**Advances in Pure Mathematics** Vol.3 No.1(2013), Article ID:27352,29 pages DOI:10.4236/apm.2013.31008

Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

Department of Applied Mathematics, Moscow State Technical University, Moscow, Russia

Email: Ludkowski@mirea.ru

Received June 28, 2012; revised September 2, 2012; accepted September 17, 2012

**Keywords:** Hypercomplex Numbers; Cayley-Dickson Algebras; Operator; Operator Algebra; Spectra; Spectral Measure

ABSTRACT

A class of meta-invariant operators over Cayley-Dickson algebra is studied. Their spectral theory is investigated. Moreover, theorems about spectra of generalized unitary operators and their semigroups are demonstrated.

1. Introduction

The Cayley-Dickson algebras are algebras over the real field R, but they are not algebras over the complex field C, since each embedding of C into with is not central. That is why the theory of algebras of operators over the Cayley-Dickson algebras with is different from algebras of operators over the complex field C. On the other hand, such theory has many specific features in comparison with the general theory of algebras of operators over R due to the graded structures of algebras. Moreover, the Cayley-Dickson algebra with can not be realized as the subalgebra of any algebra of matrices over R, since these algebras with are not associative, but the matrix algebra is associative.

The results of this paper can be used also for the development of non-commutative geometry, super-analysis, quantum mechanics over, and the theory of representations of topological and Lie groups and supergroups which may be non locally compact, for example, of the type of the group of diffeomorphisms and the group of loops or wraps of manifolds over (see [1-8]).

As examples of unbounded operators over serve differential operators, including operators in partial derivatives. For example, the Klein-Gordon-Fock or Dirac operators are used in the theory of spin manifolds [9], but each spin manifold can be embedded in the quaternion manifold [10].

The skew field of quaternions H has the automorphism of order two, where

There is the norm in H such that, consequently,.

The algebra K of octonions (octave, the Cayley algebra) is defined as the eight dimensional algebra over R with a basis, for example1) such that;

2), , , , , , , , ,;

3)

it is the law of multiplication in K for each , for quaternions with

The algebra of octonions is neither commutative, nor associative, since, , but it is distributive and the real field is its center. If , then 4) is called the adjoint element for, where. In addition we have the identities:

5) andwhere, so that 6) and is the norm in K. This implies that 7)consequently, K does not contain the divisors of zero (see also [11]). The multiplication of octonions satisfies the equations:

8)9)which define the property of alternativity of algebras. In particular,.

Preliminary results on operator theory and operator algebras over quaternions and octonions were published in [12]. This articles develops further operator theory already over general Cayley-Dickson algebras.

In the third section of this article the theory of unbounded operators is described, as well as bounded quasilinear operators in the Hilbert spaces X over the CayleyDickson algebras. At the same time the analog of the scalar product in X is defined with values in the CayleyDickson algebra. For the spectral theory of such operators there are defined and used also graded operators of projections and graded projection valued measures. The linearity of operators over the algebra with is already not worthwhile because of the non-commutativity of, therefore there is introduced the notion of quasi-linear operators.

At the same time graded projection valued measures in the general case may be non-commutative and non-associative. Because of the non-commutativity of the Cayley-Dickson algebras with commutative algebras over them withdraw their importance, since they can be only trivial, moreover, over the Cayley-Dickson algebra with the associativity may lose its importance. Therefore, in the third section the notion of quasi-commutative algebra metainvariant over is introduced. Principles of the theory of bounded and unbounded operators are given there. Spectral theory of self-adjoint and generalized unitary operators is exposed. In the third section spectral theory of normal operators is described. For algebras of normal meta-invariant over operators the theorem generalizing that of Gelfand-Mazur is proved. An analog of Stone’s theorem over for the one parameter families of unitary operators is demonstrated.

Definitions and notations of the preceding work [12] are used below.

2. Preliminaries

1) Notations and Definitions. Let denote the real Cayley-Dickson algebra with generators such that, for each, for each.

The Cayley-Dickson algebra is formed from the algebra with the help of the doubling procedure by generator, in particular, is the real field, coincides with the field of complex numbers, is the skew field of quaternions, is the algebra of octonions, is the algebra of sedenions. The algebra is power associative, that is1) for each and.

It is non-associative and non-alternative for each.

The Cayley-Dickson algebras are–algebras, that is, there is a real-linear mapping such that 2)3) for each. Then they are nicely normed, that is4) and 5) for each. The norm in it is defined by the equation:

6).

