arXiv:1204.1554v1 [math.FA] 6 Apr 2012 ahmtc ujc lsicto 00 03,1A5 77,4 17A70, 17A05, 30G35, 2010: Classification Subject Mathematics integration non-commutative alge sure, operator operator, algebra, octonion field, skew quaternion ieyue seilyi eetyasntol nmteais u also but mathematics, 16]. in 15, 8, only 10, not [5, years is applications It recent 13]. in 1, especially alg [3, real used embeddings division non-central sively has largest field the complex is the algebra which octonion in The d 9]. partial 7, of [2, and equations physics mathematical the and relatio On theoretical in of 31]. particularly problems 14, developing, 12, fast 11, is analysis [4, hypercomplex sciences hand, the in applications their different and partial tions and differential ma analysis, found functional have in field applications complex the over operators normal Unbounded Introduction 1 n pcrlter fbuddnra prtr n unbound and operators normal bounded of theory spectral and 1 npeiu ok nlssoe utrinadotnoswsdeve was octonions and quaternion over analysis works previous In e od n hae:nncmuaiefntoa nlss hy analysis, functional non-commutative phrases: and words key none omloeaosi octonion in operators normal Unbounded eosrtd pcr fubuddnra prtr r in are operators normal gated. unbounded of alg Spectra related of and demonstrated. properties their about Theorems studied. ffiitdadnra prtr notno ibr pcsar spaces Hilbert octonion in operators normal and Affiliated ibr pcsadterspectra their and spaces Hilbert 1 5Fbur 2012 February 15 ukvk S.V. Ludkovsky Abstract 1 r,seta pcrlmea- spectral spectra, bra, A0 73,47L60 47L30, 7A10, ecmlxnumbers, percomplex ba are ebras dself-adjoint ed vesti- ifferential a equa- ial ny-sided e with n inten- loped other ebra in operators was described [18, 19, 20, 21, 22]. Some results on their applications in partial differential equations were obtained [23, 24, 25, 26, 27]. This work continuous previous articles and uses their results. The present paper is devoted to unbounded normal operators and affiliated operators in octonion Hilbert spaces, that was not yet studied before. Frequently in practical problems, for example, related with partial differ- ential operators, spectral theory of unbounded normal operators is necessary. This article contains the spectral theory of unbounded affiliated and normal operators. Notations and definitions of papers [18, 19, 20, 21, 22] are used below. The main results of this paper are obtained for the first time.
2 Affiliated and normal operators
1. Definitions. Let X be a Hilbert space over the Cayley-Dickson algebra , 2 v, and let A be a von Neumann algebra contained in L (X). Av ≤ q We say that an operator Q L (X) quasi-commutes with A if the algebra ∈ q alg (Q, B) over generated by Q and B is quasi-commutative for each Av Av B A. A closed R homogeneous additive operator T with a dense ∈ Av Av vector domain (T ) X is said to be affiliated with A, when D ⊂ (1) U ∗T Ux = T x for every x (T ) and each unitary operator U L (X) quasi-commuting ∈D ∈ q with A. The fact that T is affiliated with A is denoted by T ηA. An vector subspace F (T ) is called a core of an operator T , if Av n n D F is an increasing sequenceS of graded projections and F (T ) is n Av n n D dense in (T ). S D 2. Note. Definition 1 implies that U (T ) = (T ). If V is a dense D D Av vector subspace in (T ) and T ηA, then T ηA. Indeed, lim Uyn = Uy for D |V n each unitary operator U L (X) and every sequence yn V converging to a ∈ q ∈ vector y (T ) and with lim Tyn = Ty. Therefore, the limit lim TUyn = ∈D n n lim UTyn = UTy exists. The operator T is closed, hence Uy (T ) and n ∈ D UTy = TUy. Thus (T ) U ∗ (T ). The proof above for U ∗ instead of U D ⊂ D gives the inclusion (T ) U (T ), consequently, U (T )= (T ) and hence D ⊂ D D D (U ∗T U)= (T ) and TUy = UTy for every y (T ). D D ∈D
2 For an R homogeneous additive (i.e. quasi-linear) operator A in X Av with an vector domain (A) the notation can be used: Av D ij (1) Ax = j A xj for each (2) x = Px i (A) X with x X and j j j ∈D ⊂ j ∈ j ij (3) A xPj := A(xjij) for each j =0, 1, 2, .... That is Aij (i∗πj) = Aπj, where πj : X X i is an R linear projection j → j j j j with π (x)= xjij so that j π = I. It can be lightly seen,P that Definition 1 is natural. Indeed, if A is a quasi-commutative algebra over the Cayley-Dickson algebra with 2 v, Av ≤ then an algebra A i A i is commutative for k 1 over the complex field 0 0 ⊕ k k ≥ C := R Ri , since there is the decomposition A = A i A i ...A i ... ik ⊕ k 0 0 ⊕ 1 1 ⊕ m m⊕ with pairwise isomorphic real algebras A0, A1, ..., Am, .... Let (Y, ) denote the algebra of all bounded Borel functions from a B Av topological space Y into the Cayley-Dickson algebra , let also (Y, ) Av Bu Av denote the algebra of all Borel functions from Y into with point-wise Av addition and multiplication of functions and multiplication of functions f on the left and on the right on Cayley-Dickson numbers a, b , 2 v. ∈Av ≤ 3. Lemma. If T is a closed symmetrical operator in a Hilbert space X over the Cayley-Dickson algebra , 2 v, then ranges (T MI) of Av ≤ R ± (T MI) are closed for each M , when v 3 or M i , i , ... . If T ± ∈ Sv ≤ ∈{ 1 2 } is closed and 0 < T z; z > for each z (T ), then T + I has a closed ≤ ∈ D range. Proof. Let x be a sequence in (T ) so that (T MI) x : n tends n D { ± n } to a vector y X. But < T z; z > is real for each z (T ), hence ∈ ∈D z 2 (< T z; z >2 + < z; z >2)1/2 = < (T MI)z; z > k k ≤ | ± | (T MI)z z , ≤k ± kk k consequently, x x (T MI)( x x) and x converges to some kn − m k≤k ± n − m k n vector x X. On the other hand, the sequence T x : n converges ∈ { n } to Mx + y and the operator T is closed, consequently, x (T ) and ∓ ∈ D T x = Mx + y. Indeed, ∓
∗ ∗ M (Mx)= M xj(Mij)= xjij = x, Xj Xj 3 since Mx = x and M(x i ) = x (Mi ) and M ∗(Mi ) = i for each k k k k j j j j j j x X and M for 2 v 3 or M i , i , ... . Thus (T MI)x = y, j ∈ j ∈ Sv ≤ ≤ ∈{ 1 2 } ± hence the operators (T MI) have closed ranges for such M. ± If an operator T is closed and 0 < T z; z > for each z (T ), then ≤ ∈D z 2 < z; z > + < T z; z > (T + I)z z k k ≤ ≤k kk k and analogously to the proof above we get that (T + I) has a closed range. 4. Proposition. If T is a closed symmetrical operator on a Hilbert space X over the Cayley-Dickson algebra , 2 v, then the following statements Av ≤ are equivalent: (1) an operator T is self-adjoint; (2) operators (T ∗ MI) have 0 as null space for each M , when ± { } ∈ Sv v 3 or M i , i , ... ; ≤ ∈{ 1 2 } (3) operators (T MI) have X as range for every M , when v 3 ± ∈ Sv ≤ or M i , i , ... ; ∈{ 1 2 } (4) operators (T MI) have ranges dense in X for all M with v 3 ± ∈ Sv ≤ or M i , i , ... . ∈{ 1 2 } Proof. (1) (2). We have < T x; x >=< x; Tx > R for each x ⇒ ∈ ∈ (T ), when T ∗ = T , hence D < (T ∗ MI)x; x>=< (T MI)x; x>=< T x; x> M x 2, ± ± ± k k consequently, < (T ∗ MI)x; x>= 0 only when x = 0, since Mx = x for ± k k k k each M with v 3 or M i , i , ... . Thus each operator (T ∗ MI) ∈ Sv ≤ ∈{ 1 2 } ± has 0 as null space. { } (2) (3). Ranges (T MI) are closed due to Lemma 3. Therefore, ⇒ R ± it is sufficient to show that these ranges are dense in a Hilbert space X over the Cayley-Dickson algebra . But < T x; y >= M < x; y >, when Av ∓ < (T MI)x; y >= 0 for all x (T ), consequently, y (T ∗) and ± ∈ D ∈ D T ∗y = My. Therefore, y = 0, since the operators T ∗ MI have 0 as ± ± { } null space. Thus the operators (T MI) have dense ranges for each M ± ∈ Sv when v 3 or M i , i , ... . ≤ ∈{ 1 2 } (3) (4). This follows from the preceding demonstrations. ⇔ (3) (1). When T is a closed and symmetrical operator, one has that ⇒ T T ∗ and a graph Γ(T ) is a closed subspace of the closed R-linear space ⊆ Γ(T ∗). The equality
4
For an operator T let algAv (I, T ) =: Q be a family of operators generated by I and T over the Cayley-Dickson algebra . Consider this family of Av operators on a common domain ∞(T ) := ∞ (T n). Then the family Q D ∩n=1D on ∞(T ) can be considered as an vector space. Take the decomposition D Av Q = Q i Q i ... Q i ... of this vector space with pairwise 0 0 ⊕ 1 1 ⊕ ⊕ m m ⊕ Av isomorphic real vector spaces Q , Q , ..., Q , ... and for each operator B Q 0 1 m ∈ put (1) B = jB with jB =π ˆj(B) Q i ∈ j j Xj for each j, whereπ ˆj : Q Q i is the natural R linear projection, real linear → j j spaces Qmim and imQm are considered as isomorphic. Evidently, there is the inclusion of domains of these R linear operators (kB) (B) for each k, D ⊃ D particularly, (kT ) (T ). D ⊃D 6. Lemma. Let E : n be an increasing sequence of graded {n } Av projections on a Hilbert space X over the Cayley-Dickson algebra and Av let also G be an R homogeneous additive operator with dense domain Ar E(X) =: such that G E is a bounded self-adjoint operator on X, n n E n Swhere 2 v. Then G is pre-closed and its closure T is self-adjoint. Moreover, ≤ if an operator T is closed with core and T E is a bounded self-adjoint E n operator for each n N, then T is self-adjoint. ∈
5 Proof. For each x, y there exists a natural number m so that ∈E < Gx; y >=
< T nx; ny >=< T mE nx; ny >=< nx; T mE ny >=< nx; T ny > are satisfied, consequently,
limn < T nx; ny >=< T x; y > and
limn < nx; T ny >=< x; Ty >. Thus one gets < T x; y >=< x; Ty > and that the operator T is symmetric. Mention that (T MI) E(X) = E(X) for each M with v 3 ± n n ∈ Sv ≤ or M i , i , ... , since T E is bounded and self-adjoint, for which the ∈ { 1 2 } n operator T E M E has a bounded inverse on E(X). This implies that n ± n n T MI has a dense range and it coincides with X, consequently, T is self- ± adjoint due to Lemma 3 and Proposition 4. 7. Theorem. Let µ be a σ-finite measure µ : [0, ] on a σ-algebra F → ∞ of subsets of a set S and let L2(S, , µ, ) be a Hilbert space completion F F Av of the set of all step µ measurable functions f : S with the valued →Av Av scalar product < f; g >= f(x)˜g(x)µ(dx) Z for each f,g L2(S, , µ, ), where 2 v. Suppose that A is its left ∈ F Av ≤ multiplication algebra M f = gf. Then an R-linear -additive operator g Av T is affiliated with A if and only if a measurable finite µ almost every- where on S function g : S exists so that T = M and g (t)f (t) = → Av g k j ( 1)κ(j,k)f (t)g (t) for µ almost all t S and each f L2(S, , µ, ) for − j k ∈ ∈ F Av each j, k = 0, 1, ..., where κ(j, k)=0 for j = 0 or k = 0 or j = k, while κ(j, k)=1 for each j = k 1. Moreover, an operator T ηA is self-adjoint if 6 ≥ and only if g is real-valued µ almost everywhere on S. Proof. If g : S is a µ measurable µ essentially bounded on S func- →Av tion, then M is a bounded R-linear -additive operator on L2(S, , µ, ), g Av F Av since
6 M f g f , where k g k2 ≤k k∞k k2 g = ess sup g(x) and f := √< f; f >. That is M A. k k∞ x∈S | | k k2 g ∈ Each operator G A is an ( )C combination of unitary operators in A ∈ Av i (see Theorem II.2.20 [28]) and UT T U for each unitary operator U A, ⊆ ∈ ⋆ since T ηA and A A , where Ci = R Ri. If F is an graded projection ⊆ ⊕ Av operator corresponding to the characteristic function χP of a µ-measurable subset P in S, this implies the inclusion F T T F , consequently, F f (T ) ⊆ ∈D for each f (T ). If f (T ) and F corresponds to the multiplication ∈ D ∈ D n by the characteristic function χ of the set K = x S : f(x) n , then K { ∈ | | ≤ } nF is an ascending sequence of projections in the algebra A converging to the unit operator I relative to the strong operator topology, since f is finite almost everywhere, µ x : f(x) = = 0. Therefore, F f for each { | | ∞} n ∈ E n, where denotes the set of all µ essentially bounded functions f (T ). E ∈D Moreover, the limits exist:
lim nF f = f and lim T nF f = lim nF T f = T f. n n n
Thus = F (T ), where F (T ) is dense in (T ). Thus is a E n n D n n D D E core of T . S S Each step function u : S on (S, ) has the form →Av F m
u(s)= clχBl , Xl=1 where B and c for each l = 1, ..., m, m N, where χ denotes l ∈ F l ∈Av ∈ B the characteristic function of a subset B in S so that χB(s) = 1 for each s B and χ (s) = 0 for all s outside B. A subset N in S is called µ null ∈ B if there exists H so that N H and µ(H) = 0. If consider an algebra ∈F ⊂ of subsets in S which is the completion of by µ null subsets, then each Fµ F step function in L2(S, , µ, ) may have B for each l. F Av l ∈Fµ For each functions f,g there are the equalities ∈E (1) ((f i ) kT )g =(M kT )g =( 1)κ(j,k)(kT M )g = lT f g j j k fj ij k − fj ij k ± j k =( 1)κ(j,k)M jT f =( 1)κ(j,k)g i [ jT f ], − gkik j − k k j where i i = i , κ(j, k)=0 for j =0 or k =0 or j = k, while κ(j, k)=1 j k ± l for each j = k 1, 6 ≥ f = fjij Xj 7 with real-valued functions f for each j. Let S be a sequence of pairwise j k ∈F disjoint subsets with 0 < µ(S ) < for each k and with union S = S. k ∞ k k For the characteristic function χ a sequence f k,j : j N S of real- Sk { ∈ }⊂E valued functions exists converging to χ in L2(S, , µ, ). The set S0 := Sk F Av k t S : f k,j(s)=0 j has µ measure zero, since { ∈ k ∀ }
k,j 0 0 = lim Mgf = MgχSk = g, where g = χS . j k
Put h(s) = [(T f k,j)(s)][f k,j(s)]−1 for s S S0, where j is the least natural ∈ k \ k number so that f k,j(s) = 0. Thus h is a measurable function defined µ almost 6 everywhere on S. In accordance with Formula (1) the equality
(2) f k,j(s)[T f] (s)= ( 1)κ(q,p)f (s)i [T f k,j] (s)i +f (s)i [T f k,j] (s)i l { − p p q q q q p p} p,q; Xiqip=il is accomplished for all k, j N and l =0, 1, 2, ... except for a set of measure ∈ zero, consequently, [T f](s)= h(s)f(s)= Mhf(s) almost everywhere on S so k that T f = [T f]kik for each k.
