A Uniﬁed Approach to the Decomposition Theorems in Baer ∗ -Rings

. The aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a uniﬁed approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold– S(cid:2)loci´nski decomposition is given in Baer ∗ -rings.


Preliminaries
In the recent papers [1,2] the authors noticed the important role of an algebraic structure in several results on decompositions in the algebra of bounded linear operators on Hilbert space. As a consequence they manage to generalize those results to Baer * -rings. We show a general decomposition theorem which yields the vast majority of decompositions with respect to hereditary properties. In particular, for the algebra of bounded Hilbert space operators they imply known decompositions as well as some new. Since our proofs are purely algebraic, the results are formulated in Baer * -rings.
if and only if x is p and 1 − p invariant. All the above is defined for a subset S ⊂ R by means that the respective condition holds for any x ∈ S.
Note that p decomposes S if and only if px = xp for any x ∈ S. In other words, p ∈ S (the commutant of S).
More generally we can consider a unity decomposition (factorization) {p i } n 1 ⊂R, i.e. a set of pairwise orthogonal projections such that n i=1 p i = 1. In Baer * -rings a unity decomposition may be infinite where ∞ i=1 p i := sup{p 1 + · · · + p i : i ∈ Z + }. A sequence of pairwise orthogonal projections then any x ∈ R may be decomposed as It is clear that only projections in the centre provide decompositions of R.

Theorem 2.1. The commutant of a * -subset of a Baer * -ring is a Baer *subring with unambiguous sups and infs.
In preliminaries we defined ∞ i=0 xp i := x ∞ i=0 p i for a set of pairwise orthogonal projections {p i }. However, if xp i are projections, then they are pairwise orthogonal and the left hand sum makes sense on its own. Hence, we need to check that the definition causes no ambiguity. It follows by Corollary 2.2(1) provided we check that if xp i are projections, then x is a projection commuting with all p i -s. Indeed, (xp i ) 2 = xp i = p i x * yields xp i = p i xp i and The next result follows from Theorem 2.1 and [2, equality (1)]. Precisely, [2, equality (1)] is showed for isometries, but it extends to projections.