We also denote by. Each non-zero CayleyDickson number has the multiplicative inverse given by.

The doubling procedure is as follows. Each is written in the form, where, ,. The addition is componentwise. The conjugate of a Cayley-Dickson number z is prescribed by the formula:

7). The multiplication is given by Equation 8).

for each, , .

The basis of over is denoted by

, where for each

, is the additional element of the doubling procedure of from, choose

for each,.

Put by the definition to be the imaginary part of a Cayley-Dickson number z.

It is possible to consider the Cayley-Dickson algebras of greater real dimensions.

2) Definition. Let denote the family consisting of elements such that

where

, for each s.

3) Theorem. The family has the structure of the normed power-associative left and right distributive algebra over the real field R with the external involution of order two.

Proof. Let. Then each element is the limit of the sequence, also is the limit of the sequence, where. Therefore, there exists the limit

Using the polar coordinates prove the powerassociativity. There exists the natural projection from onto for each given by the formulas:

and in the contrary case for each

where for each; then

where

Then the limit exists relative to the norm in. Therefore, for each there exists the limit

consequently, is power-associative, since each is power-associative, cos and sin are continuous functions. Evidently, is the R-linear space.

The continuity of multiplication relative to the norm follows from the inequalities

and taking the limit with tending to the infinity, since and for each and for each. The left and right distributiveness

follow from taking the limit with r tending to the infinity and such distributivity in each.

The involution in is of order two, since. It is external, since there is not any finite algebraic relation with constants in transforming the variable into. The relation

is of infinite order. The latter limit exists point-wise, but does not converge uniformly in any ball of positive radius relative to the norm in the algebra. The relations of the type in use external automorphism with, moreover, the latter relation is untrue for and instead of.

No any finite set of non-zero constants can provide the automorphism of. To prove this consider an R-subalgebra of

generated by, where,

. Since, then

, hence. If

, then it certainly can not provide the automorphism of. Consider, without loss of generality suppose. There is the scalar product in for each. Let be the projection of in a subspace of orthogonal to, then by our supposition and. Therefore, we get, consequently, is isomorphic to C. Certainly, no any proper Cayley-Dickson subalgebra, , can provide the automorphism of.

Therefore, without loss of generality suppose, that is not isomorphic to C and. If and, then and hence, since for each. Let be the projection of M_{2} in a subspace of orthogonal to relative to the scalar product. Then by our supposition and, , hence after the doubling procedure with we get, that is a subalgebra of.

Then proceed by induction. Suppose that is a subalgebra of, ,. Since can not provide the automorphism of, suppose without loss of generality, that and consider the orthogonal projection of in a subspace of orthogonal to relative to the scalar product. Then and,. Therefore, the doubling procedure with gives the algebra which is the subalgebra of, etc. As the result is the subalgebra of and can not provide the automorphism of, where due to the formula of the polar decomposition of Cayley-Dickson numbers with a real parameter and a purely imaginary parameter of the unit norm, (see also §2.2 [13]).

4) Note and Definition. Let denote a Hausdorff topological space with non-negative measure on a -algebra of all Borel subsets such that for each point there exists an open neighborhood with. Consider a set of generators with real algebra such that for each and for each, where 0 is a marked point in. Add to this set the unit such that for each and. We consider also finite products of these generators in accordance with, where each with is considered as the doubling generator.

In the case of a finite set Cayley-Dickson algebra generated by such generators is isomorphic with, where is the cardinality of the set.

For the infinite subset of generators

the construction from produces the algebra isomorphic with.

Therefore, we consider now the case. Due to the Kuratowski-Zorn lemma (see in [14]) we can suppose, that is linearly ordered and this linear ordering gives intervals being -measurable, for example, has the natural linear ordering induced by the linear ordering from and by the lexicographic ordering in the product, where, or may be an ordinal of the cardinality.

Then consider a finite partition into a disjoint union, where for each and

for, ,. The family of such partitions we denote. Let, be marked points. Then there exists a step function f_{T} such that for each, where. Consider the norm

where

for, for, for, while. To each put the element

The algebra which is the completion by the norm of the minimal algebra generated by the family of elements for f_{T} from the family of all step functions and all their ordered final products we denote by.

5) Theorem. The set is the power-associative non-commutative non-associative algebra over R complete relative to the norm with the center. Moreover, there are embeddings for. The set of generators of the algebra has the cardinality for . There exists the function

of the ordered integral product from onto.