The operator Mh is closed and affiliated with A as follows from the demon- stration above. On the other hand, M is an extension of T , consequently, h |E T M . ⊆ h The family of all functions z L2(S, , µ, ) vanishing on s S : h(s) > ∈ F Av { ∈ | | m for some m = m(z) N forms a core for M . For such a function z take } ∈ h a sequence f k of functions in tending to z in L2(S, , µ, ). Evidently E F Av f k can be replaced by yf k, where y is the characteristic function of the set s S : z(s) = 0 of all points at which z does not vanish. Thus we can { ∈ 6 } choose a sequence f k vanishing for each k when z does. Therefore,
k k (3) limk T f = limk MhMyf = MhMyz = Mhz, since the operator MhMy is bounded. For a closed operator T this implies that z (T ) and T z = M z, ∈ D h consequently, T = Mh.
If an operator T is self-adjoint, then Myh is a bounded self-adjoint oper- ator, hence the function yh is real-valued µ almost everywhere on S. For a bounded multiplication operator M the graded projections E g Av t corresponding to multiplication by characteristic function of the set s { ∈ 8 S : g(s) t forms a spectral graded spectral resolution E : t R ≤ } Av {t ∈ } of the identity for the operator T (see also Theorem 2.28 [20]). 8. Definition. Suppose that V is an extremely disconnected compact Hausdorff topological space and V W is an open dense subset in V . Ifa \ function f : V W is continuous and \ →Av lim f(x) = x→y | | ∞ for each y W , where x V W , 1 v, then f is a called a normal ∈ ∈ \ ≤ function on V . If a normal function on V is real-valued it will be called a self-adjoint function on V . The families of all normal and self-adjoint functions on V we denote by (V, ) and (V ) respectively, let also N Av Q
W+ := W+(f) := y W : lim f(x)= , { ∈ x→y ∞}
W− := W−(f) := y W : lim f(x)= { ∈ x→y −∞} for a self-adjoint function f on V . 9. Lemma. Let f and g be normal functions on V (see Definition 8) defined on V W and V W respectively so that f(x)= g(x) for each x in \ f \ g a dense subset U in V (W W ). Then W = W and f = g. \ f ∪ g f g Proof. The subset V (W W ) is dense in V , consequently, U is dense \ f ∪ g in V . If y W , then for each N > 0 there exists a neighborhood E of y in ∈ g V so that g(x) > N for each x E (V W ), hence f(x) = g(x) > N | | ∈ ∩ \ g | | | | for each x E (V (W W )), consequently, W W and symmetrically ∈ ∩ \ g ∪ f g ⊂ f W W . Thus W = W and f g is defined and continuous on V W f ⊂ g f g − \ f and is zero on U, hence f = g. 10. Lemma. Suppose that T is a self-adjoint operator acting on a Hilbert space X over either the quaternion skew field or the octonion algebra , Av 2 v 3, so that T is affiliated with some quasi-commutative von Neumann ≤ ≤ algebra A over , where A is isomorphic to C(Λ, ) with an extremely Av Av disconnected compact Hausdorff topological space Λ. Then there is a unique self-adjoint function h on Λ such that hˆe C(Λ, ) and a function hˆe · ∈ Av · represents T E, when E is an graded projection for which T E L (X) Av ∈ q 9 is a bounded operator on X, a function e C(Λ, ) corresponds to E, ∈ Av hˆe(x)= h(x) for e(x)=1, while hˆe(x)=0 otherwise. There exists an · · Av graded resolution of the identity E : b so that ∞ F (X) is a core for {b } n=1 n T , where F := E E and S n n − −n n (1) T x = −n dbE.bx for every x R F (X) and each n in the sense of norm convergence of Rie- ∈ n mann sums. Proof. Take Y = X iX, where i is a generator commuting with i ⊕ j for each j such that i2 = 1. Take an extension T onto (T ) (T )i so − D ⊕D that T (x + yi) = T x +(Ty)i for each x, y (T ). Then (T + iI) = Y ∈ D R and (T iI) = Y , where (T iI)=(T iI)Y , ker(T iI) = 0 R − R ± ± ± { } and inverses B := (T iI)−1 are everywhere defined on Y and of norm not ± ± exceeding one in accordance with II.2.74 and Proposition II.2.75 [28]. Then § the equalities < B (T + iI)x;(T iI)y >=< x;(T iI)y >=< (T + iI)x; y >=< + − − (T + iI)x, B (T iI)y > − − are satisfied for each x and y (T ), since the operator T is self-adjoint, ∈ D ∗ hence B− = B+. Then an arbitrary vector z X has the form z = B B x for the cor- ∈ − + responding vector x (T ) with T x (T ), since (B ) = Y . In the ∈ D ∈ D R ± latter case we get B B x =(T T iIT + T iI + I)x = B B x, since to F − + − + − n a real-valued function f C(Λ, R) corresponds for each n. On the other n ∈ ∗ hand, B− = B+, consequently, the operator B+ is normal. Consider a quasi-commutative von Neumann algebra A˘ over the complex- ified algebra ( )C containing I, B and B A. If U is a unitary operator Av i + − ∈ in A⋆ (see II.2.71 [28]), then §
Ux = UB+(T + iI)x =(B+)U(T + iI)x hence
(T + iI)Ux = U(T + iI)x, consequently, T is affiliated with A˘ by Definition 1, since (iI)Ux = U(iI)x for a unitary operator U as follows from the definition of a unitary operator and the valued scalar product on X extended to the ( )C valued scalar Av Av i product on Y :
10 < a + bi; c + qi >=(< a; c> + < b; q >)+(< b; c> < a; q >)i − for each vectors a, b, c, q X. ∈ The algebra A˘ has the decomposition A˘ = A0 A1i, where A0 and A1 are ⊕ quasi-commutative isomorphic algebras over either the quaternion skew field or the octonion algebra with 2 v 3. Av ≤ ≤ In view of Theorem 2.24 [20] A˘ is isomorphic with C(Λ, ( )C ) for some Av i extremely disconnected compact Hausdorff topological space Λ, since the 0 1 generator i has the real matrix representation i = −1 0 and the real field is the center of the algebra with 2 v 3. Letf and f be functions Av ≤ ≤ + − corresponding to B+ and B− respectively. Put h± = 1/f± at those points where f± does not vanish. Therefore, these functions h± are continuous on 1 their domains of definitions as well as h = 2 (h+ + h−). The function h corresponds to T . It will be demonstrated below that h is real-valued and then the Av graded spectral resolution of T will be constructed.
∗ ∗ At first it is easy to mention that f+ = f−, since B+ = B−, consequently, h is real-valued. The functions f+ and f− are continuous and conjugated, −1 −1 hence the set W := f+ (0) = f− (0) is closed. If the set W contains some non-void open subset J, then cl(J) W so that cl(J) is clopen (i.e. closed ⊂ and open simultaneously) in Λ, consequently, W is nowhere dense in Λ. The projection P corresponding to this subset cl(J) would have the zero product B P = 0 contradicting the fact that ker(B )= 0 . Thus each point x W + + { } ∈ is a limit point of points y Λ W , but h and h are defined on the latter ∈ \ + − set. Therefore, the set Λ W is dense in Λ and the functions h and h are \ + − defined on Λ W . \ Then T B B y =(T +iI iI)B B y = B y (iI)B B y for each y Λ, + − − + − − − + − ∈ hence T B B = B (iI)B B and analogously T B B = B +(iI)B B + − − − + − − + + − + and inevitably (2) 2(iI)B B = B B and + − − − + 1 (3) T B+B− = 2 (B+ + B−).
From Formula (2) and the definitions of functions f± and h one gets (h(y)+ i)−1 = f (y) and (h(y) i)−1 = f (y) for each y Λ W , since + − − ∈ \ (h(y)+i)f (y)= 1 + 1 f +if = 1 + w−i 1 +i 1 =1 2i 1 +i 1 = 1, + 2 2f− + + 2 2 w+i w+i − 2 w+i w+i 11 where w i = 1 corresponds to (T iI). But f (y) tends to zero when ± f± ± + y Λ W tends to x W , consequently, lim h(y) = . This means ∈ \ ∈ y→x | | ∞ that h is a self-adjoint function on Λ. We put U := x Λ W : h(x) > b and V = U W (h) for b R b { ∈ \ } b b ∪ + ∈ (see Definition 8). The function h is continuous on the open subset Λ W \ in Λ, hence the subset U is open in Λ. For a point x W there exists an b ∈ + open set Q containing x such that h(y) > max(b, 0) for each y Q (Λ W ), ∈ ∩ \ since lim h(y) = , where W = W (h) (see Definition 8). Therefore, y→x | | ∞ + + this implies the inclusion Q (Λ W ) U V . ∩ \ ⊂ b ⊂ b Suppose that there would exist a point z Q W , where W = W (h), ∈ ∩ − − − then a point y Q with h(y) < 0 would exist contradicting the choice of Q. ∈ This means that Q W W and Q V , consequently, V is open in Λ. ∩ ⊂ + ⊂ b b We consider next the subset Λ := Λ cl(V ). In accordance with 2.24 b \ b § [20] the set Λ contains every clopen subset K := y Λ: h(y) b , b b { ∈ ≤ } where h(y) = for y W so that W K for each b R. To −∞ ∈ − − ⊂ b ∈ demonstrate this suppose that y V , then y Λ K and cl(V ) Λ K ∈ b ∈ \ b b ⊂ \ b and K Λ cl(V )=Λ , since Λ K is closed. Moreover, we have h(y) b b ⊂ \ b b \ b ≤ for each y Λ (Λ W ), since y / U , while the set Λ is clopen in Λ. ∈ b ∩ \ ∈ b b Therefore, y W W and h(y) b for each y Λ W and W = W W . ∈ \ + ≤ ∈ b ∩ − \ + Thus Λ is the largest clopen subset in Λ so that h(y) b and Λ W = W . b ≤ b ∩ − Denote by eb the characteristic function of the subset Λb and bE be an graded projection operator in A corresponding to e , where b R (see Av b ∈ also 2.24 [20]). The subset W is nowhere dense in Λ, consequently, e =1 § ∨b b and e = 0 such that E = 1 and E = 0. Thus E : b is the ∧b b ∨b b ∧b b {b } Av graded resolution of the identity so that bEs =(sI) bE = bE(sI) corresponds to e s = se for each marked Cayley-Dickson number s . This resolution b b ∈Av of the identity is unbounded when h / C(Λ, ). ∈ Av Put F = E E for a < b R, hence e e =: u is the characteristic b − a ∈ b − a function of Λ Λ corresponding to F . Therefore, the inclusion Λ Λ Λ b \ a b \ a ⊂ \ W follows and f (y)f (y) = 0 when u(y) = 1, since Λ W =Λ W = W . + − 6 b ∩ a ∩ − Then f + f (4) h(y)= + − (y) 2f+f−
12 for each y Λ W . The function f f is continuous and vanishes nowhere ∈ \ + − on the clopen subset Λ Λ , consequently, a positive continuous function ψ b \ a on Λ exists so that ψf+f− = u and ψu = ψ. Consider an element Ψ of the algebra A corresponding to ψ, hence
(5) ΨB+B− = F . On the other hand, from the construction above it follows that a h(y) ≤ ≤ b for each y Λ Λ and from Formula (4) one gets ∈ b \ a af f u (f++f−)u bf f u and + − ≤ 2 ≤ + − aψf f u = au (f++f−)ψu = (f++f−)ψ bψf f u = bu, since the real + − ≤ 2 2 ≤ + − field is the center of the Cayley-Dickson algebra . Thus Av (6) aF (B++B−)Ψ bF . ≤ 2 ≤ Therefore, Formulas (3, 5, 6) imply that (7) aF T F bF . ≤ ≤ Therefore, the operator T F is bounded and the element hˆu C(Λ, ) · ∈ Av corresponds to it due to (3 5). If E is an graded projection belonging − Av to A so that T E L (X) and U is a unitary operator in A⋆ such that ∈ q U −1T EU = U −1T UE = T E one has T E A (see II.2.71 [28]). Let each ∈ § function g C(Λ, ) be corresponding to the graded projection F n ∈ Av Av n and Λ := g−1(1). Then Λ is dense in Λ, since F = I. If a function n n n n ∨n n e C(Λ, ) correspondsS to E, then (hˆ g)e corresponds to the operator ∈ Av · n T nF E, where nF E = E nF . Suppose that e(y) = 1forsome y Λ. For each (open) neighborhood H of ∈ y there exist n N and x Λ such that x H, hence ((hˆ g)e)(x)= h(x) ∈ ∈ n ∈ · n and h(x) T E F T E . Thus y / W and h(y) T E and hence | |≤k n k≤k k ∈ | |≤k k hˆe C(Λ, ). Then one gets also (h ˆ g )e = h ˆ( g e )=(h ˆe ) g · ∈ Av j· n k l j· n k l j· l n k for each j,k,l. If s C(Λ, ) represents T E, then s g represents T E F , ∈ Av n n that is s g =(hˆe) g for every n and s = hˆe. n · n · On the other hand, (2 F I)T (2 F I) = T , since T and F are R- n − n − n linear operators, also ( F I)(X)=(I F )(X) and X = F (X) ( F n − − n n ⊕ n − I)(X), hence F T T F . Moreover, lim F x = x and n ⊂ n n n limn T nF x = limn nF T x = T x hence n nF (X) is a core for T . InS view of Theorem I.3.9 [28] we get that E E : b R is an { b |E(X) ∈ } 13 graded resolution of the identity on E(X), since (hˆe)ee bee and Av · b ≤ b b(e ee ) (hˆe)(e ee ). Applying Theorem I.3.6 [28] for F in place of E − b ≤ · − b n for T F and F E : b leads to the formula n |nF (X) {n b |nF (X) } n T x = dbE.bx Z−n for each x F (X) and every n N. ∈ n ∈ 11. Remark. Lemma 10 means that
n n ∞ lim dbE.bx = lim dbE.b nF x = lim T nF x = T x = dbE.bx n n n Z−n Z−n Z−∞ for each x (T ) interpreting the latter integral as improper. ∈D Under conditions imposed in Lemma 10 one says that the function h ∈ (Λ) represents an affiliated operator T ηA. Q Mention that an graded projection operator E is R-homogeneous Av b and -additive, hence R-linear. Therefore, in the particular case of a real- Av valued Borel function h(b) one has dbE.h(b)x = h(b)dbE.1x or simplifying notation h(b)dbEx. 12. Lemma. Suppose that A is a quasi-commutative von Neumann algebra over , where 2 v 3, so that A is isomorphic to C(Λ, ) for Av ≤ ≤ Av some extremely disconnected compact Hausdorff topological space Λ. Then each function h (Λ) represents some self-adjoint operator T affiliated ∈ Q with A. Proof. From 10 it follows that a self-adjoint function h determines an § graded resolution of the identity E : b R in A. Moreover, hˆg Av {b ∈ } · n ∈ C(Λ, ), where g = e e , e C(Λ, ) represents E. Consider an Av n n − −n n ∈ Av n operator T corresponding to hˆg . Certainly one has (hˆg )g = hˆg for n · n · m n · n each n m, consequently, T F = T , where F corresponds to g . Put ≤ m n n n n Gx = T x for every vector x F (X) and n N. Therefore, G is an R n ∈ n ∈ linear additive operator on ∞ F (X) =: . Therefore, the operator Av n=1 n K G is pre-closed and its closure ST is a self-adjoint operator with core as K Lemma 6 asserts. For a unitary operator U in A⋆ and x F (X) we get ∈ n Ux F (X), hence ∈ n T Ux = nT Ux = U nT x. Therefore, T ηA due to Definition 1 and Remark 2.