Corollary 2.2. Infinite sums admit the following properties:
(1) If a projection p commutes with pairwise orthogonal projections Let us generalize a definition of a hereditary property, which is known in the algebra of Hilbert space operators, to Baer * -rings. The property may concern a single element like unitarity or several elements like commutativity. Definition 2.5. Let P denotes a property of elements in a Baer * -ring.
• The property P is called hereditary if for any {x k : k ∈ K} ⊂ R having the property P and any p ∈R ∩ {x k : k ∈ K} the set of compressions {px k : k ∈ K} have the property P relative to the corner ring pRp. • A set of elements {x k : k ∈ K} completely does not have the property P (is completely non-P) if and only if the only p ∈R ∩ {x k : k ∈ K} such that the set of compressions {px k : k ∈ K} have the property P relative to the corner ring pRp is p = 0.
In Definition 2.5 the property P for the set of compressions is considered in the corner pRp, not in R. Let us explain the difference on examples. Let u ∈ R be unitary, so uu * = u * u = 1. Let p be a projection commuting with u. Then (pu) * pu = pu(pu) * = p. Since p is a unity in the corner pRp the element pu is unitary relative to pRp. However, if p = 1 then pu is not unitary relative to R. Some properties means the same relative to R and pRp. Then the ambiguity appears in the second part of Definition 2.5. As an example, if x ∈ pRp is normal relative to pRp (i.e. xx * = x * x), then it is normal relative to R. On the other hand, let x ∈ pRp be completely non-normal relative to pRp. Thus 1 − p commutes with x and (1 − p)x = 0, so (1 − p)x is normal relative to (1 − p)R (1 − p). Hence x is not completely non-normal relative to R for p = 1.
Similarly like examples above, many properties may be defined by algebraic expressions. Let us introduce some description which may be used to the most of hereditary properties. Definition 2.6. Let it be given a Baer * -ring R. Let R n (where n is not necessarily finite) be an R-module with natural operations for any x ∈ R n and any q ∈R ∩ x , where x is the commutant of x viewed as the commutant of a subset of R. We say that x has the property P F relative to R if x ∈ i∈I ker F i .
The property P F should be viewed as a property of elements in R rather than a property of an element in R n . From (2.1) it is a hereditary property. Moreover, by the condition (2.1) functions F i preserve commutativity with q. The functions defined using algebraic expressions of elements in x, their adjoints or possibly operations of taking the left or the right projection satisfy (2.1). Now we state the main theorem of the paper. Let us explain the reason why in the result below, px is claimed to have the property P F relative to the both R and pRp, but (1 − p)x is completely non-P F relative to (1 − p)R(1 − p) only. By Definition 2.6 clearly x ∈ pRp has the property P F relative to pRp if and only if it has the property P F relative to R. On the other hand, note that by (2.1) we get F i (0) = 0. Hence any x ∈ pRp (p = 1) is not completely non-P F relative to R since (1 − p)x = 0. Theorem 2.7. Let R be a Baer * -ring and P F be a property defined by a family F as in Definition 2.6.
For any x ∈ R n there is a unique projection p ∈ x such that: • px has the property P F relative to R and pRp, In conclusion px has the property P F relative to R and so relative to pRp. Let r ∈ x be a projection such that the compression rx has the property P F . Let us show that r ≤ p. The element p+r is not necessarily an idempotent, but it is selfadjoint, it commutes with elements in x, and by the assumptions, x has the property P F . Since projections in a corner are precisely projections in the whole ring majorised by the projection generating the corner For uniqueness of p assume that r is a projection that decomposes x between objects having the property P F and completely not having it. Since rx has the property P F , by the previous part of the proof r ≤ p. Hence p and r commute, for any i ∈ I. In other words, the compression of (1 − r)x given by (p − r) have the property P F . However, by assumption on r the only compression of (1 − r)x having the property P F is the trivial one. In conclusion p − r = 0, so p is unique.
The uniqueness of the projection p in Theorem 2.7 implies that p and 1 − p are the maximal projections with the respective properties. Precisely: Corollary 2.8. Let R be a Baer * -ring, P F be a property defined by a family F as in Definition 2.6, x ∈ R n and p ∈ x be the unique projection obtained in Theorem 2.7

. Then
• p is the maximal projection commuting with x such that the compression px has the property P F relative to R and pRp, Proof. The maximality of p follows directly from its definition in the proof of Theorem 2.7.
For the maximality of 1−p consider an arbitrary q ∈ x such that (1−q)x is completely non-P F relative to (1−q)R(1−q). Let r be the unique projection obtained by Theorem 2.7 for the compression qx in the corner ring qRq, so rqx = rx has the property P F relative to qRq and rqRqr = rRr and (q −r)x is completely non-P F relative to (q −r)R(q −r). We may use rx as a compression since r ∈ x . Indeed, since r ≤ q and r commutes with qx we get rx = rqx = qxr = xqr = xr.
Clearly, rx has the property P F also relative to R. By the maximality of p we get r ≤ p and so p − r is a projection commuting with x. Since p − r ≤ p and the property P F is hereditary the compression (p − r)x has the property It has left to show that (1 − r)x is completely non-P F relative to (1 − r)R(1 − r). Assume s is a projection commuting with x, majorized by 1 − r and such that sx has the property P F relative to sRs. Note that qsq is not necessarily a projection, but it is selfadjoint. Moreover, since q, s ∈ x we get qsq ∈ x and by Corollary 2.4 also [qsq] ∈ x . Moreover, by (2.1) we get we get [qsq] = 0 which yields 0 = qsq = qs 2 q = (qs)(qs) * . Since the involution in Baer * -rings is proper we get qs = 0. Consequently, s ≤ 1 − q and since sx has the property P F in sRs and (1 − q)x is completely non-P F relative to (1 − q)R(1 − q) we get s = 0.
One can consider F J = {F i } i∈J where J ⊂ I and the corresponding property P FJ . Then, by Theorem 2.7 there is the respective projection p J . In the following Proposition 2.9 we show that projections p J corresponding to various sets J commute to each other. By this commutativity we are able to get decompositions among more than two summands and so we gain more detailed descriptions. In the next section we show several applications of this fact. More precisely, we extend several classical results known in the algebra of bounded linear operators on a Hilbert space B(H) to Baer * -rings. Moreover, we get some new results also in B(H).