Proof. For the algebra is isomorphic with or in accordance with. Thus it remains to consider the case.

For each it can be defined the ordered integral exponential product

with corresponding to the left order of brackets. Thus the embeddings of into exist. Then.

The completion of the family contains all functions of the type

where is the disjoint union of, each is -measurable, and

Since for each

with and

, for each there exists

relative to. From the identity for each it follows, that the family of all elements of the type, contains all generators of the embedded subalgebra generated by the countable subfamily.

The completion of the family by the norm is the infinite dimensional linear subspace over in.

All possible final ordered products from and the completion of their R-linear span by the norm produces. Then for each element from there exists the representation in the form of the ordered integral exponential product. Since is the algebra over R and, then the family of generators of the algebra has the cardinality.

6) Weakened Topology. Suppose that X is a real topological vector space. Let denote the vector space of all continuous R linear functionals on X. Each continuous R linear functional defines a seminorm. The topology generated by the semi-norms family is called the weak or weakened topology for X and it is usually denoted by.

7) Non-Commutative Riemann Sphere. For the Cayley-Dickson algebra is the locally compact topological space relative to the norm topology. In the finite dimensional over case, the weak and norm topologies on are equivalent.

Henceforward, we denote by also, where the cardinality of is equal to the topological weight of the Cayley-Dickson algebra relative to the norm topology, , where. Without loss of generality can be considered as an ordinal due to the Kuratowski-Zorn lemma. Generally embed into the non-commutative unit sphere

over the real field R such that is a singleton, where denotes a continuous bijective mapping, or may be . To realize an embedding one considers their union embedded into such that they intersect by the set

.

As a mapping one takes the stereographic projection so that

.

As the R linear normed space the Cayley-Dickson algebra is isomorphic either with the Euclidean space for or with the real Hilbert space of the topological weight. Consider on the weak topology provided by the family of all continuous R linear functionals on. In view of the Alaoglu-Bourbaki Theorem (9.3.3) [15] the unit sphere is compact relative to the weak topology on inherited from the weak topology. Therefore, relative to the weak or weakened topology induced by the weak topology of has the one-point compactification realized topologically as (see also Theorem 3.5.11 about one-point Alexandroff compactification in [14]). That is,.

The non-commutative analog of the Riemann sphere can be supplied with the topology inherited from the norm topology on. That is on two topologies were considered above the weakened topology and the norm topology.

8) Definitions. We say that a real vector space Z is supplied with a scalar product if a bi-R-linear bi-additive mapping is given satisfying the conditions:

1) if and only if;

2);

3) for all real numbers and vectors.

Then an vector space X is supplied with an valued scalar product, if a bi-R-linear bi--additive mapping is given such that

where, each is a real linear space with a real valued scalar product, is real linear isomorphic with

and for each j, k. The scalar product induces the norm:

.

An normed space or an vector space with scalar product complete relative to its norm will be called an Banach space or an Hilbert space respectively.

9) Meta-Invariant Operators. If a topology on an vector space X is such that the addition of vectors and the multiplication of vectors on Cayley-Dickson numbers from on the left and on the right are continuous relative to the topology on X and the norm topology on, then X is called a topological vector space.

Let X and Y be topological vector spaces over the Cayley-Dickson algebra. An R-linear -additive operator we call meta-invariant for, if a family of CayleyDickson sub-algebras isomorphic with exists such that

and

and

and

and

for each, where ^{t}X and ^{t}Y are vector sub-spaces over in X and Y respectively (may be up to a continuous with its inverse isomorphism of topological vector spaces as two-sided modules), denotes the closure of a subset H in a topological space G. In another words for it is complex meta-invariant, for or such operator will be called quaternion or octonion meta-invariant correspondingly.

We say that a family of operators is metainvariant, if Conditions are fulfilled for each and decompositions are the same for the entire family.

One example, is the following. If or is a right or left linear operator for normed spaces X and Y over such that with for each j, then

or

correspondingly. Therefore, (up to a continuous with its inverse isomorphism of normed spaces as two-sided modules) such operator C is meta-invariant for any.

Let X be a Hilbert space over the Cayley-Dickson algebra, then there exists an underlying Hilbert space over R, i.e. a real shadow. The scalar product on X with values in from induces the scalar product in with values in R.