14 If u (Λ) represents T , then uˆg represents T F in accordance with ∈ Q · n n Lemma 10, consequently, hˆg = uˆg for each n. In view of Lemma 9 h = u, · n · n since h and u are consistent on a dense subsets in Λ. Thus the function h represents the self-adjoint operator T . 13. Lemma. Suppose that E : b is an graded resolution of the {b } Av identity on a Hilbert space X over the Cayley-Dickson algebra , 2 v Av ≤ ≤ 3, also A is a quasi-commutative von Neumann algebra over so that A Av contains E : b , where 2 v 3. Then there exists a self-adjoint operator {b } ≤ ≤ T affiliated with A so that
n (1) T x = dbE.bx Z−n for each x F (X) and every n N, where F = E E, and E : b ∈ n ∈ n n − −n {b } is the graded resolution of the identity for T given by Lemma 10. Av Proof. Take a function e C(Λ, ) corresponding to E and a subset b ∈ Av b Λ = e−1(1) clopen in Λ. Consider the subsets W := Λ Λ and W := b b + \ b b − S b Λb. From their definition it follows that W+ and W− are closed in Λ. These Ttwo subsets W are nowhere dense in Λ, since E = 0 and E = I, ± ∧b b ∨b b hence W := W W is nowhere dense in Λ. For a point x Λ W we + ∪ − ∈ \ put h(x) := inf b : x Λ . Given a positive number ǫ> 0 and y Λ W { ∈ b} ∈ \ so that h(y) = b one gets h(z) h(y) ǫ for each z Λ Λ . This | − | ≤ ∈ b+ǫ \ b−ǫ means that the function h is continuous on Λ W . Then lim h(y)= \ y→x ±∞ for x W , where y Λ W . Thus the function h is self-adjoint h (Λ) ∈ ± ∈ \ ∈ Q and by Lemma 12 corresponds to a self-adjoint operator T affiliated with A. Certainly we get that E : b is the graded resolution of the identity for {b } Av the operator T and Formula (1) is valid, since Λb is the largest clopen subset in Λ on which the function h takes values not exceeding b. If Ψ is another such clopen subset and e is its characteristic function, E A corresponds to e, then Ψ Λ for each c b. This leads to the ∈ ⊂ c ≥ conclusion that E E and Ψ Λ . ≤ ∧c>b c ⊂ b 14. Lemma. Suppose that T is a closed operator on a Hilbert space X over either the quaternion skew field or the octonion algebra with 2 Av ≤ v 3 and E : b is an graded resolution of the identity on X, where ≤ {b } Av
15 := F (X) is a core for T while F = E E and E n n n n − −n S n (1) T x = dbE.bx Z−n for each x E(X) and all n, then T is self-adjoint and E is the graded ∈ n b Av resolution of the identity for T .
Proof. Formula (1) implies that T nF is bounded and everywhere defined and is the strong operator limit of finite real-linear combinations of E : b . {b } Thus bE(T nF )=(T nF ) bE and T nF is self-adjoint, hence T is self-adjoint by Lemma 6. For each vector x (T ) there exists a subsequence n : p ∈D { p } of natural numbers and a sequence x : p of vectors such that {p }
limppx = limp np F px = x and limp T px = T x, since is a core for the operator T . Therefore, F T T F for each n, E n ⊆ n since
nF T x = limp nF (T np F ) px = limp(T np F )nF px = limp T nF px = T nF x. On the other hand, the limits exist:
limn (T nF ) bEx = limn bE(T nF )x = bET x and limn nF bEx = bEx. The operator T is closed, hence Ex (T ) and T Ex = ET x. This leads b ∈D b b to the conclusion that ET T E and (2 E I)T (2 E I) = T , since b ⊆ b b − b − these operators are R linear and X = E(X) ( E I)(X). Therefore, b ⊕ b − E(B )=(B ) E for each b R. b ± ± b ∈ Take the quasi-commutative von Neumann algebra G = G(T ) = cl[ alg B , B ; E : b R alg B , B ; E : b Av { − + b ∈ } ⊕ Av { − + b ∈ R i]. } In view of Lemma 10 the operator T is affiliated with G. But Lemma 13 means that there exists a self-adjoint operator H affiliated with G so that
n Hx = dbE.bx Z−n for each x F (X) and every natural number n. Therefore, H = T and ∈ n E : b is the graded resolution of the identity for T , since H = T {b } Av |E |E and is a core for T and H simultaneously. E 15. Note. The quasi-commutative von Neumann algebra G = G(T ) generated by B and B in 14 with which the self-adjoint operator T is − + § affiliated will be called the von Neumann algebra generated by T .
16 16. Theorem. Let A be a quasi-commutative von Neumann algebra acting on a Hilbert space X over either the quaternion skew field or the octonion algebra , 2 v 3, let also (A) be a family of all self-adjoint Av ≤ ≤ Q operators affiliated with A. Suppose that A is isomorphic to C(Λ, ) for an Av extremely disconnected compact Hausdorff topological space Λ and (Λ) is Q the family of all self-adjoint functions on Λ. Then (a) there exists a bijective mapping φ from (A) onto (Λ) which is an ex- Q Q tension of the isomorphism of A with C(Λ, ) for which φ(T )ˆe corresponds Av · to T E for each projection E in A with T E A, where e C(Λ, ) corre- ∈ ∈ Av sponds to E and ((φ(T )ˆe)(y)=(φ(T ))(y) for e(y)=1 while (φ(T )ˆe)(y)=0 · · for e(y)=0; (b) an graded resolution E : b of the identity exists in the quasi- Av {b } commutative von Neumann subalgebra G generated by an operator T in (A) Q so that
n (1) T x = dbE.bx Z−n for each x F (X) and every natural number n, where F = E E ∈ n n n − −n and F (X) =: is a core for T ; n n E (cS) if Q : b R is an graded resolution of the identity on X so {b ∈ } Av that
n (2) T x = dbQ.bx Z−n for each x P (X) and every natural number n, where P = Q Q ∈ n n n − −n and n nP (X) is a core for T , then bE = bQ for all b; (dS) if E : b R is an graded resolution of the identity in A, then {b ∈ } Av there exists an operator T (A) for which Formula (1) is valid; ∈ Q (e) if a function e C(Λ, ) corresponds to E and Λ = e−1(1), then b ∈ Av b b b Λb is the largest clopen subset of Λ on which φ(T ) takes values not exceeding b in the extended sense. The latter theorem is the reformulation of the results obtained above in this section. 17. Definition. A closed densely defined operator T in a Hilbert space
17 X over the Cayley-Dickson algebra will be called normal when two self- Av adjoint operators T ∗T and T T ∗ are equal, where 2 v. ≤ 18. Remark. For an unbounded R homogeneous additive operators Av the following properties are satisfied: (1) if A B and C D, then A + C B + D; ⊆ ⊆ ⊆ (2) if A B, then CA CB and AC BC; ⊆ ⊆ ⊆ (3) (A + B)C = AC + BC and CA + CB C(A + B). ⊆ The latter inclusion does not generally reduce to the equality. For exam- ple, an operator C may be densely defined, but not everywhere, one can take A = I and B = I. This gives C(A + B)=0on X, but CA + CB is zero − only on (C). D These rules imply that if CA AC for each C in some family , then ⊆ Y T A AT for each sum of products of operators from . Apart from algebras ⊆ Y of bounded operators a family may be not extendable to an algebra, since Y a distributive law generally may be invalid, as it was seen above. Another property is the following: (4) if T : b Ψ is a net of bounded operators in L (X) so that {b ∈ } q lim T = T in the strong operator topology and T A B T for each b in b b b ⊂ b a directed set Ψ, where B is a closed operator, then T A BT . ⊆ Indeed, if x (A), then T x (B) and lim B T x = lim T Ax = ∈ D b ∈ D b b b b T Ax and hence lim T x = T x. Then T x (B) and BT x = T Ax, since b b ∈ D the operator B is closed, from which property (4) follows. We sum up these properties as the lemma. 19. Lemma. Suppose A is a closed operator acting in a Hilbert space X over the Cayley-Dickson algebra and CA AC for every operator C in Av ⊆ a self-adjoint subfamily of L (X), where 2 v. Then T A AT for each Y q ≤ ⊆ operator T in the von Neumann algebra over generated by . Av Y 20. Definition. When A is a closed R homogeneous additive oper- Av ator in a Hilbert space X over the Cayley-Dickson algebra , 2 v, and Av ≤ E is an graded projection on X so that EA AE and AE is a bounded Av ⊆ operator on X, we say that E is a bounding graded projection for A. Av An increasing sequence E : n N such that each E is a bounding {n ∈ } n graded projection for A and E = I will be called a bounding Av ∨n n Av 18 graded sequence for A. 21. Lemma. Let E be a bounding graded projection for a closed Av densely defined operator T in a Hilbert space over the Cayley-Dickson algebra , 2 v. Then E is bounding for T ∗, T ∗T and T T ∗ and Av ≤ (1) (T E)∗ = T ∗E∗. Moreover, if E : n is a bounding sequence for T , then E(X) is a {n } n n core for T and T ∗ and T ∗T and T T ∗. S Proof. The conditions of this lemma mean that ET T E and T E ⊆ is bounded, hence the operator ET is pre-closed and densely defined and bounded. Therefore, the operator ET has the closure T E, also (2) (T E)∗ = (ET )∗ is closed and hence the operators (T E)∗ and (ET )∗ are closed by Theorem I.3.34 [28]. For each vectors x E(X) and y (T ) ∈ ∈D the equality < Ty; x >=< y;(E∗T )∗x > is satisfied so that x (T ∗) ∈ D and T ∗x = (ET )∗x, consequently, T ∗E = (E∗T )∗E. At the same time we ∗ have (I E)E∗T = 0 and hence (E∗T )∗ = (E∗T ) = (E∗T )∗E = T ∗E. − Therefore, E∗T ∗ (T E)∗ and E is bounding for T ∗ in accordance with ⊆ Equality (2). Analogously E is bounding for T ∗T and similarly for T T ∗, since E(T ∗T ) (T ∗T )E. ⊆ This implies that E : n is a bounding sequence for T ∗, T ∗T and T T ∗ {n } if it is such for T . Then lim Ex = x, Ex (T ) and lim T Ex = n n n ∈ D n n limnnET x = T x for every x (T ). Thus ∞ E( (T )) is a core for T , consequently, ∈ D n=1 n D ∞ E(X) is a core for T Sand T ∗, T ∗T and T T ∗ as well, since E(X) n=1 n n ⊆ S (T ) for each natural number n. D 22. Remark. Mention that in accordance with Theorem 2.28 [20] an Av graded projection operator can be chosen self-adjoint E∗ = E for a quasi- commutative von Neumann algebra over , since E corresponds to a char- Av acteristic function e which is real-valued. 23. Theorem. Suppose that A is a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra , Av 2 v 3, acting on a Hilbert space X over , also T and BηA. Then ≤ ≤ Av (1) each finite set of operators affiliated with A has a common bounding
19 sequence in A; (2) an operator B + T is densely defined and pre-closed and its closure is B+ˆ T ηA; (3) BT is densely defined and pre-closed with the closure BˆT ηA affiliated · with A; (4) jBˆ kT =( 1)κ(j,k) kTˆ jB for each j, k, also B∗B = B∗ˆB = BB∗; · − · · (5) ((bI)B+ˆ T )∗ = B∗(b∗I)+ˆ T ∗ for each quaternion or octonion number b ; ∈Av (6) (BˆT )∗ = T ∗ˆB∗; · · (7) if B T , then B = T ; if B is symmetric, then B∗ = B; ⊆ (8) the family (A) of all operators affiliated with A forms a quasi- N commutative -algebra with unit I under the operations of addition +ˆ and ∗ multiplication ˆ given by (2, 3). · Proof. Take an arbitrary unitary operator U in the super-commutant A⋆ of A (see II.2.71 [28]). From T U = UT it follows that T ∗U = UT ∗, § hence T ∗ηA. Then (T ∗T )U = U(T ∗T ), since U ∗U = UU ∗ = I, consequently, (T ∗T )ηA. For an graded projection E in A an operator (2E I) is unitary Av − such that T (2E I)=(2E I)T , − − i.e. T (2E I)x = (2E I)T x for each x (T ) X, hence ET T E. − − ∈ D ⊂ ⊆ In view of Theorem I.3.34 [28] the operator T ∗T is self-adjoint. Take an Av graded resolution E : b of the identity for T ∗T and put F = E E {b } n n − −n for each natural number n. From Theorem 16 the inclusion E A follows. b ∈ ∗ The operator T T nF is bounded and everywhere defined, consequently, the operator T nF is everywhere defined and closed, since T is closed and nF is bounded. In accordance with the closed graph theorem 1.8.6 [12] for R linear operators one gets that T nF is bounded. It can lightly be seen also from the estimate T F x 2 =< F x; T ∗T Fx > T ∗T F x 2. A sequence F : n k n k n n ≤k n kk k {n } of projections is increasing with least upper bound I for which F T T F . n ⊆ n Therefore, the limits exist:
limn nF x = x and limn T nF x = limnnF T x = T x for each vector x in the domain (T ), consequently, ∞ F (X) is a core D n=1 n S 20 for T and F is a bounding graded sequence in A for the operator T . n Av Let now E : n be a bounding sequence in A for T : p =1, ..., m {n } {p − 1 A and let F be a bounding sequence in A for T A. Then } ⊂ n m ∈ E F : n is a bounding sequence in A for T, ..., T , particularly, ∞ E F (X) {n n } 1 m n=1 n n is a common core for 1T, ..., mT . This implies that two operators T S+ B and T ∗ + B∗ are densely defined, but T ∗ + B∗ (T + B)∗, consequently, (T + B)∗ ⊆ is densely defined and T + B is pre-closed (see also Theorem I.3.34 [28]). As soon as E : n is a bounding sequence in A for T , B, T ∗ and {n } B∗, the inclusions are satisfied ET T E and EB B E and hence n ⊆ n n ⊆ n E(T B) (T B) E and T EB E T B E E. The operators T E and n ⊆ n n n ⊆ n n n B nE are bounded and defined everywhere, consequently,
(9) T nEB nE = T BnE (see also Theorem 2.28 [20]). This means that E : n is a bounding {n } sequence for T B and analogously for BT and B∗T ∗. That is, the operator B∗T ∗ is densely defined. Since B∗T ∗ (T B)∗, one gets that the operator ⊆ (T B)∗ is densely defined and T B is pre-closed. From Formula (9) we infer that kT jB E =( 1)κ(j,k) jB kT E n − n for each j, k, since A is quasi-commutative. Therefore, the operators TˆB · and BˆT agree on their common core ∞ E(X) and inevitably · n=1 n S (10) jBˆ kT =( 1)κ(j,k) kTˆ jB. · − ·
The operators T ∗T and T T ∗ are self-adjoint, consequently, T ∗T = T ∗ˆT and · T T ∗ = TˆT ∗ and T ∗ˆT = TˆT ∗. Then U ∗x and Ux (T + B) for each · · · ∈ D x (T ) (B)= (T + B), consequently, U (T + B) = (T + B) and ∈D ∩D D D D (T + B)U = U(T + B). Thus (T +ˆ B)U = U(T +ˆ B), as well as (T +ˆ B)ηA. For each vector y (T B) the inclusions follow y (B) and By ∈ D ∈ D ∈ (T ). Therefore, Uy (B) and BUy = UBy (T ), consequently, D ∈ D ∈ D Uy (T B). Then we get U (T B) = (T B), since U ∗y (T B). But ∈ D D D ∈ D (T B)Uy = U(T B)y, hence (TˆB)Uy = U(TˆB)y, and inevitably (TˆB)ηA. · · · Having a bounding graded sequence E : n for T and T ∗ one gets Av {n } ET ∗ T ∗ E and E∗T ∗ (T E)∗, consequently, T ∗ E and (T E)∗ n ⊆ n n ⊆ n n n
21 ∗ ∗ ∗ are bounded everywhere defined extensions of operators nET and nE T ∗ ∗ ∗ respectively. In view of Lemma 21 the equality (T nE) = nE T follows. We now consider a bounding graded sequence E : n for T , T ∗, Av {n } B, B∗, ((bI)T +ˆ B), ((bI)T +ˆ B)∗, (TˆB), (TˆB)∗ and T ∗ˆB∗. From the · · · preceding demonstration we get the equalities
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (T (b I)+ˆ B ) nE =(T (b I)) nE+ˆ B nE = [((bI)T +ˆ B) nE ] = ((bI)T +ˆ B) nE and (TˆB)∗ E = [(TˆB) E∗]∗ =(B∗ˆT ∗) E · n · n · n due to (9, 10). The operators ((bI)T +ˆ B)∗ and (T ∗(b∗I)+ˆ B∗) agree on their
∞ ∗ ∗ ∗ ∗ common core n=1 nE(X), consequently, ((bI)T +ˆ B) =(T (b I)+ˆ B ). Anal- ogously we inferS that (TˆB)∗ = B∗ˆT ∗. · · Then T E B E and hence T E = B E as soon as T B and n ⊆ n n n ⊆ E : n is a bounding graded sequence in A for T and B. Therefore, {n } Av ∞ the operators T and B are consistent on their common core n=1 nE(X) and hence T = B. If T is symmetric, then T T ∗ and from theS preceding ⊆ conclusion one obtains T = T ∗. For any three operators T,B,C (A) we take a common bounding ∈N Av graded sequence E : n and get {n } (TˆB)ˆC = Tˆ(BˆC), since · · · · [(TˆB)ˆC] E = [Tˆ(BˆC)] E · · n · · n for all n (see also [30]). From this Statement (8) of the theorem follows. 24. Lemma. Suppose that F : n is a bounding graded sequence {n } Av for the closed operator T on a Hilbert space X over the Cayley-Dickson al- gebra , 2 v, and T F is normal for each natural number n. Then T Av ≤ n is normal.
∗ ∗ ∗ Proof. In view of Lemma 21 we have the equalities (T nF ) = nF T ∗ ∗ ∗ ∗ ∗ ∗ and nF T = T nF and T T nF = T nF T nF and (T T ) nF = (T T ) nF . Therefore, the self-adjoint operators T ∗T and T T ∗ agree on their common
∞ ∗ ∗ core n=1 nF (X), consequently, T T = T T , i.e. the operator T is normal. 25.S Remark. The condition T ∗T = T T ∗ is equivalent to
[(T ∗)ik T ij T ij (T ∗)ik ]=0on (T ∗T ) = (T T ∗), since πj = I j,k − D D j (seeP2). P § If BT T B in a Hilbert space X over the Cayley-Dickson algebra, ⊆ 22 then BT TB for B = iB = 0 B and T = iT = 0 T defined on ⊆ −B 0 −T 0 (B)= (B) (B)i and (T)= (T ) (T )i with i = 0 1 (see also D D ⊕D D D ⊕D −1 0 18). § 26. Lemma. Let BT T B and (T ) (B), let also T be a self- ⊆ D ⊆ D adjoint operator and let B be a closed operator in a Hilbert space X over either the quaternion skew field or the octonion algebra , 2 v 3. Av ≤ ≤ Then EB B E for each E in the spectral graded resolution E : b b ⊆ b b Av {b } of T . Proof. There is the decomposition alg (I,B,T, iI, iT, iT )= alg (I,B,T ) Av Av ⊕ alg (I,B,T )i and this family is defined in the Hilbert space X Xi. From Av ⊕ Formula 18(3) we have the inclusion (BT + BiI) B(T + iI). Now we ⊆ consider these operators with domains in X Xi denoted by (T ), (B), ⊕ D D etc. Take an arbitrary vector x (B(T + iI)), then x (T ) and ∈ D ∈ D T x + ix (B). But from the suppositions of this lemma we have the inclu- ∈D sion (T ) (B), consequently, T x (B) and hence x (BT + B(iI)) D ⊆D ∈D ∈D and B(T +iI)x = BT x+B(ix). Thus B(T +iI) BT +B(iI), consequently, ⊆ B(T + iI)= BT + B(iI). Denote by Q and Q the bounded everywhere defined inverses to (T iI) − + − and (T + iI) respectively. In view of 18(1 3) and the preceding proof we − infer Q B = Q B(T iI)Q = Q (BT B(iI))Q Q (T B B(iI))Q = ± ± ± ± ± ± ± ⊆ ± ± ± Q (T iI)BQ and hence (Q )B B(Q ). From 10 one has Q =(Q )∗ ± ± ± ± ⊆ ± § + − and applying Lemma 19 one gets QB BQ for each element Q in the ⊆ von Neumann algebra over generated by Q and Q . To an graded Av + − Av projection operator E on X an operator E E on X Xi corresponds. b b ⊕ b ⊕ Particularly, this means the inclusion EB B E in X for each b. b ⊆ b 27. Theorem. Suppose that T is an operator on a Hilbert space X over either the quaternion skew field or the octonion algebra , 2 v 3. An Av ≤ ≤ operator T is normal if and only if it is affiliated with a quasi-commutative von Neumann algebra A over . Moreover, there exists the smallest such Av algebra 0A. Proof. In view of Theorem 23 if an operator is affiliated with a quasi- commutative von Neumann algebra A over either the quaternion skew field
23 or the octonion algebra with 2 v 3, then it is normal. Suppose that Av ≤ ≤ an operator T is normal. Then Lemma 26 is applicable, since T T ∗T = T ∗T T and (T ∗T ) (T ). D ⊆D For an graded spectral resolution E : b R this implies that Av {b ∈ } ET T E for each b, hence F T T F for each natural number n, b ⊆ b n ⊆ n where F = E E. Then we also have T ∗T ∗T = T ∗T T ∗ and (T ∗T )= n n − −n D (T T ∗) (T ∗), consequently, F T ∗ T ∗ F . From 23 it follows that the D ⊆D n ⊆ n § ∗ operators T nF and nF T are bounded, since such the operator T T nF = T T ∗ F is. But then F ∗T ∗ (T F )∗ and hence both operators (T F )∗ n n ⊆ n n ∗ ∗ and T nF are bounded extensions of the densely defined operator nF T when choosing bE and hence nF self-adjoint for each b and n. Thus one gets ∗ ∗ ∗ ∗ the equalities (T nF ) = T nF and (T nF ) = T nF . On the other hand, there are the inclusions T F T F T T F and T F T F T T F for n m ⊆ n m n ⊆ n each n m. ≤ The operators T nF T mF and T mF T nF are everywhere defined, con- sequently, T nF T mF = T T nF = T mF T nF. There are the equalities ∗ ∗ ∗ ∗ T mF T nF = T T nF = T T nF = T nF T mF , hence
A := cl[alg F, T F, T ∗ F : n N ] 0 Av {n n n ∈ } is a quasi-commutative von Neumann algebra over the algebra , since there Av is the inclusion alg F, T F, T ∗ F : n N L (X). Av {n n n ∈ } ⊂ q Moreover, one has that ∞ F (X) =: is a core for T , since ∞ F = n=1 n Y ∨n=1 n I and F T T F . If USis a unitary operator in ( A)⋆ and x , then n ⊆ n 0 ∈ Y the equalities T Ux = T U nF x = T nF Ux = UT nF x = UT x are fulfilled ∗ for some natural number n. In accordance with Remark 2 T η 0A and T η 0A also. If T η , then T ∗η and T ∗T η are affiliations as well. From Note 15 R R R it follows that the self-adjoint operator T ∗T generates a quasi-commutative von Neumann algebra A contained in , consequently, F and hence R n ∈ R T F, T ∗ F . This implies the inclusion A . n n ∈ R 0 ⊂ R 28. Definition. The algebra 0A from the preceding section will be called the von Neumann algebra over the Cayley-Dickson algebra with 2 v Av ≤ generated by the normal operator T . 29. Theorem. Suppose that A is a quasi-commutative von Neumann
24 algebra over either the quaternion skew field or the octonion algebra , Av 2 v 3, also φ is an isomorphism of A onto C(Λ, ), where Λ is a ≤ ≤ Av compact Hausdorff topological space, T ηA. Then there exists a unique normal function φ(T ) on Λ so that φ(T E) = φ(T )ˆφ(E), when E is an graded · Av bounding projection in A for T , where (φ(T )ˆφ(E))(z) = φ(T )(z)φ(E)(z), · if φ(T )(z) is defined and zero in the contrary case, z Λ. If (Λ, ) ∈ N Av is the family of all valued normal function on Λ and f,g (Λ, ), Av ∈ N Av then there are unique normal functions f˜, sf, fs, f+ˆ g and fˆg so that · f˜(z) = f(z), (sf)(z) = sf(z), (fs)(z) = f(z)s, (f+ˆ g)(z) = f(z)+ g(z) and (fˆgg)(z) = f(z)ˆg(z), when f and g are defined at z Λ, s . · · ∈ ∈ Av Endowed with the operations f f˜ and (s, f) sf and (f,s) fs and 7→ 7→ 7→ (f,g) f+ˆ g and (f,g) fˆg, the family (Λ, ) is a quasi-commutative 7→ 7→ · N Av algebra with unit 1 and involution f f˜, it is associative over the quaternion 7→ skew field H = and alternative over the octonion algebra O = . The A2 A3 natural extension of φ is a -isomorphism of (A) onto (Λ, ). ∗ N N Av Proof. In view of Theorem 23 an operator T affiliated with A has an graded bounding sequence E : n in A. If φ has the properties de- Av {n } scribed above, if also f and g are normal functions defined at a point z and corresponding to φ(T ), then