Proposition 2.9. Suppose we have functions
Let x ∈ R n and let p 12 , p 1 , p 2 be projections decomposing x with a correspondence to the properties P F , P F1 , P F2 , respectively, as in Theorem 2.7.
The projection p 1 commutes with p 2 and p 1 p 2 = p 12 .
Proof. It is clear that p 12 ≤ p 1 by which they commute and so p 1 (1 − p 12 ) is a well defined projection. Consider an arbitrary projection q ∈ x such that q ≤ p 1 (1 − p 12 ). Note that if q ≤ 1 − [F j (x)] for all j ∈ I 1 ∪ I 2 , then, since p 12 is defined as the supremum of such projections (see proof of Theorem 2.7), we get q ≤ p 12 . However, we assumed q ≤ p 1 (1 − p 12 ), so in particular q ≤ 1−p 12 . Hence, either q = 0 or there is j 0 ∈ I 1 ∪I 2 such that q ≤ 1−[F j0 (x)] (equivalently q[F j0 (x)] = 0). On the other hand, since q ≤ p 1 the compression qx has the property P F1 so we get 0 = F i (qx) = qF i (x) for any i ∈ I 1 . In other , then by the left annihilators equality q[F j0 (x)] = 0 which is not true. Since q was arbitrary, p 1 (1 − p 12 )x completely does not have the property P F2 . Consequently where the last equality follows by p 12 ≤ p 1 , p 12 ≤ p 2 . Hence p 2 p 1 = p 12 and since the product of projections is a projection if and only if they commute we get p 1 p 2 = p 2 p 1 .
Note that if I 1 ∩ I 2 = ∅, then we may considerF = {F i } i∈I1∩I2 and F 1 = {F i } i∈I1\I2 and the corresponding projectionsp,p 1 . Clearly p 1 ≤p, p 2 ≤ p, p 1 ≤p 1 and p 1 =p 1p . Consequently p 1 p 2 =p 1 p 2 . In other words it is enough to consider disjoint sets I 1 , I 2 or more precisely disjoint families F 1 , F 2 .

Applications
In this section we derive several decompositions from Theorem 2.7. The condition pF i (x) = F i (px) makes P F a hereditary property. By hereditarity, (1−p)x completely does not have the property P F relative to (1 − p)R(1 − p) if and only if p is the maximal (so unique) projection such that px has the property P F . Hence the existence of the maximal projection in some statements in this section is equivalent to the existence of the corresponding decomposition. Let us give a little leeway that for non hereditary property the maximality does not imply the uniqueness of the corresponding decomposition -there may exist different decompositions between a part having the property and the one completely not having it. The reason is that the maximality of the projection may be considered only as a maximal element of some chain without uniqueness. For example, the property of being a bilateral shift is non hereditary. There may exist different maximal bilateral shift parts of the same unitary operator on a Hilbert space. We skip details since it requires Spectral Theorem and is far from the subject of the article. In this section we recall or adopt from the algebra of bounded linear operators on Hilbert spaces several properties of Baer * -ring elements. Let us start with the basic ones. Recall that an element x ∈ R is called normal, a partial isometry, an isometry, a unitary element if xx Proof. The result follows from Theorem 2.7 where: Note that, Theorem 3.1 states the respective properties only relative to the corners while Theorem 2.7 yields the result relative to the both, the whole ring and the corresponding corner. Since Theorem 3.1 follows from Theorem 2.7 one may find it strange. However, the interpretation of the property P F depends on the ring and in the case of the whole ring it may not be the one considered in Theorem 3.1. In the cases of p n and p p the interpretations are the same relative to the whole ring and the corner. Indeed, p n x is normal and p p x is a partial isometry also relative to the whole ring R. In the case of p i (and similarly p u ) the interpretation differs. Note that F (p i (x, 1)) = F (p i x, p i ) = 0 yields (p i x) * p i x = p i . Hence, the corresponding property P F may be interpreted as p i x is an isometry only if p i is the unity which is the case of the corner ring p i Rp i . Relative to the whole rind P F means that p i x is a partial isometry but not an isometry (unless p i = 1). The next result is formulated for a general element of a Baer * -ring, but it can be viewed in the context of Halmos-Wallen-Foiaş result on power partial isometries [7,Theorem].

