better (from the quantum logic point of view) characterization of
the projection ortholattices of (real or complex, always excluding type
I$_2$ cases) finite factors as continuous geometries with transition probability'' is due to von Neumann (and then unfortunately well forgotten by modern quantum logicians). Form this, two generalizations are obtainable with standard methods: (1) to the decomposable case, using Boolean valued analysis (a decomposable case is the same as indecomposable object of a boolean valued universe); (2) to the semifinite (instead of finite) case, using the fact that in the semifinite case the join-dense ideal of finite elements completely determines $L$ (a standard method to adjoin 1 to a generalized
orthomodular lattice'', due to Janowitz, produces the lattice of all
finite and cofinite elements; then the Dedekind completion produces
$L$).

[Digression. In particoular, this gives a characteriazion of Hilbertian
logics (of type I factors) that is physically much better than the
characterization that modern quantum logicians deduce from Soler's
theorem (which is however mathematically wonderful). The modern theorem
must exclude all finite dimensional factors (why a finite dimensional
irreducibly quantum logic should be automatically embeddable in a
infinite dimensional one? von Neumann's method instead excludes only the
spin factors'', which are not really quantum since they are the only factors with nonclassical logic with noncontextual hidden variables'',
and the nonarguesian planes, which cannot be embedded in any larger
logic except by direct product, which means that these exceptional
components can have only classical, not quantum, interactions with the
other components) and must presuppose together a complete lattice and
orthomodularity without physical reasons (orthomodularity is justified
by restricting only to certain propositions, and the "complete
lattice" property is justified by enlarging using completions, like
Dedekind completions. Unfortunately this only produces two possibly
different structures, a restricted orthomodular one and a a larger
complete lattice; almost no known mathematical theorem produces
automatically a orthomodular completion. The only exception is precisely
von Neumann's method when applied to type I cases: the only completeness
axiom which is not trivially satisfied in the finite dimensional case is
used only in the last step, to show that an already constructed
Hilbertian representation is surjective; so, were this last completeness
axiom not satisfied, one can always take as completion the bicommutant
of the algebra in the Hilbertian representation: one has proved that a
completion exists, a conceptual case analogous to the well known proof
that, assuming the archimedean axiom for the measures of physical
quantities, then one can assume that the measures are real numbers: the
archimedean axiom, involving two magnitudes and a simple arithmetic
progression, is experimentally falsifiable at least ideally, but the
completeness axions for real numbers, with arbitrary infinite sets, is
physically hopeless). End digression.]

