---

Spectral and Polar Decomposition in AW*-Algebras

arXiv:funct-an/9303001v1 11 Mar 1993Spectral and Polar Decomposition in AW*-AlgebrasM. FrankWe show the possibility and the uniqueness of polar decomposition of elements of arbitraryAW*-algebras inside them.

We prove that spectral decomposition of normal elements of normalAW*-algebras is possible and unique inside them. The possibility of spectral decomposition ofnormal elements does not depend on the normality of the AW*-algebra under consideration.Key words: Operator algebras, monotone complete C*-algebras, AW*-algebras, spectral decom-position and polar decomposition of operatorsAMS subject classification: 46L05, 46L35, 47C15The spectral decomposition of normal linear (bounded) operators and the polar decom-position of arbitrary linear (bounded) operators on Hilbert spaces have been interestingand technically useful results in operator theory [3, 9, 13, 20].

The development of theconcept of von Neumann algebras on Hilbert spaces has shown that both these decompo-sitions are always possible inside of the appropriate von Neumann algebra [14]. New lighton these assertions was shed identifying the class of von Neumann algebras among allC*-algebras by the class of C*-algebras which possess a pre-dual Banach space, the W*-algebras.

The possibility of C*-theoretical representationless descriptions of spectral andpolar decomposition of elements of von Neumann algebras (and may be of more generalC*-algebras) inside them has been opened up. Steps to get results in this direction weremade by several authors.

The W*-case was investigated by S. Sakai in 1958-60, [18, 19].Later on J. D. M. Wright has considered spectral decomposition of normal elements ofembeddable AW*-algebras, i.e., of AW*-algebras possessing a faithful von Neumann typerepresentation on a self-dual Hilbert A-module over a commutative AW*-algebra A (on aso called Kaplansky–Hilbert module), [23, 24]. But, unfortunately, not all AW*-algebrasare embeddable.

In 1970 J. Dyer [5] and O. Takenouchi [21] gave (∗-isomorphic) examplesof type III, non-W*, AW*-factors, (see also K. Saitˆo [15]). Polar decomposition insideAW*-algebras was considered by I. Kaplansky [12] in 1968 and by S. K. Berberian [3] in1972.

They have shown the possibility of polar decomposition in several types of AW*-algebras, but they did not get a complete answer. In the present paper the partial resultof I. Kaplansky is used that AW*-algebras without direct commutative summands andwith a decomposition property for it’s elements like described at Corollary 5 below allowpolar decomposition inside them, [3, §21: Exerc.

1] and [12, Th. 65].

For a detailledoverview on these results we refer to [3].The aim of the present paper is to show that both these decompositions are possibleinside arbitrary AW*-algebras without additional assumptions to their structures.

Recall that an AW*-algebra is a C*-algebra for which the following two conditions aresatisfied (cf. I. Kaplansky [10]):(a) In the partially ordered set of projections every set of pairwise orthogonal projec-tions has a least upper bound.

(b) Every maximal commutative ∗-subalgebra is generated by its projections, i.e., itis equal to the smallest closed ∗-subalgebra containing its projections.An AW*-algebra A is called to be monotone complete if every increasingly directed,norm-bounded net of self-adjoint elements of A possesses a least upper bound in A. AnAW*-algebra A is called to be normal if the supremum of every increasingly directed netof projections of A being calculated with respect to the set of all projections of A is it’ssupremum with respect to the set of all self-adjoint elements of A at once, cf. [25, 16].For the most powerful results on these problems see [4] and [17].To formulate the two theorems the following definitions are useful.Definition 1 (J. D. M. Wright [24, p.264]): A measure m on a compact Hausdorffspace X being valued in the self-adjoint part of a monotone complete AW ∗-algebra iscalled to be quasi-regular if and only ifm(K) = inf{m(U) : U−open sets in X, K ⊆U}for every closed set K ⊆X .