Corollary 3.2. For any x in a Baer * -ring there is a unity decomposition
Proof. Indeed, let p u , p i be as in Theorem 3.1 for x while p ci be an isometric projection calculated for x * in Theorem 3.1. By Proposition 2.9 p i p ci = p ci p i = p u . Hence p pi = p i (1−p u ) and p pci = p ci (1−p u ) are well defined and orthogonal to each other. It remains to define p r = 1−p u −p pi −p pci . Since p r is orthogonal to p i and p ci it compress x and x * to completely non-isometric elements.
For power partial isometries the result is finer, the last part is described as truncated shifts [1,7]. Theorem 2.7 may be successfully applied to pairs (more generally sets) of elements. It works well, nevertheless the considered property describes a relation between/among elements (f.e. commutativity) or characterizes elements (f.e. normality). The following result on the double commutativity may be modified to the commutativity. Recall that elements in a pair (x, y) doubly commute if x ∈ {y, y * } . ((1 − p)x, (1 − p)y) completely do not doubly commute.

Theorem 3.3. For any pair (x, y) of arbitrary elements in R there is a unique projection p ∈ {x, y} such that elements in (px, py) doubly commute and elements in
Proof. It follows from Theorem 2.7 for F 1 (x, y) = xy − yx, F 2 (x, y) = xy * − y * x.
Much wider class are compatible pairs. The concept of compatibility was introduced for isometries on Hilbert spaces by Horák and Müller in [8]. It naturally extends to general pairs of elements in Baer * -rings. The following corollary is obvious for isometries in B(H), while in Baer * -rings it follows from Lemma 2.3.

Corollary 3.5. Any doubly commuting pair is compatible.
An example of compatible, completely non doubly commuting pair is (x, x) where x is a non unitary isometry. Other examples can be found in papers on operators on Hilbert spaces [4,5,8].
The next result shows a decomposition of an arbitrary pair between a compatible pair and a completely non compatible pair.  The compatibility does not imply the commutativity. We give an example. Recall that two projections are equivalent if there is a partial isometry having them as the left and the right projection.
One may ask about a quaternary decomposition with respect to commutativity and compatibility as in Theorem 3.9 below. The answer is affirmative, but not obvious even in B(H). It follows from Proposition 2.9. Let P be a property characterizing individual elements (f.e. normality). Recall that a set S completely does not have the property P (f.e. is completely not normal) if for any 0 = p ∈ S there is at least one x ∈ S such that pxp does not have the property (at least one pxp is not normal). We extend results of Theorem 3.1 on subsets. We show the decomposition with respect to normality. Other results may be proved similarly. Proof. It is enough to take F s : R S x → x * s x s − x s x * s for any s ∈ S and apply Theorem 2.7.
Let us finish this section by a generalization of Wold, Helson-Lowdenslager, Suciu result [10, Theorem 3]. For those reason we extend the concept of a quasi-unitary semigroup of isometries to Baer * -rings.
Note that by Corollaries 2.2 and 2.4, pF (x) = F (px) for any projection p ∈ x . Hence, by Theorem 2.7 we get a projection p qu which is the maximal one compressing the semigroup to a quasi-unitary semigroup. Similarly like in Corollary 3.10 we consider a family of functions F s : and get a projection p u . It is clear that p u ≤ p qu and so p pqu = p qu (1 − p u ) is a well defined projection compressing the semigroup to a purely quasiunitary semigroup. Clearly p s = 1 − p qu compress the semigroup to a strange semigroup.