Really, the theorem (but not Dye's own proof) also holds for real AW$^*$
algebras with no abelian or type I$_2$ component (or even more generally
for Rickart real $C^*$-algebras of matrix order at least 3 and
$C^*$-direct sums of such algebras). An explicit reconstruction of the
Rickart $C^*$-algebra $A$ from its projection ortholattice $L$ is the
following: first note that it is sufficient to recontruct the $*$-ring
$M$ of affiliated locally measurable operators'' (defined by Berberian and Saito using strongly dense domains'' in $L$, but algebraically it
is the ring of classical quotients of $A$); infact, $A$ is the subring
of $M$ generated by its projections (or also the $*$-subring of bounded
elements in the algebraic sense first used by von Neumann). Then $M$,
being a direct product of matrix rings of order at least 3, is generated
as ring by its idempotents $e,f,\dots$ using (besides idempotence) the
relations given by a restriction of the classical ``circle operation''
to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent
when $fe=0$; moreover, these generators and relations depend only by
lattice theory: idempotents are identified with complementary ordered
pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of
right ideals of $M$, and the partial circle operation becomes
$(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$
(moreover, the join is a independent join and dually for the meet). (All
this follows from the easy part of von Neumann's coordinatization, in
any ring even without regularity conditions). Finally: the above pairs
$(K,I)$ and the circle partial operation on them only depends upon the
lattice $L$ (which is the same for $A$ and $M$; it is the lattice
associated to these Rickart rings): these are exactly the complementary
and modular pairs in the lattice (by $O$-symmetry of such ortholattices,
all known reasonable modularity conditions for pairs of elements are
equivalent), with join and meet computed in $L$; lastly, the projections
(as opposed to generic idempotents) are the pairs $(K,I)$ which are
orthocomplementary (as opposed to only modular complementary) in $L$;
the involution in $A$ is the only one that makes such projections (that
ring generate $A$) self-adjoint (and then the involution is also unique
on the classical quotient ring $M$). Note that $A$ contains only some of
the idempotents of $M$; precisely, the idempotents corresponding to
pairs $(K,I)$ which are "nonasymptotic" (for this classical concept
see Topping, Bures [with improvements by S. Maeda in the interaction
with lattice theory], and more recently M. Anoussis, A. Katavolos, I. G.
Todorov math/0601003v2 on arxiv.org). In the von Neumann algebra case, a
(external) lattice description of "nonasimptoticy" is "absolute
modularity": for one normal embedding of $L$ in a Hilbert lattice (as
its own bicommutant; note that the commuting of projections is
ortholattice definable), the pair is modular in the larger lattice (then
the same happens for each normal embedding of $L$ in any projection
ortholattice of a von Neumann algebra). [Since the ortholattice $L$
determines $A$, it also determines the orthosymmetric structure that $A$
defines on $L$; really, Mayet's othosymmetric structure on $L$ is unique
since for each element $e$ of $L$ there is only one involutory
automorphisms of $L$ i.e. of $A$ that fixes exactly the projections that
commute with $e$ (see for example lemma 2.4 in euclid.cmp/1103859692 on projecteuclid.org ; it is sufficient the even more folklore case of
factors: then apply a subdirect decomposition into factors for the
general case)]

You obtain an answer since "factorial" and "type III" are expressed
in ortholattice terms (see the Loomis - Maeda dimension theory, in
particoular the last version [Maeda, 1961] where equidimensionality is
identified with lattice semi-projectivity).

A much better (from the quantum logic point of view) characterization of
the projection ortholattices of (real or complex, always excluding type
I$_2$ cases) finite factors as "continuous geometries with transition
probability" is due to von Neumann (and then unfortunately well
forgotten by modern quantum logicians). Form this, two generalizations
are obtainable with standard methods: (1) to the decomposable case,
using Boolean valued analysis (a decomposable case is the same as
indecomposable object of a boolean valued universe); (2) to the
semifinite (instead of finite) case, using the fact that in the
semifinite case the join-dense ideal of finite elements completely
determines $L$ (a standard method "to adjoin 1 to a generalized
orthomodular lattice", due to Janowitz, produces the lattice of all
finite and cofinite elements; then the Dedekind completion produces
$L$).

[Digression. In particoular, this gives a characteriazion of Hilbertian
logics (of type I factors) that is physically much better than the
characterization that modern quantum logicians deduce from Soler's
theorem (which is however mathematically wonderful). The modern theorem
must exclude all finite dimensional factors (why a finite dimensional
irreducibly quantum logic should be automatically embeddable in a
infinite dimensional one? von Neumann's method instead excludes only the
"spin factors", which are not really quantum since they are the only
factors with nonclassical logic but with "noncontextual hidden variables",
and the nonarguesian planes, which cannot be embedded in any larger
logic except by direct product, which means that these exceptional
components can have only classical, not quantum, interactions with the
other components) and must presuppose together a complete lattice and
orthomodularity without physical reasons (orthomodularity is justified
by restricting only to certain propositions, and the "complete
lattice" property is justified by enlarging using completions, like
Dedekind completions. Unfortunately this only produces two possibly
different structures, a restricted orthomodular one and a a larger
complete lattice; almost no known mathematical theorem produces
automatically a orthomodular completion. The only exception is precisely
von Neumann's method when applied to type I cases (and, analogoulsy, the metric completion of pre-Hilbert spaces): the only completeness
axiom which is not trivially satisfied in the finite dimensional case is
used only in the last step, to show that an already constructed
Hilbertian representation is surjective; so, were this last completeness
axiom not satisfied, one can always take as completion the bicommutant
of the algebra in the Hilbertian representation: one has proved that a
completion exists, a conceptual case analogous to the well known proof
that, assuming the archimedean axiom for the measures of physical
quantities, then one can assume that the measures are real numbers: the
archimedean axiom, involving two magnitudes and a simple arithmetic
progression, is experimentally falsifiable at least ideally, but the
completeness axions for real numbers, with arbitrary infinite sets, is
physically hopeless). End digression.]