We remark that this condition is equivalent to the condition:m(U) = sup{m(K) : K−closed sets in X, K ⊆U}for every open set U ⊆X.Further, if m(E) = inf{m(U) : U−open sets in X, E ⊆U} for every Borel set E ⊆X,then the measure m is called to be regular.Definition 2 (M. Hamana [8, p.260] (cf. [1], [22])): A net {aα : α ∈I} of ele-ments of A converges to an element a ∈A in order if and only if there are bounded nets{a(k)α: α ∈I} and {b(k)α: α ∈I} of self-adjoint elements of A and self-adjoint elementsa(k) ∈A, k = 1, 2, 3, 4, such that(i) 0 ≤a(k)α −a(k) ≤b(k)α , k = 1, 2, 3, 4, α ∈I,(ii) {b(k)α: α ∈I} is decreasingly directed and has greatest lower bound zero,(iii)P4k=1(i)ka(k)α= aα for every α ∈I ,P4k=1(i)ka(k) = a (where i = √−1).We denote this type of convergence by LIM{aα : α ∈I} = a.By [6, p.260] the order limit of {aα : α ∈I} does not depend on the special choiceof the nets {a(k)α: α ∈I}, {b(k)α: α ∈I} and of the elements a(k), k = 1, 2, 3, 4.

If Ais a commutative AW ∗-algebra, then the notion of order convergence defined above isequivalent to the order convergence in A which was defined by H.Widom [22] earlier.Note that (cf. [8, Lemma 1.2]) if LIM{aα : α ∈I} = a , LIM{bβ : β ∈J} = b, then(i) LIM{aα + bβ : α ∈I, β ∈J} = a + b,(ii) LIM{xaαy : α ∈I} = xay for every x, y ∈A,(iii) LIM{aαbβ : α ∈I, β ∈J} = ab,(iv) aα ≤bα for every α ∈I = J implies a ≤b,(v) ∥a∥A ≤lim sup{∥aα∥A : α ∈I}.Furthermore, we need the following lemma describing the key idea of the present paperand being of interest on its own.

Lemma 3 : Let A be an AW*-algebra and B ⊆A be a commutative C*-subalgebra.Then the monotone closures ˆB(D), ˆB(D′) of B inside arbitrary two maximal commutativeC*-subalgebras D, D′ of A which contain B, respectively, are ∗-isomorphic commutativeAW*-algebras. Moreover, all monotone closures ˆB(D) of B of this type coincide as C*-subalgebras of A if A is normal.Proof: Let D be a maximal commutative C*-subalgebra of A containing B. Bydefinition D is generated by its projections.

Let p ∈ˆB(D) be a projection. Supposep ̸∈B.

Then p is the supremum of the set P = {x ∈B+h ⊆D : x ≤p} by [7, Lemma 1.7].In particular, (1A −p) is the maximal annihilator projection of P inside D. But, P2 = Pand, hence, the supremum of P being calculated inside D′ is a projection p′ again, and(1A −p′) is the maximal annihilator projection of P in D′. Changing the possitions ofD and D′ one finds a one-to-one correspondence between the projections of ˆB(D) andˆB(D′).Moreover, the product projection p1p2 of two projections p1, p2 ∈ˆB(D) correspondsto the supremum of the intersection set of the two appropriate sets P1 and P2 of elementsof B, and hence, to the product projection p′1p′2 of the corresponding two projectionsp′1, p′2 ∈ˆB(D′).

That is, the found one-two-one correspondence between the sets of pro-jections of ˆB(D) and of ˆB(D′) preserves the lattice properties of these nets. Since ˆB(D)and ˆB(D′) are commutative AW*-algebras (i.e.

both they are linearly spanned by theirprojection lattices as linear spaces and as Banach lattices), this one-to-one correspondenceextends to a ∗-isomorphism of ˆB(D) and ˆB(D′).Now, fix such a set P ⊆B. By [10] there exists a global maximal annihilator pro-jection (1A −q) of P in A.

The problem arrising in this situation can be formulated asfollows: Does q commute with B, i.e. is q an element of D and, hence, of every maximalcommutative C*-subalgebra D′ of A containing B?

Obviously, (1A −q) is the supremumof the set of all those annihilator projections {(1A −p)} which we have constructed above,but only in the sense of a supremum in the net of all projections of A since monotonecompleteness or normality of A are not supposed, in general. So we have to assume thatA is normal, cf.

[25, 16], to be sure in our subsequent conclusions. Then there follows that(1A −q) has to be the supremum of the set of the set of all those projections {(1A −p)}in the self-adjoint part of A.

Hence, q commutes with B since each of the projectionsp do. This means that q belongs to every maximal commutative C*-subalgebra D of Acontaining B because of their maximality, and q = p for every p ∈D since q ≤p, and pwas the supremum of P inside D+h .Since P was fixed arbitrarily one concludes that ˆB(D) does not depend on the choiceof D inside normal AW*-algebras A.Theorem 4 (cf.