Then for an -vector subspace Y in X the orthogonal complement can be created relative to the scalar product. At the same time the equation for, defines the R-linear operator acting on a Hilbert space X. That is, is the projection on Y parallel to. Moreover, one has1), also 2).

Mention, that is the orthogonal projection from X on, since and

for each and.

Consider the decomposition of a Hilbert space over, where is the family of standard generators of the Cayley-Dickson algebra, are pairwise isomorphic Hilbert spaces over.

A family of operators with we call an -graded projection operator on, if it satisfies conditions below:

1) for each, where 2) for each3) for each4) for each and5) for each and6) for each and7) on X_{0} for eachwhere, for each, denotes the adjoint operator.

3. Normal Operators

1) Definition. An operator T in a Hilbert space X over the Cayley-Dickson algebra is called normal, if.

An operator T is unitary, if and.

An operator T is called symmetrical, if for each.

An operator T is self-adjoint, if. Further, for it is supposed that is dense in.

2) Remark. In the more general definition than previously of a unitary operator is given without any restriction on either meta-invariance or leftor right--linearity.

On a space of real-valued continuous functions on a Hausdorff topological space B one considers a lattice structure:

1) and 2) for each.

Therefore, one gets the decomposition:, where and so that a function is the difference of two positive functions with supports

,

.

Suppose that, and are positive or nonnegative continuous functions on B. Put and, consequently, and so that is a positive or nonnegative continuous function. If for some, then, since and

. This contradicts the hypothesis. Thus. Therefore, the decomposition with and is valid, which is called Riesz’ decomposition.

Consider now a space of all continuous valued functions on a Hausdorff topological space B. An R linear additive functional p on we call Hermitian if for each.

This induces a lattice structure on the set of all Hermitian functionals on, since the restriction is a real-valued function.

For an R-linear additive functional t on a conjugate functional is defined by the formula 3) for each.

Then the functional is Hermitian and is skew Hermitian, that is

for each. Therefore, the functional t generates the family of Hermitian functionals for each, where for each.

Each Hermitian functional induces an valued scalar product by the formula:

4).

If a topological space B is compact, then . If p is a Hermitian state, then 5) due to Cauchy-Schwartz’ inequality.

Thus Hermitian state p is continuous and has norm.

3) Lemma. Let be a normal metainvariant operator for a Hilbert space X over the CayleyDickson algebra with. Let also A be a minimal von Neumann algebra generated by T, and I over. Suppose that is a sub-algebra in isomorphic with for each and

are generators of (see). Then the restricted sub-algebra

is associative for r = 2 and alternative for r = 3 for each.

Proof. In view of Theorem 2.22 [12] an algebra A is quasi-commutative. Compositions are considered as point-wise set-theoretic compositions of mappings so that is the n + m times composition of T with itself for each, where are natural numbers. Moreover, Ix = x for each. Then consider as a formal variable so that. This variable is power associative for all, since T is metainvariant. Moreover, and commute, , since T is the normal operator,.

Therefore, the minimal algebra over generated by and I consists of formal polynomials in variables with coefficients from. This algebra is R linear and left and right module over. Relative to addition and multiplication of polynomials the algebra is associative for r = 2 and alternative for r = 3, since the quaternion skew field is associative and the octonion algebra is alternative and is isomorphic with for each.

Consider the algebra relative to the topology inherited from the von Neumann algebra (see also Definitions 2.22.1 and Theorem 2.22 [12]).

The completion of gives the von Neumann algebra up to isomorphism of algebras. The operations of addition and multiplication are continuous relative to a norm in a -algebra. Thus the restricted sub-algebra is associative for r = 2 and alternative for r = 3 for each.

4) Theorem. Suppose that is a selfadjoint meta-invariant operator on a Hilbert space X over the Cayley-Dickson algebra with and, A is a von Neumann algebra containing T. Then there is a family of - graded projection operators on X such that 1) for, while for;

2) for;

3);

4) and for each;

5) and this integral is the limit of Riemann sums converging relative to the operator norm.

Considering a restricted algebra

isomorphic to with an extremely disconnected compact Hausdorff topological space let f and in correspond to and in A respectively, then is the characteristic function of the largest clopen subset on which f takes values not exceeding b, where

.

Proof. An operator T is -meta-invariant, consequently, a von Neumann subalgebra over contains the restriction of T on Y up to an isomorphism of von Neumann algebras over, where .

In accordance with Theorem 2.22 [12] we have that an algebra is isomorphic with for some extremely disconnected compact Hausdorff topological space.