Multiple Canonical Decomposition
Consider a property of a single element. Assume there is a pair (x, y) such that each of its elements admits a decomposition between the summand having the property and the one completely not having it. We may usually find also a decomposition of the pair (x, y) between the pair having the property and the one completely not having it as in Corollary 3.10 for example. However, the fact that the pair completely does not have the property does not say much about individual elements in the pair. Indeed, consider as an example the property of being normal. A normal element and a completely non-normal element as well as two completely non-normal elements form completely non-normal pairs. Hence a pair completely not having a given property requires finer description. Wold, Helson-Lowdenslager, Suciu result recalled in the previous section is one of the first attempts to give a characterization of this type. The best result would be a quaternary decomposition, as defined: Definition 4.1. A canonical decomposition of a pair (x, y) with respect to a property P characterizing single elements x, y is a quaternary decomposition p 11 + p 10 + p 01 + p 00 = 1 where p 11 , p 10 , p 01 , p 00 ∈ {x, y} are such that • each of p 11 x, p 11 y has the property P, • p 10 x has the property P, p 10 y completely does not have the property P, • p 01 x completely does not have the property P, p 01 y has the property P, • each of p 00 x, p 00 y completely does not have the property P.
Unfortunately, a general pair may not admit a canonical decomposition. Let us explain why Proposition 2.9 does not work for canonical decompositions. Consider once again the property of being normal. By Proposition 2.9 projections p x , p y corresponding (in the sense of Theorem 2.7) to F x (x, y) = x * x − xx * , F y (x, y) = y * y − yy * do commute. Hence p x p y is a projection. It can be checked that it is the maximal projection where both compressions are normal. However, p x (1 − p y ) compress x to a normal element but p x (1 − p y )y is not necessarily a completely non-normal element. Indeed, there may exist a projection 0 = q ≤ p x (1 − p y ) commuting with y where qy in normal but q does not commute with x. To be precise, in the decomposition of x we consider the projection corresponding to F x (x) = x * x − xx * instead of F x (x, y) = x * x − xx * . The formula is the same. The difference is that the respective supremum is taken in the set of projections commuting only with x in the first case and with both (x, y) in the second case. Hence the projection corresponding to F x (x) may majorize the one corresponding to F x (x, y). Let us formulate a result similar to [6,Corollary (2.3)].

Proposition 4.2.
Let R be a Baer * -ring and let P F be a hereditary property defined by a family F := {F i : R → R} i∈I satisfying (2.1) as in Definition 2.6.
Then: • there are maximal projections p x ∈ {x} and p y ∈ {y} such that p x x, p y y have the property P F , • there are maximal projections q x , q y ∈ {x, y} such that q x x, q y y have the property P F , • q x ≤ p x , q y ≤ p y .
Moreover, the following conditions are equivalent: • (x, y) admits a canonical decomposition with respect to the property P F , • p x , p y ∈ {x, y} , • p x = q x , p y = q y .
Proof. In fact the first part has been explained before the proposition. Precisely, the existence of p x , p y follows from Theorem 2.7 for {F i } i∈I . Define Note that q x ≤ 1 − [F i1 (x, y)] = 1 − [F i (x)] and obviously q x ∈R ∩{x} . Hence, q x belongs to the set above, so q x ≤ p x . Similarly q y ≤ p y .
For the second part, denote by p 11 + p 10 + p 01 + p 00 = 1 the canonical decomposition of the pair (x, y). If it exists, then p x = p 11 + p 10 , p y = p 11 + p 01