Really, the theorem (but not Dye's own proof) also holds for real AW$^*$
algebras with no abelian or type I$_2$ component (or even more generally
for Rickart real $C^*$-algebras of matrix order at least 3 and
$C^*$-direct sums of such algebras). An explicit reconstruction of the
Rickart $C^*$-algebra $A$ from its projection ortholattice $L$ is the
following: 

first note that it is sufficient to recontruct the $*$-ring
$M$ of "affiliated locally measurable operators" (defined by Berberian
and Saito using "strongly dense domains" in $L$, but algebraically it
is the ring of classical quotients of $A$); infact, $A$ is the subring
of $M$ generated by its projections (or also the $*$-subring of bounded
elements in the algebraic sense first used by von Neumann). 

Then $M$,
being a direct product of matrix rings of order at least 3, is generated
as ring by its idempotents $e,f,\dots$ using (besides idempotence) the
relations given by a restriction of the classical ``circle operation''
to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent
when $fe=0$; moreover, these generators and relations depend only by
lattice theory: idempotents are identified with complementary ordered
pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of
right ideals of $M$, and the partial circle operation becomes
$(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$
(moreover, the join is a independent join and dually for the meet). (All
this follows from the easy part of von Neumann's coordinatization, in
any ring even without regularity conditions). 

Finally: the above pairs
$(K,I)$ and the circle partial operation on them only depends upon the
lattice $L$ (which is the same for $A$ and $M$; it is the lattice
associated to these Rickart rings): these are exactly the complementary
and modular pairs in the lattice (by $O$-symmetry of such ortholattices,
all known reasonable modularity conditions for pairs of elements are
equivalent), with join and meet computed in $L$; lastly, the projections
(as opposed to generic idempotents) are the pairs $(K,I)$ which are
orthocomplementary (as opposed to only modular complementary) in $L$;
the involution in $A$ is the only one that makes such projections (that
ring generate $A$) self-adjoint (and then the involution is also unique
on the classical quotient ring $M$). 

Note that $A$ contains only some of
the idempotents of $M$; precisely, the idempotents corresponding to
pairs $(K,I)$ which are "nonasymptotic" (for this classical concept
see Topping, Bures [with improvements by S. Maeda in the interaction
with lattice theory], and more recently M. Anoussis, A. Katavolos, I. G.
Todorov math/0601003v2 on arxiv.org). In the von Neumann algebra case, a
(external) lattice description of "nonasimptoticy" is "absolute
modularity": for one normal embedding of $L$ in a Hilbert lattice (as
its own bicommutant; note that the commuting of projections is
ortholattice definable), the pair is modular in the larger lattice (then
the same happens for each normal embedding of $L$ in any projection
ortholattice of a von Neumann algebra). 

[Since the ortholattice $L$
determines $A$, it also determines the orthosymmetric structure that $A$
defines on $L$; really, Mayet's othosymmetric structure on $L$ is unique
since for each element $e$ of $L$ there is only one involutory
automorphisms of $L$ i.e. of $A$ that fixes exactly the projections that
commute with $e$ (see for example lemma 2.4 in euclid.cmp/1103859692 on projecteuclid.org ; it is sufficient the even more folklore case of
factors: then apply a subdirect decomposition into factors for the
general case)]