f(z)φ(nE)(z)= φ(T nE)(z)= g(z)φ(nE)(z) for every natural number n. Thus if φ(nE)(z) = 1 for some natural number n, then certainly f(z) = g(z). Applying Lemma 9 we obtain, that f = g, since the sequence nE is monotone increasing to I, while f and g agree on a dense subset of Λ. Thus an isomorphism φ is unique. On the other hand, Theorem 16 provides φ(T ) with the required prop- erties for each T (A), where (A) denotes a family of all self-adjoint ∈ Q Q operators affiliated with A. This means that φ is defined on (A). Q For any pair of operators T , B in (A) it is possible to choose a bounding Q sequence E : n for both T and B by Theorem 23. Therefore, the opera- {n } tors T nE, B nE, (T +B) nE and T B nE belong to the quasi-commutative von Neumann algebra A over the Cayley-Dickson algebra so that Av (T +ˆ B) nE = T nE + B nE and (TˆB) E = T B E. Therefore, we deduce the equalities · n n 25 φ(T +ˆ B)(z)φ(nE)(z)= φ((T +ˆ B)nE)(z)= φ(T )(z)φ(nE)(z)+φ(B)(z)φ(nE)(z) and φ(TˆB)(z)φ( E)(z)= φ((TˆB) E)(z)= φ(T )(z)φ(B)(z)φ( E)(z), · n · n n when φ(T ), φ(B), φ(T +ˆ B) and φ(TˆB) are defined at z. The pairs · [φ(T +ˆ B); φ(T )+ φ(B)] and [φ(TˆB); φ(T )φ(B)] agree on dense subsets of · Λ. Therefore, images φ(T +ˆ B) and φ(TˆB) are finite, when φ(T ) and φ(B) · are defined, hence φ(T +ˆ B) and φ(TˆB) are normal extensions of φ(T )+φ(B) · and φ(T )φ(B) correspondingly. Then φ(T )+ˆφ(B) and φ(T )ˆφ(B) are defined · as normal extensions of φ(T )+ φ(B) and φ(T )φ(B) respectively. Each func- tion w (Λ) corresponds to some operator T (A) and by Theorem 16 ∈ Q ∈ Q the operations +ˆ and ˆ are applicable to all functions in (Λ). An iterated · Q application of Lemma 9 shows that (Λ) endowed with these operations is Q a commutative algebra over the real field R with unit 1, since each function in (Λ) is real valued. Moreover, φ is an isomorphism of (A) onto (Λ). Q Q Q If v is finite, for each j = 0, ..., 2v 1 the R-linear projection operator − πˆj :=π ˆj : A A i is expressible as a sum of products with generators and v → j j j real constants due to Formulas (1, 2) below so thatπ ˆ (A)= ijAj = Ajij: v (1)π ˆj(A)=( i (Ai ) (2v 2)−1 A + 2 −1 i (Ai∗) )/2 − j j − − {− k=1 k k } for each j =1, 2, ..., 2v 1, P − 2v−1 (2)π ˆ0(A)=(A + (2v 2)−1 A + i (Ai∗) )/2, − {− k k } kX=1 where 2 v N, ≤ ∈ (3) A = j Ajij, A A forP each j, A A. j ∈ j ∈ If A is embedded into L (X) for a Hilbert space X over the algebra , q Av we can consider projections πj : A A i and get decomposition (3) so that → j j A A for each j. The function φ(A )i is defined on Λ W when j ∈ j j j \ j j φ(A ) is defined on X W . But ΛP W is everywhere denseS in Λ, since j \ j \ j j z = z2 is the norm on . If p S W , then | | j j Av ∈ j qP 2 lim φ(A)(x) = φ(Aj) (x)= . x∈Λ\ Wj , x→p | | ∞ j sXj S That is φ(A )i is an element ˆ φ(A )i = φ(A) (Λ, ). j j j j j j ∈N Av P P 26 In accordance with Theorem 6.2.26 [6] a topological space Λ is extremely disconnected if and only if for each pair of nonintersecting open subsets U and V in Λ their closures always cl(U) cl(V )= do not intersect. By Theorem ∩ ∅ 8.3.10 [6] if (S, ) is a uniform space and (Y, ) is a complete uniform space, U V then each uniformly continuous mapping f :(P, ) (Y, ), where P is an UP → V everywhere dense subset in S relative to a topology induced by a uniformity , has a uniformly continuous extension f :(S, ) (Y, ). U U → V A set ξ−1(0) is closed in Λ, when ξ (Λ, ). Indeed, a function ξ is ∈ N Av continuous on Λ W , consequently, ξ−1(0) is closed in Λ W . At the same \ ξ \ ξ time the limit lim ξ(x) = x→p, x∈Λ\Wξ | | ∞ is infinite for each p W , hence p / cl[ξ−1(0)]. This fact implies that ∈ ξ ∈ the interior K := Int[ξ−1(0)] is clopen in Λ. Thus the function g defined such that g(x)=1on K, while g(x) := ξ(x)/ ξ(x) on Λ (ξ−1(0) W ), | | \ ∪ ξ is continuous on [Λ (ξ−1(0) W )] K. This function g has a continuous \ ∪ ξ ∪ extension q C(Λ, ) by Theorems 6.2.26 and 8.3.10 [6], since Λ ( ξ−1(0) ∈ Av \ { \ K W ) is a dense open subset in Λ. } ∪ ξ Indeed, Cantor’s cube Dm is universal for all zero-dimensional spaces of topological weight m , where D is a discrete two-element space (see ≥ ℵ0 Theorem 6.2.16 [6]). If m < , then Λ is discrete and this case is trivial. ℵ0 Therefore, if m , the compact topological space Λ has an embedding into ≥ℵ0 Dm as a closed subset. Then a uniformity compatible with its topology U can be chosen non-archimedean. That is, each pseudo-metric ρ in a family P of all pseudo-metrics inducing a uniformity on Λ satisfies the inequality U ρ(x, y) max(ρ(x, z); ρ(z,y)) for each x, y, z Λ. ≤ ∈ The function g(x) has values in the unit sphere in , which is closed in Av the Cayley-Dickson algebra and hence is complete. If z (ξ−1(0) W ) K Av ∈ ∪ ξ \ and x : α Σ [Λ (ξ−1(0) W )] K is a Cauchy net converging to { α ∈ } ⊂ \ ∪ ξ ∪ z, then x : α Σ can be chosen such that the limit lim g(x ) exists, { α ∈ } α α where Σ is a directed set. For any other Cauchy net y : α Σ [Λ { α ∈ } ⊂ \ (ξ−1(0) W )] K converging to z the limit of the net g(y ): α Σ exists ∪ ξ ∪ { α ∈ } and will be the same for g(x ): α Σ , since each pseudo-metric ρ P is { α ∈ } ∈
27 non-archimedean and the function g is continuous on [Λ (ξ−1(0) W )] K \ ∪ ξ ∪ (see also Propositions 1.6.6 and 1.6.7 and Theorem 8.3.20 [6]). For a point x Λ W the equality q(x) ξ(x) = ξ(x) is fulfilled. Since ∈ \ ξ | | ξ (Λ), one gets also normal extensions q ˆ ξ for ξ defined on Λ W . | | ∈ Q j·| | j \ ξ Thus the functions ξ,ξ are in (Λ, ) for each j. Choosing A (A) j N Av j ∈ Q as φ(Aj) = ξj leads to the equalities φ(ˆ jAjij)= ˆ jξjij = ξ. Thus this function φ maps (A) onto (Λ, ). N N Av P P As soon as functions f,g (Λ, ) are defined on Λ W and Λ W ∈N Av \ f \ g respectively, then components fj and gj have normal extensions for each j as follows from the proof above. Therefore, their sum f + g and product fg defined on Λ (W W ) have the normal extensions ˆ (f +ˆ g )i = f+ˆ g \ f ∪ g j j j j and ˆ (f ˆg )i i = fˆg. j,k j· k j k · P The set (Λ, ) supplied with these operations is the algebra over the P N Av Cayley-Dickson algebra with 2 v. Moreover, the algebra (Λ, ) Av ≤ N Av is quasi-commutative by its construction and with unit 1 and adjoint op- eration f f˜. This algebra (Λ, ) is associative over the quaternion 7→ N Av skew field H = and alternative over the octonion algebra O = , since A2 A3 the multiplication of functions is defined point-wise, while the quaternion skew field H is associative and the octonion algebra O is alternative. There- fore, the mapping φ has an extension up to a -isomorphism of (A) onto ∗ N (Λ, ). For T ηA and a bounding graded projection E in A for T we N Av Av get φ(T E)= φ(T ˆ E)= φ(T ) ˆ φ(E). · · 30. Definition. The spectrum sp(T ) of a closed densely defined operator T on a Hilbert space X over the Cayley-Dickson algebra with 2 v is Av ≤ the set of all those Cayley-Dickson numbers z for which an operator ∈ Av (T zI) is not a bijective R linear additive mapping of (T ) onto X, − Av D where (T ) as usually denotes a vector domain of definition of T . D Av 31. Remark. For an operator T from Definition 30 if z / sp(T ), then ∈ (T zI) is bijective from (T ) onto X and has an R linear additive − D Av inverse B = (T zI)−1 : X (T ). The graph of B is closed, since it is − → D such for (T zI). In accordance with the closed graph theorem 1.8.6 [12] − this inverse operator B is bounded. Thus we get z / sp(T ) (T zI) has ∈ ⇔ − a bounded inverse from X onto (T ). If T ηA for some quasi-commutative D 28 von Neumann subalgebra in L (X) and z / sp(T ), then B A. From the q ∈ ∈ boundedness of B and closedness of (T zI) it follows that − I =(T zI)B =(T zI) ˆ B = B ˆ (T zI). − − · · − Therefore, z / sp(T ) is equivalent to: (T zI) has an inverse B in the ∈ − algebra (Λ, ) and B A. N Av ∈ 32. Proposition. Let T be a normal operator affiliated with a quasi- commutative von Neumann algebra A over either the quaternion skew field or the octonion algebra with 2 v 3. Then sp(T ) coincides with the range Av ≤ ≤ of φ(T ), where φ denotes the isomorphism of (A) onto (Λ, ) extending N N Av the isomorphism of A with C(Λ, ) (see 29). Av § Proof. In accordance with Definition 30 z / sp(T ) if and only if there is ∈ a bounded B inverse to (T zI). For a unitary operator U in A⋆ one has the − equality U ∗(T zI)U =(T zI), since (T zI)ηA. Therefore, U ∗BU = B − − − for each such unitary operator U and B A. But an operator B A exists ∈ ∈ so that (T zI) ˆ. B = I. Thus the equality φ(T zI) ˆ. φ(B) = 1 is satisfied − − if and only if z / sp(T ). This means that there exists such φ(B) C(Λ, ) ∈ ∈ Av if and only if a Cayley-Dickson number z does not belong to the range of φ(T ), hence sp(T ) is the range of φ(T ). 33. Definition. A self-adjoint operator T is called positive when < T x; x> 0 ≥ for each vector x (T ) in its domain. ∈D 34. Note. A function f z1 for f (Λ, ) and z has not an − ∈ N Av ∈ Av inverse in (Λ, ) if and only if f z1 vanishes on some non-void clopen N Av − subset in Λ, where 2 v 3. Considering this in (A) we get a non-zero ≤ ≤ N graded projection operator F A so that (T zI) ˆ F = 0. The latter is Av ∈ − · equivalent to the existence of a non-zero vector x on which (T zI)x = 0. − In this situation one says that z is in the point spectrum of T . Thus the spectrum of T relative to (A) is its point spectrum. N 35. Proposition. Let T be a self-adjoint operator in a Hilbert space X over either the quaternion skew field or the octonion algebra. Then T is positive if and only if z 0 for each z sp(T ). ≥ ∈ Proof. In view of Lemma 10 it is sufficient to consider the variant, when T is affiliated with a quasi-commutative von Neumann algebra A. There
29 exists an isomorphism φ of (A) onto (Λ, ) extending an isomorphism N N Av of A onto C(Λ, ) (see Theorem 29), where Λ is an extremely disconnected Av compact Hausdorff topological space. Whenever φ(T ) is defined on Λ W \ φ(T ) and φ(T )(y) < 0 for some point y Λ W , there exists a non-void clopen ∈ \ φ(T ) subset P containing y and a negative constant b< 0 so that P Λ W ⊂ \ φ(T ) and φ(T )(x)
3 Normal operators and homomorphisms
37. Definition. A subset P of a topological space W is called nowhere dense in W if its closure has empty interior. A subset B in W is called meager or ∞ of the first category if it is a countable union B = j=1 Pj of nowhere dense subsets Pj in W . S There is said that the mapping B(Φ, R) f f(T ) L (X) with the ∋ 7→ ∈ q monotone sequential convergence property is σ-normal. 38. Lemma. Suppose that Λ is an extremely disconnected compact Haus- dorff topological space. Then each Borel subset of Λ differs from a unique clopen subset by a meager set. Each bounded Borel function g from Λ into the Cayley-Dickson algebra with card(v) differs from a unique con- Av ≤ℵ0
30 tinuous function f on a meager set. There exists a conjugation-preserving σ-normal homomorphism θ from the algebra (Λ, ) of bounded Borel func- B Av tions onto C(Λ, ). Its kernel ker(θ) consists of all bounded Borel functions Av vanishing outside a meager set. Proof. As card(v) the Cayley-Dickson algebra is separable and ≤ ℵ0 of countable topological weight as the topological space relative to its norm
n topology, since card( N )= . n∈ ℵ0 ℵ0 Consider the familyS of all subsets contained in Λ which differ from a F clopen subset by a meager set. Take an arbitrary V and a clopen subset ∈F Q so that (V Q) (Q V ) =: P is meager, i.e. Λ V and Λ Q differ by this \ ∪ \ \ \ same meager set. Since Λ Q is clopen, one gets (Λ V ) . In addition \ \ ∈ F each open set U in Λ belongs to , since cl(U) is clopen and cl(U) U is F \ nowhere dense in Λ. For a sequence V , Q ,P : j =0, 1, 2, ... of such sets { j j j } the inclusion is valid:
∞ ∞ ∞ ∞ ∞ [( V ) ( Q )] [( Q ) ( V )] P . j \ j ∪ j \ j ⊆ j j[=1 j[=1 j[=1 j[=1 j[=1 The set ∞ P is meager and the set ∞ Q is open, hence ( ∞ V ) j=1 j j=1 j j=1 j ∈ . ThusS the family contains the σ-algebraS generated by openS subsets, F F consequently, contains the Borel σ-algebra (Λ) of all Borel subsets in Λ. F B ∞ Theorem 3.9.3 [6] tells that the union j=1 Kj of a sequence of nowhere dense subsets Kj in a Cˇhech-complete topologicalS space W is a co-dense subset, i.e. its complement set W ∞ K is everywhere dense in W . On the \ j=1 j other hand, each topological spaceS metrizable by a complete metric is Cˇhech- complete, also each locally compact Hausdorff space is Cˇhech-complete [6]. This implies that the complement of a meager set is dense in Λ. Therefore, two continuous functions agree on the complement of a meager set only if they are equal. This means that there exists at most one continuous function agreeing with a given bounded Borel function on the complement of a meager set.