[24, Th.3.1 and Th.3.2]): Let A be a normal AW*-algebra anda ∈A be a normal element.Let B ⊆A be that commutative C*-subalgebra in Abeing generated by the elements {1A, a, a∗}, and denote by ˆB the smallest commutativeAW*-algebra inside A containing B and being monotone complete inside every maximalcommutative C*-subalgebra of A. Then there exists a unique quasi-regular ˆB-valuedmeasure m on the spectrum σ(a) ⊂C of a ∈A, the values of which are projections in ˆBand for which the integralZσ(a) λ dmλ=aexists in the sense of order convergence in ˆB ⊆A.

If A is not normal, then for every maximal commutative C*-subalgebra D of A con-taining B there exists a unique spectral decomposition of a ∈A inside the monotone clo-sure ˆB(D) of B with respect to D. But, it is unique only in the sense of the ∗-isomorphyof ˆB(D) and ˆB(D′) for every two different maximal commutative C*-subalgebras D, D′of A containing B.If A is a W*-algebra, then m is regular and the integral exists in the sense of normconvergence.Proof: By the Gelfand–Naimark representation theorem the commutative C*-subalge-bra B ⊆A being generated by the elements {1A, a, a∗} is ∗-isomorphic to the commutativeC*-algebra C(σ(a)) of all complex-valued continuous functions on the spectrum σ(a) ⊂Cof a ∈A. Denote this ∗-isomorphism by φ, φ : C(σ(a)) −→B.

The isomorphism φ isisometric and preserves order relations between self-adjoint elements and, hence, positivityof self-adjoint elements. Therefore, φ is a positive mapping.Selecting an arbitrary maximal abelian C*-subalgebra D of A containing B one cancomplete B to ˆB(D) with respect to the order convergence in D. Note that ˆB(D) ⊆Adoes not depend on the choice of D by the previous lemma if A is normal.Now, by [24] , [23, Th.4.1] there exists a unique positive quasi-regular ˆB(D)-valuedmeasure m with the property thatZσ(a) f(λ) dmλ=φ(f)for every f ∈C(σ(a)).

Since φ−1(a)(λ) = λ for every λ ∈σ(a) ⊂C by the definition ofφ one getsZσ(a) λ dmλ=a.Moreover, since the extension ˆφ of φ to the set of all bounded Borel functions on σ(a)fulfils ˆφ(χE)2 = ˆφ(χ2E) = ˆφ(χE) for the characteristic function χE of every Borel setE ∈σ(a) the measure m is projection-valued, cf. [24].

One finishes refering to Lem-ma 3The following corollary is essential to get the polar decomposition theorem.Corollary 5 : Let A be an AW*-algebra and x ∈A be different from zero. Thenthere exists a projection p ∈A+h , p ̸= 0, and an element a ∈A+h such that a, p and(xx∗)1/2 commute pairwise, and a(xx∗)1/2 = (axx∗a)1/2 = p.Proof: Consider the commutative C*-subalgebra B of A being generated by the ele-ments {1A, xx∗}.

By the spectral theorem there exists a unique positive quasi-regularmeasure m on the Borel sets of σ((xx∗)1/2) ⊂R+ being projection-valued in the mono-tone closure ˆB(D) ⊆A of B with respect to an arbitrarily chosen, but fixed, maximalcommutative C*-subalgebra D of A containing B, and satisfying the equalityZσ((xx∗)1/2) λ dmλ = (xx∗)1/2in the sense of order convergence in ˆB(D) ⊆A. Now, if (xx∗)1/2 is a projection, then seta = 1A, p = xx∗.

If (xx∗)1/2 is invertible in A, then set p = 1A, a = (xx∗)−1/2. Otherwiseconsider a number µ ∈σ((xx∗)1/2), 0 < µ < ∥x∥, and set K = [0, µ] ∩σ((xx∗)1/2).

The

value m(K) ∈ˆB(D) is a projection different from zero. It commutes with every spectralprojection of (xx∗)1/2 and with (xx∗)1/2 itself.

Since m is a quasi-regular measure one hasZσ((xx∗)1/2)\K λ d(mλ(1A −m(K))) = (1A −m(K))(xx∗)1/2.Therefore, one finds p = (1A −m(K)) and a = ((1A −m(K))(xx∗))−1/2, where the inverseis taken inside the C*-subalgebra (1A −m(K)) ˆB(D) ⊆A. Since µ < ∥x∥the projectionp is different from zero.

The existence of a ∈A+h is guaranteed by 0 < µNow we go on to show the polar decomposition theorem for AW*-algebras using resultsof S. K. Berberian and I. Kaplansky. Previously we need it for commutative AW*-algebras.Note that the proof of the following lemma works equally well for all monotone completeC*-algebras.Lemma 6 : Let A be a commutative AW*-algebra.

For every x ∈A there exists aunique partial isometry u ∈A such that x = (xx∗)1/2u and uu∗is the range projection of(xx∗)1/2.Proof: Throughout the proof we use freely the order convergence inside monotonecomplete C*-algebras as defined at Definition 2.First, suppose x to be self-adjoint. The sequence {un = x(1/n + |x|)−1 : n ∈N} isbounded in norm by the representation theory.

It consists of self-adjoint elements, and thesequences {|x|(1/n+|x|)−1} and {(|x|−x)(1/n+|x|)−1} are monotone increasing. Hence,the sequence {un} is order converging inside A, LIM un = u, and u ∈Ah.

Furthermore,the sequence {un|x| : n ∈N} converges to x in order, i.e., x = u|x|. From the equalityx2 = |x|u∗u|x| one draws u∗u ≥rp(|x|) (where rp(|x|) denotes the range projection of |x|being an element of A).

Hence, u∗u = rp(|x|) by construction and u is a partial isometry.Now suppose x ∈A to be arbitrarily chosen. Consider again the sequence {un} ofelements of A as defined at the beginning.

One has to show the fundamentality of it withrespect to the order convergence. Since A is monotone complete the existence of it’s orderlimit u inside A will be guaranteed in this case.

For the self-adjoint part of the elementsof {un} the inequality0≤[x((1/n + |x|)−1 −(1/m + |x|)−1) + ((1/n + |x|)−1 −(1/m + |x|)−1)x∗]2≤2[x((1/n + |x|)−1 −(1/m + |x|)−1)2x∗++((1/n + |x|)−1 −(1/m + |x|)−1)x∗x((1/n + |x|)−1 −(1/m + |x|)−1)]is valid for every n, m ∈N. The expression of the right side converges weakly to zero asn, m go to infinity in each faithful ∗-representation of A on Hilbert spaces.

Therefore, itis bounded in norm and converges in order to zero as n, m go to infinity because of thepositivity of the expression. Since taking the square root preserves order relations betweenpositive elements of a C*-algebra and since self-adjoint elements have polar decompositioninside A the order fundamentality of the sequence {1/2(un+u∗n) : n ∈N} turns out.

Theorder convergence of the anti-self-adjoint part of the sequence {un}, {1/2i · (un −u∗n)},can be shown in an analogous way. Hence, there exists LIM un = u inside A.Now, from the existence of LIMun|x| = x one derives the equality x = u|x|.

Theequality x∗x = |x|u∗u|x| shows that u∗u ≥rp(|x|) and, consequently, u∗u = rp(|x|) byconstruction, i.e., u is a partial isometry.To show the uniqueness of polar decomposition inside A suppose x = v|x| for a partialisometry v with v∗v = rp(|x|). Then v|x| = u|x|, i.e., v = v · rp(|x|) = u

Theorem 7 : Let A be an AW*-algebra. For every x ∈A there exists a uniquepartial isometry u ∈A such that x = (xx∗)1/2u and uu∗is the range projection of(xx∗)1/2.Proof: By [12, Th.

65] polar decomposition is possible inside of all AW*-algebraswithout direct commutative summands under the supposition that every element of ithas the property of Corollary 5, (see also [3, §21: Exerc. 1]).

Since polar decompositionworks separately in every direct summand we have only to compare Corollary 5, Lemma6 and the result of I. KaplanskyThe result of Theorem 7 is of interest also because monotone completeness was notnecessary to show it, what is a little bit surprising.REFERENCES[1] Azarnia, N.: Dense operators on a KH-module. Rend.

Circ. Mat.

Palermo (2) 34 (1985),105 - 110. [2] Berberian, S. K.: Baer ∗-rings.