Let now V be a Borel subset of Λ, let also g = χV be the characteristic function of V . Take a clopen subset Q in Λ so that (Q V ) (V Q) =: P is \ ∪ \ a meager subset in Λ. The characteristic function f = χQ of Q is continuous and (g f) is zero on Λ P . Therefore, there exists at most one clopen − \ 31 subset in Λ differing from V by a meager set. Therefore, a step function being a finite vector combination of characteristic functions of disjoint Av Borel subsets in Λ differs from a continuous function on the complement of a meager set. The set of step functions is dense in the vector space (Λ, ) relative Av B Av to the supremum-norm. This means that if g is a Borel function g :Λ , →Av g := sup g(x) < , then there exists a sequence of Borel step functions k k x∈Λ | | ∞ g so that lim g g = 0. Let f be a sequence of continuous functions n n→∞ kn − k n agreeing with ng on the complement of a meager set Pn. Therefore, the inequality f f g g if fulfilled, since f f and g g kn − m k≤kn − m k n − m n − m agree on the complement of a meager set P P . The set P P is meager n ∪ m n ∪ m and f(x) f(x) g g for each x Λ (P P ). Therefore, |n − m |≤kn − m k ∈ \ n ∪ m f : n is a Cauchy sequence converging in supremum norm to a continuous {n } function f C(Λ, ). Therefore, f and g agree on Λ ( ∞ P ), where a ∈ Av \ n=1 n ∞ S set ( n=1 Pn) is meager. IfS functions g1 and g2 (Λ, ) differ from f 1 and f 2 C(Λ, ) ∈ B Av ∈ Av on meager sets P 1 and P 2, theng ˜1, bg1 + g2, g1b + g2 and g1g2 differ from f˜1, bf 1 + f 2, f 1b + f 2 and f 1f 2 on a subset of P 1 P 2. Thus ∪ the mapping θ : (Λ, ) g f C(Λ, ) is a conjugate preserving B Av ∋ 7→ ∈ Av surjective homomorphism so that θ(g) = 0 if and only if g vanishes on the complement of a meager set. ֒ (Particularly, if g : n is a monotone increasing sequence in (Λ, R {n } B → (Λ, ) of bounded Borel real-valued functions tending point-wise to the B Av bounded Borel function g, each continuous real-valued function f C(Λ, ) n ∈ Av differs from g on the meager set P , then f(x) f(x) for each x n n n ≤ n+1 ∈ Λ (P P ) in a dense set so that f f. Therefore, f f \ n ∪ n+1 n ≤ n+1 n ≤ for each natural number n, while f differs from g on the meager set P . Thus the sequences g(x): n and f(x): n tend to f(x) for each {n } {n } x Λ (P ( ∞ P )). This means that the function f is the least up- ∈ \ ∪ n=1 n per bound in CS(Λ, R) of the sequence f : n and the homomorphism θ is {n } σ-normal. 39. Corollary. Suppose that U is an open dense subset in an extremely disconnected compact Hausdorff topological space Λ and f is a continuous
32 bounded function f : U , card(v) . Then there exists a unique → Av ≤ ℵ0 continuous function ξ :Λ extending f from U onto Λ. →Av Proof. Put g(x)= f(x) for each x U, while g(x) = 0foreach x Λ U, ∈ ∈ \ hence g is a bounded Borel function on Λ. From Lemma 38 we know that a unique continuous function ξ :Λ exists so that ξ and g differ on → Av the complement of a meager set. If ξ(x) = f(x) for some x U, then by 6 ∈ continuity of f ξ on U we get that f(x) = ξ(x) on a non-void clopen subset − 6 W of U. But W is not meager. This contradicts the choice of ξ. Thus ξ is the continuous extension of f from U on Λ. 40. Remark. If Λ is metrizable, the condition card(v) can be ≤ ℵ0 dropped in Lemma 38 and Corollary 39 in accordance with 31 in chapter § 2 volume 1 [17]. We denote by (Λ, ) the family of all Borel functions Bu Av f :Λ . →Av 41. Lemma. Suppose that Λ is an extremely disconnected compact Haus- dorff space. Let either Λ be metrizable or card(v) . Then a conjugation- ≤ℵ0 preserving surjective homomorphism ψ : (Λ, ) (Λ, ) exists so Bu Av → N Av that its kernel ker(ψ) consists of all those functions in (Λ, ) vanishing Bu Av on the complement of a meager set. Proof. As f and g are normal functions defined on Λ W and Λ W \ f \ g respectively and f(x)= g(x) for each x Λ (W W P ), where P is a ∈ \ f ∪ g ∪ meager subset of Λ, then Wf = Wg and f = g in accordance with Lemma 9, since the set (W W P ) is meager in Λ, while Λ (W W P ) is f ∪ g ∪ \ f ∪ g ∪ dense in Λ. This implies that at most one normal function can be agreeing with any function on the complement of a meager set. For each Borel function g :Λ and each ball B( ,y,q) := z → Av Av { ∈ : z y q with center y and radius q > 0, its inverse image Av | − | ≤ } g−1(B( ,y,q)) is a Borel subset V of Λ. In view of Lemma 38 a clopen Av q subset Q exists so that (V Q ) (Q V ) is meager in Λ. Take the n n \ n ∪ n \ n Borel function g such that g is equal to g on V and is zero on Λ V , n n n \ n consequently, it satisfies the inequality g n. Applying Lemma 38 one kn k ≤ gets a continuous function f :Λ so that that agrees with g on the n → Av n complement of a meager subset Pn of Λ. The function f vanishes on the subset Λ (V P ), since g vanishes n \ n ∪ n n 33 on Λ V . Certainly the set Λ (V P ) contains the subset (Λ Q ) \ n \ n ∪ n \ n \ (P (V Q ) (Q V )). The set (P (V Q ) (Q V )) is meager n ∪ n \ n ∪ n \ n n ∪ n \ n ∪ n \ n and f is continuous on Λ, hence this mapping f vanishes on Λ V . The n n \ n inclusion V V for each natural number n implies the inequality e (x) n ⊆ n+1 n ≤ en+1(x) for each point x in Λ outside a meager set, where en := χQn is the characteristic function of Q . But continuity gives e e and hence n n ≤ n+1 Q Q for every n N. The functions g and g are consistent on n ⊆ n+1 ∈ n+1 n Vn, consequently, continuous functions n+1f and nf agree on Qn. Then the inequality n f(x) for each x Q Q and n < m is fulfilled, since ≤ |m | ∈ m \ n n g(y) when y V V . The equality ∞ V = Λ leads to the ≤ |m | ∈ m \ n n=1 n inclusion S W =Λ ( ∞ Q ) ∞ (V Q ), f \ n=1 n ⊆ n=1 n△ n where A B :=S (A B) (BS A) for two sets. Therefore, the set W is closed △ \ ∪ \ f and meager, consequently, Wf is nowhere dense in Λ. If f(x) = nf(x) for each x Q and n N, then the mapping f is continuous on Λ W . ∈ n ∈ \ f For a point x W there exists a natural number n so that x Λ Q . ∈ f ∈ \ n When y Λ (W Q ) the inequality n f(y) is valid. Thus the function ∈ \ f ∪ n ≤| | f is normal. By the construction above two functions f and g agree on the complement of W ( ∞ P ), where the latter set is meager. This induces f ∪ n=1 n a conjugation-preservingS surjective homomorphism ψ : (Λ, ) g f B Av ∋ 7→ ∈ (Λ, ) with kernel consisting of those Borel functions vanishing on the N Av complement of a meager set in Λ. 42. Proposition. Let A be a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra with 2 Av ≤ v 3, let also T be a sequence of operators in (A) with upper bound T ≤ n Q 0 in (A). Then T : n =1, 2, 3, ... has a least upper bound T in (A). Q {n } Q Proof. An algebra A is -isomorphic with C(Λ, ), where Λ is a com- ∗ Av pact extremely disconnected Hausdorff space (see Theorem I.2.52 [28]). Take normal functions f in (Λ, ) representing T . Then B+ˆ T is the least n N Av n 1 upper bound of T : n , when B is the leat upper bound of T +ˆ T : n . {n } {n − 1 } This means that without loss of generality we can consider T 0 for each n ≥ n N. If a normal function f is defined on Λ W , then W = (W ) , ∈ n \ n n n + hence W W for each n. If x / W := ∞ W , then f(x) is de- n ⊆ n+1 ∈ n=0 n n S 34 fined for each natural number n and the sequence f : n has an upper {n } bound f(x) so that f h for each h ψ−1(φ( T )) (see Proposition 32 and 0 0 ≤ ∈ 0 Lemma 41). Thus f : n converges to some g(x). Put g to be zero on {n } W , then g is a Borel function on Λ. Applying Lemma 41 one gets a normal function f (Λ, ) agreeing with g on Λ P , where P is a meager subset ∈N Av \ in Λ. Thus the sequence f(x): n converges to f(x) on the dense subset {n } Λ (P W ). This means that f is the least upper bound for f : n . Then \ ∪ {n } an element T in (A) represented by f (Λ) is the least upper bound of Q ∈ Q T : n =1, 2, ... . {n } 43. Notes. Using Proposition 42 we can extend our definition of a σ- normal homomorphism to (A), (Λ, ), ( , ) and (Λ, ). N N Av Bu Av Av Bu Av In view of Lemma 41 this homomorphism from (Λ, ) onto (Λ, ) is Bu Av N Av σ-normal. Consider an increasing sequence g : n of Borel functions on Λ tending {n } point-wise to the Borel function 0g and take the σ-normal functions nf cor- responding to g. Then the sequence f : n has the least upper bound f n {n } 0 in (Λ, ). Indeed, the functions g and f agree on the complement of a N Av n n meager set Pn. Therefore, the limit exists limn nf(x)= 0f(x) for each point x Λ ( ∞ P ) and the subset Λ ( ∞ P ) is dense in Λ. If a function h is ∈ \ n=0 n \ n=0 n an upperS bound for f : n =1, 2, 3,S ... , then f(x) h(x) for each natural {n } n ≤ number n = 1, 2, 3, ... and points x Λ ( ∞ P ) in the complement of ∈ \ n=0 n the meager set ( ∞ P ). Thus f(x) h(x)S for all x in the complement of n=0 n 0 ≤ this meager set,S hence the mapping h+ˆ f has non-negative values on a − 0 dense set. Thus f h and f is the least upper bound in (Λ, ) of the 0 ≤ 0 N Av sequence f : n =1, 2, 3, ... . {n } Using Lemma 41 one can define g(T ) for an arbitrary Borel function g on sp(T ) for any normal operator T in a Hilbert space either over the quaternion skew field or the octonion algebra with 2 v 3. In accordance with Av ≤ ≤ Theorem 27 operators T , T ∗ and I generate a quasi-commutative von Neu- mann algebra A over the algebra with 2 v 3 such that T is affiliated Av ≤ ≤ with A. It is known from Theorems 2.24 and 2.28 [20] that A is isomorphic with C(Λ, ) for some extremely disconnected compact Hausdorff topolog- Av ical space Λ. In accordance with Theorem 29 there exists an isomorphism φ
35 of (A) onto (Λ, ). As φ(T ) is defined on Λ W , the function q defined N N Av \ to be zero on W and g φ(T )onΛ W is Borel, q (Λ, ). But Lemma ◦ \ ∈ Bu Av 41 says that a function h exists so that h (Λ, ) and h agrees with q ∈ N Av on the complement of a meager set in Λ. We now define g(T ) as φ−1(h). It may happen that sp(T ) is a subset of a Borel set V (in , for example) Ar and g is a Borel function on V , then g(T ) will denote g (T ), where g |sp(T ) |B denotes the restriction of g to a subset B V . ⊂ As usually (Y, ) denotes the algebra of all bounded Borel functions B Av from a topological space Y into the Cayley-Dickson algebra , while (Y, ) Av Bu Av denotes the algebra of all Borel functions from Y into . Av 44. Theorem. Let A be a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra , 2 v 3, Av ≤ ≤ so that A is generated by a normal operator T acting on a Hilbert space X over the algebra . Then the mapping θ : g g(T ) of the algebra Av 7→ (sp(T ), ) into (A) is a σ-normal homomorphism such that θ(1) = I Bu Av N and θ(id) = T . Moreover, the mapping V E(V ) of Borel subsets in 7→ into A is an graded projection-valued measure on a Hilbert space X, Av Av where E(V )= χ (T ), χ denotes the characteristic function of V ( ). V V ∈ B Av Suppose that h : is a Borel bounded function, ( , ) denotes Av → Av B Av Av the algebra over of all such Borel bounded functions, then Av (1) h(T ) sup h(y) =: h and k k ≤ y∈Av | | k k
(2) < h(T )x; x>= dµx(y).h(y) ZAv for each vector x X and every Borel bounded function h ( , ), ∈ ∈ B Av Av where µ (V ).h(y) :=< E(V ).h(y)x; x >. For f ( , ) a vector x x ∈ Bu Av Av belongs to a domain (f(T )) if and only if D 2 2 (3) dµx(y). f(y) =: f(T )x < ZAv | | k k ∞ and Formula (2) is valid with f in place of h. Let φ be an extension on (A) of the isomorphism ψ of A with C(Λ, ), N Av let also νx be a regular Borel measure on Λ so that
(4) < Bx; x>= dνx(t).(φ(B))(t) ZΛ 36 for each B A, then x (Q) with Q (A) if and only if ∈ ∈D ∈N
2 2 (5) dνx(t). (φ(Q))(t) =: Qx < . ZΛ | | k k ∞ If additionally T is a self-adjoint operator, its spectral resolution is E : b {b ∈ R , where E = I E((b, )), and x (f(T )) if and only if } b − ∞ ∈D ∞ 2 (6) d< bE. f(b) x; x >< Z−∞ | | ∞ and for such vector x (f(T )) in a domain of f(T ) the equality ∈D ∞ (7) < f(T )x; x>= d< bE.f(b)x; x> Z−∞ is valid. Proof. In accordance with Remark 43 there is a σ-normal homomor- phism ω from (sp(T )) into (Λ, ). On the other hand, the map- Bu Bu Av ping assigning h (Λ, ) to ω(g) is a σ-normal homomorphism, where ∈ N Av g (sp(T )). Therefore, the mapping g g(T ) is a σ-normal homo- ∈ Bu 7→ morphism from ( , ) into (A). When g = 1 is the constant unit Bu Av Av N function, the function h is also constant unit on Λ, consequently, g(T )= I. When g = id, then h = φ(T ), so that id(T ) = T , where id(x) = x for each x . ∈Av As V is a Borel subset of and g = χ is its characteristic function, Av V the identities are satisfied: g(T )∗ =g ˜(T )= g(T ) and g(T )= g2(T )= g(T )2. This means that g(T ) is an graded projection E(V ) in A. Particularly, Av χ = 0 and E( ) = 0, χ 1, χ (T )= I and E( )= I. ∅ ∅ Av ≡ Av Ar Recall that graded projection valued measures were defined in I.2.73 Av §§ and I.2.58. We use in this section the simplified notation E instead of Eˆ. For any countable family V : j of disjoint Borel subsets in and { j } Av their characteristic functions jg := χVj , their sums nh := 1g + ... + ng form an increasing sequence tending point-wise to the characteristic function h := χV , where V = ∞ V . Then n E(V ): n has the least upper bound E(V ) j=1 j { j=1 j } so that E(VS) = ∞ E(V ),P since g g(T ) is a σ-normal homomorphism. j=1 j 7→ Using our notationP we get:
< h(T )x; x>=< E(V )x; x>= µx(V )= dµx(t).h(t) ZAv 37 hence Equation (2) is valid for Borel step functions h. For any bounded Borel function h on with values in we have the Av Av inequality ω(h) h = sup h(x) k k≤k k x∈Ar | | and the function f (Λ, ) corresponding to ω(h) belongs to C(Λ, ) ∈ N Av Av by Lemma 38. We put f = θ(h). Then we infer, that
h(T ) = φ−1(f) = f h , k k k k k k≤k k since f ω(h) h . Each bounded Borel function in ( , ) is k k≤k k≤k k B Av Av a norm limit of Borel step functions, consequently, Equality (2) is valid for each h ( , ). ∈ B Av Av We consider now a self-adjoint operator T and the characteristic function g = χ of(b, ). Then θ(g) is the characteristic function of φ(T )−1((b, )) (b,∞) ∞ ∞ which is an open subset contained in Λ W and hence in Λ. Moreover, the \ function in C(Λ, ) corresponding to θ(g) is the characteristic function 1 e Av − b of cl[φ(T )−1((b, ))]. This means that E((b, )) = g(T ) A corresponds to ∞ ∞ ∈ 1 e and from Theorem 16 one gets E = I E((b, )). − b b − ∞ This implies that < ( E E)x; x>= µ ((b, c]) for any b c and c − b x ≤ ∞ 2 2 d< bE. f(b) x; x>= dµx(b). f(b) Z−∞ | | Z | | for each Borel function f ( , ), consequently, the last assertions of ∈ Bu Av Av this theorem reduce to Formulas (2, 3).
Denote by qn := χVn the characteristic function of the subset Vn := f −1([0, n]), where f ( , ) is a Borel function, we put F := q (T ). | | ∈ Bu Ar Av n n Therefore, nf(T ) = f(T ) nF due to the first part of the proof, where nf = fqn, since the functions f and qn commute, fqn = qn f. This implies that for a vector x X we get the equalities: ∈
2 2 2 (6) f(T ) nF x =< fn (T )x; x>= dµx(t). f(t) and k k | | ZVn | |
2 2 (7) f(t) nF x f(t) mF x = dµx(t). f(t) . k − k ZVn\Vm | | The sequence F : n is increasing with least upper bound I, since q : n {n } { n } is the increasing net of non-negative Borel functions tending point-wise to
38 1. Then we infer that nF f(T )x = f(T ) nF x for each vector x in a domain (f(T )), since F f(T ) f(T ) F . This implies that the limit exists D n ⊆ n
lim f(T ) nF x = f(T )x n and Formula (3) follows from (6). Vise versa, if the integral dµ (t). f(t) 2 converges, then one gets a Av x | | Cauchy sequence f(T ) F x : R n due to Formula (7) converging to some { n } vector in the Hilbert space X over the Cayley-Dickson algebra . But Av x (f(T )), since lim F x = x and the function f(T ) of the operator T ∈ D n n is closed. Quite analogous demonstration leads to Formula (5). We have a non- negative measure
µ (V ).1=< E(V ).1x; x>=< E2(V ).1x; x>=< E(V ).1x; E∗(V ).1x> 0 x ≥ for each Borel subset V . As x is a vector in the domain (f(T )), the func- D tion f belongs to the Hilbert space L2( ,µ , ), while L2( ,µ , ) Av x Av Av x Av ⊂ L1( ,µ , ), since µ .1 is a finite non-negative measure, where Lp( ,µ , ) Av x Av x Av x Av with 1 p < denotes an vector space which is the norm completion ≤ ∞ Av of the family of all step Borel functions u from into , where the norm Av Av is prescribed by the formula:
p p u := dµx(t). u(t) . k k sZAv | | Finally one deduces that
< f(T )x; x>= lim < f(T ) nF x; x>= lim qn(t)dµx(t).f(t)= dµx(t).f(t). n n ZAv ZAv 45. Remark. The graded projection E(V ) of the preceding theorem Av will also be referred as the spectral graded projection for T corresponding Av to the Borel subset V of , where 2 v 3. Av ≤ ≤ 46. Theorem. Suppose that T is a normal operator affiliated with a von Neumann algebra acting on a Hilbert space over either the quaternion skew field or the octonion algebra with 2 v 3 and ψ is a σ-normal Av ≤ ≤ homomorphism of the algebra ( , ) of Borel functions into (A) so Bu Av Av N that ψ(1) = I and ψ(id) = T , where id : denotes the identity Av → Av 39 mapping id(t) = t on . Then ψ(f) = f r(T ) for each f ( , ), Av ∈ Bu Av Av where f r = f denotes the restriction of f to sp(T ). |sp(T ) Proof. This homomorphism ψ is adjoint preserving, since ψ is σ-normal. Positive elements of ( , ) have positive roots, hence ψ is order pre- Bu Av Av serving. Moreover, ψ : ( , ) A and does not increase norm, since B Av Av → ψ(1) = I.
At first we consider the case when T is bounded. Put 0g := χAv\B, where B = B( , 0, 2 T ) is the closed ball in with center 0 and radius 2 T . Av k k Av k k Then 0 (2 T )n g id n for each natural number n =1, 2, 3, .... There- ≤ k k 0 ≤| | fore, ψ( id n)= T n such that 0 (2 T )nψ( g) T n, hence ψ( g) 2−n | | ≤ k k 0 ≤| | k 0 k ≤ for each natural number n, consequently, ψ(0g) = 0.