Berlin - Heidelberg - New York: Springer–Verlag 1972. [3] Berberian, S.K.

: Notes on spectral theory (Van Nostrand Mathematical Studies: Vol.5). Princeton: D. van Nostrand Company, Inc.

1966. [4] Christensen, E. and G. K. Pedersen: Properly infinite AW*-algebras are monotonesequentially complete.

Bull. London Math.

Soc. 16 (1984), 407-410.

[5] Dyer, J.: Concerning AW*-algebras. Notices Amer.

Math. Soc.

17 (1970), 788. [6] Hamana, M.: Regular embeddings of C∗-algebras in monotone complete C∗-algebras.

J.Math. Soc.

Japan 33 (1981), 159 - 183. [7] Hamana, M.: The centre of the regular monotone completion of a C*-algebra.

J. LondonMath. Soc.

(2) 26 (1982), 522 - 530. [8] Hamana, M.: Tensor products for monotone complete C*-algebras.

Japan. J.

Math. 8(1982), 259 - 283.

[9] Hilbert, D.:Grundz¨uge einer allgemeinen Theorie der linearen Integralgleichungen(Fortschritte der mathematischen Wissenschaften in Monographien: Vol. 3).

Leipzig–Berlin:Teubner–Verlag, 1912. [10] Kaplansky, I.: Projections in Banach algebras.

Ann. Math.

53 (1951), 235 - 249. [11] Kaplansky, I.: Modules over operator algebras.

Amer. J.

Math. 75 (1953), 839 - 858.

[12] Kaplansky, I.: Rings of operators. New York: Benjamin 1968.

[13] Lengyel, B. A. and M.H.

Stone: Elementary proof of the spectral theorem. Ann.

Math.37 (1936), 853 - 864. [14] Murray, F. J. and J. von Neumann: On rings of operators.

Ann. of Math.

(2) 37 (1936),116 - 229. [15] Saitˆo, K.: AW*-algebras with monotone convergence property and type III, non-W*,AW*-factors.

In: C*-algebras and Applications to Physics. Proc.

Second Japan–USA Semi-nar Los Angeles, April 18-22, 1977 (eds. : H. Araki, R. V. Kadison).

Lect. Notes Math.

650(1978), 131 - 134.

[16] Saitˆo, K.: On normal AW*-algebras. Tˆohoku Math.

J. 33 (1981), 567-576.

[17] Saitˆo, K. and J. D. M. Wright: All AW*-factors are normal. J. London Math.

Soc. (2)44 (1991), 143-154.

[18] Sakai, S.: The theory of W*-algebras (Lecture Notes Series). New Haven, USA: YaleUniversity 1962.

[19] Sakai, S. C*-algebras and W*-algebras (Ergebnisse der Mathematik und ihrer Grenzgebi-ete: Bd.60). Berlin–Heidelberg–New York: Springer–Verlag 1971.

[20] Sz.-Nagy, B.: Spektraldarstellungen linearer Transformationen des Hilbertschen Raumes(Ergebnisse der Mathematik: Vol. 5, Teil 5).

Berlin: Springer–Verlag 1942. [21] Takenouchi, O.: A non-W*, AW*-factor.

In: C*-algebras and Applications to Physics.Proc. Second Japan–USA Seminar Los Angeles, April 18-22, 1977 (eds.

: H. Araki, R. V.Kadison). Lect.

Notes Math. 650 (1978), 135 - 139.

[22] Widom, H.: Embedding in algebras of type I. Duke Math. J.

23 (1956), 309 - 324. [23] Wright, J. D. M.: Stone-algebra-valued measures and integrals.

Proc. London Math.Soc.

(3) 19 (1969), 107 - 122. [24] Wright, J. D. M.: A spectral theorem for normal operators on a Kaplansky–Hilbertmodule.

Proc. London Math.

Soc. (3) 19 (1969), 258 - 268.

[25] Wright, J. D. M.: Normal AW*-algebras. Proc.

Royal Soc. Edinburgh 85A (1980),137-141.Received 18.9.91; in revised version 28.4.92Dr.

Michael FrankUniversit¨at LeipzigMathematisches InstitutAugustusplatz 10D(Ost)–7010 Leipzigfrank@mathematik.uni-leipzig.dbp.de

---

출처: arXiv:9303.001 • 원문 보기