As 1g is the characteristic function of the ball B, we get ψ(1g)= I, such that ψ( gh)= ψ(h) for each Borel function h ( , ). 1 ∈ Bu Av Av Consider the restriction h = h of h to B and put ψ0( h) = ψ(h). 0 |B 0 Therefore, this mapping ψ0 is a σ-normal homomorphism of (B, ) into Bu Av (A) with the properties: ψ0(χ )= I and ψ0( id)= T and ψ0 : (B, ) N B 0 B Av → A. At the same time it is known from the exposition presented above that C(B, ) is a C∗-algebra over the algebra with unit being the constant Av Av 0 function χB. Therefore, by Theorem 1.3.17 [28] one gets that ψ (f) = ψ0(f( id)) = f(T ) for each continuous function f C(B, ). It is known 0 ∈ Av from I.3.17 that the characteristic function h of the open subset B sp(T ) § 1 \ of B is the point-wise limit of an increasing sequence f : n of positive {n } 0 functions nf on B, consequently, ψ (1h) is the least upper bound in A of the sequence ψ0( f): n , since the homomorphism ψ0 is σ-normal. Each func- { n } 0 tion nf is continuous and vanishes on sp(T ), consequently, ψ (nf) = nf(T ) 0 and nf(T ) = 0 and hence ψ (1h) = 0. For the characteristic function χsp(T ) of sp(T ) B one obtains the equalities: ψ0(χ )= I and ψ0(χ h)= ⊂ sp(T ) sp(T ) 0 ψ0( h) for each h (B, ). 0 0 ∈ Bu Av If put ψ1(q )= ψ0(q) for each q (B, ), then ψ1 : (sp(T ), ) |sp(T ) ∈ Bu Av Bu Av → (A) is a σ-normal homomorphism so that ψ1(1) = I and ψ1(id) = T and N ψ1( (sp(T ), )) A. In view of Theorem I.3.21 [28] applied to to the B Av ⊂ restriction ψ1 the equality ψ1(f) = f(T ) is valid for each Borel |B(sp(T ),Av) 40 bounded function f (sp(T ), ). ∈ B Av On the other hand, each positive function g is the point-wise limit of an increasing sequence of positive functions in (sp(T ), R) (sp(T ), ). B ⊂ B Av Therefore, ψ1(g) = g(T ) for each positive Borel function g (sp(T ), R), ∈ Bu since the homomorphism ψ1 : (sp(T ), ) (A) is σ-normal. Using Bu Av → N the decomposition h = h i of each Borel function h (sp(T ), ) j j j ∈ Bu Av with real-valued Borel functionsP h and h = h+ h− with non-negative j j j − j + − 1 Borel functions hj and hj we infer that ψ (h) = h(T ). This implies that if q ( , ) and q = q and qr = q , then ψ(q)= ψ0( q)= ψ1(qr)= ∈ Bu Av Av 0 |B |sp(T ) 0 qr(T ). We take now an arbitrary normal operator T (A) and a bounding ∈N Av graded projection E for T in A. The mapping φ posing (YˆE) (AE) · |E(X) ∈N acting on E(X) to Y (A) is a σ-normal homomorphism of (A) into ∈ N N (AE). Taking the composition of φ with ψ yields a σ-normal homomor- N phism ψ2 : ( , ) (AE) mapping 1 onto E and id onto T . Bu Av Av →N |E(X) |E(X) But the composition of φ with the mapping f f r(T ) of ( , ) into 7→ Bu Av Av (AE) is another homomorphism. The restriction T is bounded, con- N |E(X) sequently, from the first part of this proof we infer that
(ψ(f)ˆE) = ψ2(T )=(f r(T )ˆE) . · |E(X) |E(X) · |E(X)
Theorem 23 states that there exists a common bounding graded sequence Av E : n for T and ψ(f) and f r(T ), where f is a given element of ( , ). {n } Bu Av Av Then we deduce that
(ψ(f)ˆ E) =(ψ(f) E) =(f r(T )ˆ E) =(f r(T ) E) · n |nE(X) n |nE(X) · n |nE(X) n |nE(X)
r r so that ψ(f) nE = f (T ) nE for each n. Therefore, ψ(f) = f (T ), since ∞ r n=1 nE(X) is a core for both ψ(f) and f (T ). S 47. Note. The procedure of 43 assigning g(T ) A can be applied to § ∈ (Ω, ), where Y is another quasi-commutative von Neumann algebra over N Av either the quaternion skew field or the octonion algebra with 2 v 3 so Av ≤ ≤ that T is affiliated with Y and Y = C(Ω, ), Ω is an extremely disconnected ∼ Av compact Hausdorff topological space. The operator in (A) formed in this N way is g(T ) (Y) by Theorem 46. ∈N 41 48. Corollary. Let T be a normal operator satisfying conditions of Theorem 46 and let f and g be in ( , ), where 2 v 3. Then Bu Av Av ≤ ≤
(1) (f g)(T )= f(g(T )). ◦
Proof. Consider the von Neumann quasi-commutative von Neumann algebra over the algebra with 2 v 3 generated by T , T ∗ and I, then Av ≤ ≤ g(T ) (A) and f f g is a σ-normal homomorphism φ of ( , ) into ∈N 7→ ◦ Bu Av Av ( , ). Taking the composition of φ with the σ-normal homomorphism Bu Av Av h h(T ) of ( , ) into (A) leads to a σ-normal homomorphism 7→ Bu Av Av N ξ : f (f g)(T ) of ( , ) into (A) with ξ(1) = I and ξ(id)= g(T ). 7→ ◦ Bu Av Av N Then Formula (1) follows from Theorem 46. 49. Proposition. Suppose that ψ is a σ-normal homomorphism of (A) into (B) so that ψ(I)= I, where A and B are von Neumann quasi- N N commutative algebras over either the quaternion skew field or the octonion algebra , where 2 v 3. Then ψ(f(T )) = f(ψ(T )) for each T (A) Av ≤ ≤ ∈N and each f ( , ). ∈ Bu Av Av Proof. Each quasi-commutative algebra A over the algebra with Av 2 v 3 has the decomposition A = A i , where A and A are real ≤ ≤ j j j j k isomorphic commutative algebras for eachL j, k = 0, 1, 2, ..., 2v 1. On the − other hand, the minimal subalgebra algR(i , i , i ) for each 1 j = k is 0 j k ≤ 6 isomorphic with the quaternion skew field. Therefore, knowing restrictions
ψ A A A A for each 1 j 42 , where 2 v 3. Let also E be an graded projection in A and T ηA Av ≤ ≤ Av and f ( , ). Then the identity f((TˆE) )=(f(T )ˆE) is ∈ Bu Av Av · |E(X) · |E(X) fulfilled. Proof. Consider the mapping B (BˆE) which is a σ-normal 7→ · |E(X) homomorphism ψ of (A) onto (AE ) so that ψ(I) = E . But N N |E(X) |E(X) E is the identity operator on E(X). Then Proposition 49 implies that |E(X) f((TˆE) )= f(ψ(T )) = ψ(f(T ))=(f(T )ˆE) . · |E(X) · |E(X) 51. Note. Consider a situation when T and B are normal operators, where T and B may be unbounded operators whose spectra are contained in the domain of a Borel function g ( , ) and g has an inverse Borel ∈ Bu Av Av function f, where 2 v 3. Then g(T ) = g(B) if and only if T = B. ≤ ≤ To demonstrate this mention that if g(T ) = g(B), then T = (f g)(T ) = ◦ f(g(T )) = f(g(B))=(f g)(B)= B due to Corollary 48. ◦ Observe particularly, that if T 2 = B2 with positive operators T and B, then T = B. This means that a positive operator has a unique positive square root. Let y be a non-zero vector in X so that Ty = by for some b for a ∈ Av normal operator T , let also f ( , ) be a Borel function whose domain ∈ Bu Av Av contains sp(T ). Suppose in addition that T is strongly right linear, that Av is by our definition T (xu)=(T x)u for each x X and u . Take an ∈ ∈Av Av graded projection E with range x : T x = bx , which is closed since T is a { } closed operator. Therefore, T E = bE on X, since ET T E, and hence ⊆ f(T )y = [(f(T )ˆE) ]y = f((TˆE) )y = f(bE )y = f(bI)y · |E(X) · |E(X) |E(X) in accordance with Corollary 50. 52. Example. Let X be a separable Hilbert space over either the quaternion skew field or the octonion algebra with 2 v 3 and an Av ≤ ≤ orthonormal basis e : n N , let also A be the algebra of bounded { n ∈ } diagonal operators T e = t e , where t for each natural number n n n n ∈ Av n N = 1, 2, 3, ... . Then A is isomorphic with C(Λ, ), where Λ = βN is ∈ { } Av the Stone-Cˇhech compactification of the discrete space N due to Theorems 3.6.1, 6.2.27 and Corollaries 3.6.4 and 6.2.29 [6], since N Λ and Λ is ⊂ extremely disconnected. Take points sn corresponding to the pure states 43 T < T e ; e > of this algebra A, where n N. Then the set s : n N 7→ n n ∈ { n ∈ } is dense in Λ, since from < T en; en >= 0 for every n it follows that T = 0 and each continuous function f :Λ vanishing on s : n is zero. → Av { n } Consider the characteristic function χsn of the singleton sn. The projection corresponding to e lies in A and a continuous function corresponds to Av n this projection χ C(Λ, ). Thus s is an open subset of Λ. Therefore, sn ∈ Av n s : n is an open dense subset in Λ and its complement Z =Λ s : n { n } \{ n } is a closed nowhere dense subset in Λ. One can define the function h(s ) = t with lim t = , hence n n ∈ Av n | n| ∞ this function h is normal and defined on Λ Z. \ As t = nξ so that ξ with ξ = 1 for each n we get a normal n n n ∈ Ar | n| function f corresponding to an operator Q affiliated with A and Qen = nξnen. Choosing t =(n1/4 n)ξ we obtain a normal function g corresponding to n − n an operator B affiliated with A so that Be =(n1/4 n)ξ e . Take the vector n − n n x = ∞ n−1z e with z 1, ξ , ξ∗ for each n, then x X and n=1 n n n ∈ {± ± n ± n} ∈ ν ( sP )= n−2. Thus one gets x { n} 2 dνx(b). f(b) = and ZΛ | | ∞ ∞ 2 −3/2 dνx(b). (f + g)(b) = n < . Λ | | ∞ Z nX=1 The function f + g is normal when defined on Λ Z, since (f + g)(s ) = \ | n | n1/4 and f+ˆ g corresponds to Q+ˆ B. In view of Theorem 44 this vector →∞ x does not belong to (Q), but x (Q+ˆ B). Thus Q + B = Q+ˆ B. D ∈D 6 −3/4 Take now f, Q and x as above and put h(sn) = n ξn. The operator C corresponding to h is bounded and hf is normal when defined on Λ Z. \ Thus the functions hf corresponds to the product CˆQ and · ∞ 2 −3/2 dνx(b). (hf)(b) = n < , Λ | | ∞ Z nX=1 hence x (CˆQ). Contrary x / (CQ), since x / (Q). Thus CQ = CˆQ, ∈D · ∈D ∈D 6 · consequently, the operator CQ is not closed. In accordance with Section 23 this product in the reverse order QC is automatically closed. 53. Note. Let T and Q be two positive operators affiliated with a quasi- commutative von Neumann algebra A, which acts on a Hilbert space X over 44 either the quaternion skew field or the octonion algebra , where 2 v 3, Av ≤ ≤ so that A is isomorphic with C(Λ, ). Positive normal functions f and g Av correspond to these operators T and Q such that f and g are defined on Λ W and Λ W respectively. Therefore, their sum f + g is defined on \ f \ g Λ (W W ) and is normal and corresponds to T +ˆ Q, hence \ f ∪ g 2 2 0 dνx(t). f(t) dνx(t). f(t)+ g(t) < ≤ ZΛ | | ≤ ZΛ | | ∞ for each vector x (T +ˆ Q). This implies that x (T ) and symmetrically ∈D ∈D x (Q), consequently, x (T + Q) and T + Q = T +ˆ Q. ∈D ∈D If drop the condition that T and Q are positive, but suppose additionally that W W = , then two disjoint open subsets U and U in Λ exist f ∩ g ∅ f g containing W and W correspondingly. Therefore, cl(U ) Λ U and the f g f ⊂ \ g clopen set cl(Uf ) contains Wf , since a Hausdorff topological space Λ is ex- tremely disconnected. Thus f is bounded on X cl(U ) and g is bounded \ f on cl(U ). Take the graded projection in A corresponding to the char- f Av acteristic function χ of cl(U ). Then two operators QE and T (I E) cl(Uf ) f − belong to Lq(X). But the operator T E is closed, since T is closed and E is bounded, consequently, T E = TˆE. We deduce that · (T +ˆ Q)E = T E+ˆ QE = T E + QE =(T + Q)E and (T +ˆ Q)(I E)=(T + Q)(I E). − − Take a vector x (T +ˆ Q), then Ex and (I E)x are in the domain ∈ D − (T +ˆ Q), hence Ex and (I E)x are in (T +Q), consequently, x (T +Q) D − D ∈D and inevitably we get that T +ˆ Q = T + Q. 54. Proposition. Suppose that A is a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra with Av 2 v 3 and T ηA. Let ≤ ≤ (1) P (z)= a zn1 ...a znk { s,n1 s,nk }q(2k) k,s, n1+X...+nk≤n be a polynomial on with coefficients a , where k,s N, 0 n Z Av Av s,m ∈ ≤ l ∈ for each l, n is a marked natural number, z0 := 1, z , q(m) is a vector ∈Av indicating on an order of the multiplication of terms in the curled brackets, 45 a ...a =0 for n + ... + n = n and constants c> 0 and R> 0 exist so 1,n1 1,nk 6 1 k that (2) c z n a zn1 ...a znk | | ≤| { s,n1 s,nk }q(2k)| k,s, n1+X...+nk=n for each z > R. Then the operator | | a T n1 ...a T nk { s,n1 s,nk }q(2k) k,s, n1+X...+nk≤n is closed and equal to P (T ). Proof. We have that A is isomorphic with C(Λ, ), where Λ is a Haus- Av dorff extremely disconnected compact topological space (see Theorem I.2.52 [28]). An operator T corresponds to some normal function f defined on Λ W so that a set W is nowhere dense in Λ. Therefore, the composite \ f f function P (f(z)) is defined on Λ W and normal. Thus P (f(z)) corresponds \ f to the polynomial P (T ) of the operator T . On the other hand, a vector x is in (P (T )) if and only if the integral D 2 (3) dνx(t). P (f(t)) < ZΛ | | ∞ converges. Consider the sets Λ := cl t : f(t) < m . Since m { | | } lim P (z) = |z|→∞ | | ∞ and due to Condition (2) there exists a positive number m> 0 such that 1 1 (4) c f(t) n a f(t)n1 ...a f(t)nk P (f(t)) 2 | | ≤ 2| { s,n1 s,nk }q(2k)|≤| | k,s, n1+X...+nk=n t Λ (W Λ ), consequently, ∀ ∈ \ f ∪ m 2n (5) dνx(t). f(t) < . ZΛ\(Wf ∪Λm) | | ∞ A measure νx is non-negative and finite on Λ, the function f is bounded on Λ , consequently, f L2n(Λ, ν , ), hence f Lk(Λ, ν , ) for each m ∈ x Av ∈ x Av 1 k 2n. The power f n of f is defined on Λ W and is normal, hence ≤ ≤ \ f f n represents T n. In view of Theorem 44 the inclusion x (T k) is fulfilled ∈D for each k =1, ..., n, particularly, x (T )= (T ). ∈D D We have T = T . Suppose that T k = T k for k =1, ..., j 1. As y (T j), − ∈D then y (T ). Take a bounding graded sequence E in A for T j−1 and ∈D Av l 46 T . Therefore, we deduce that T j E = (T j−1ˆT ) E = T j−1 ET E, m · m m m j j j−1 j−1 consequently, mET y = T mEy = T mET mEy = T mETy. On the j j j j other hand, the limits exist limm mET y = T y and limm mET y = T y. The operator T j−1 is closed, consequently, Ty (T j−1) and T j−1Ty = T jy, ∈D hence T j T j−1T . By our inductive assumption T j−1 = T j−1. Therefore, ⊆ T j T j and hence T j = T j. 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