Recall that a subset B of a topological space W is called clopen, if it is closed and open simultaneously,.

If a function corresponds to an operator, then a set is clopen in. On this set the function has values not exceeding. If is another clopen subset so that , then and. On the other hand, the set is clopen, consequently, . Therefore is the largest clopen subset in so that.

The characteristic function of by its definition takes value 1 on and zero on its complement. It is continuous, since the subset is clopen in. Thus an operator in A corresponding to is a projection, ,.

Then to an operator

on X_{j} corresponds, where. Over the CayleyDickson algebra the projection induces an graded projection in accordance with, since .

Statements (1, 2, 4) are evident from the definition of the subsets and the properties of the isomorphism of with. Indeed, the function f is continuous and

.

For proving Statement (3) we mention that , since the topological space is extremely disconnected and each is clopen in. This implies that is the infimum of the family

, consequently, , since

in accordance with Propositions 2.22.7 and 2.22.9 and Corollary 2.22.13, Definition 2.23 and Theorem 2.24 [12] with.

It remains to prove Statement. Take

and a partition of the segment

so that. If

let be a point of this intersection, otherwise let when the intersection is empty,. Therefore,

when this intersection is emptysince a spectrum of T is a range of f, consequently, and.

We call two characteristic functions and orthogonal, if identically, that is.

Thus characteristic functions are mutually orthogonal for different values of j. We consider the function. This induces the disjoint partition with, that is and for each. If, then and, hence 6).

We consider Riemann integral sums

.

If is an isomorphism of quasicommutative algebras, then 7).

On the other hand,

since the operator is meta-invariant and hence is meta-invariant as well for each, since and and is a graded projection operator (see). Therefore, Formulas (6), (7) imply Statement (5) in the sense of norm convergence in.

5) Definition. A family of graded projections indexed by satisfying Conditions (2, 3) of the preceding theorem and two conditions below 1) and 2)

is called an graded spectral resolution of the identity. An graded spectral resolution of the identity is called bounded, when two finite numbers exist such that for every and for each. Otherwise we say that a resolution of the identity is unbounded.

6) Definitions. A^{*}-homomorphism of a C^{*}-algebra A into is called a representation of A on a Hilbert space X over the Cayley-Dickson algebra, that is and

and for all

and, if A contains a unit element, then.

If additionally is one-to-one, then such a mapping will be called a faithful representation of A onto B, where.

A representation is said to be essential if the union of ranges of graded projection operators (elements) of A in the image is the unit operator I.

Let denote the subspace of all continuous vanishing at infinity functions so that

and.

7) Theorem. Suppose that is an graded resolution of the identity and is a bounded self-adjoint operator such that

and for every

or for each, where is a marked positive number,. Then is the resolution of the identity for, where A is a quasi-commutative von Neumann algebra generated by T and I over the Cayley-Dickson algebra,

, denotes a completion of a sub-algebra B in relative to the weak operator topology.

Proof. The inequality implies that

is self-adjoint. Therefore, restrictions and

commute on X_{0} for each b. Hence the algebra

is quasi-commutative. In view of Theorem 3 the restricted algebra is isomorphic with, where is an extremely disconnected compact Hausdorff space and.

If is a resolution of the identity for an operator, a function f in corresponds to T, while corresponds to, then is the characteristic function of the largest clopen subset in on which f takes values not exceeding b. On the other hand, if a function corresponds to, then is the characteristic function of a clopen subset in on which f takes values not exceeding b, since, consequently,

The equality implies that is the largest clopen subset contained in. By the conditions of this theorem and

for any t > b. Hence for each, consequently, for each t > b and inevitably is a clopen subset contained in. Therefore, , since is the largest clopen subset contained in. Thus and for each.

The resolution of the identity for the operator T in A satisfies Condition (4) of Theorem 4 and.

Now, if

for each, where is a marked positive number, , we take for each an arbitrary partition of the segment so that

for an operator

with for each. This implies that

and

consequently, We have also

and

consequently, for every. Take a limit in the strong operator topology with tending to the infinity:, consequently, for each, since is arbitrary, where for every in accordance with the first part of the proof.

8) Lemma. Let T be an meta-invariant operator on an vector space X and let. Then T is meta-invariant.

Proof. The Cayley-Dickson algebra has also the structure of the vector space for, since is the sub-algebra in. Therefore, the decomposition from satisfying Conditions generates analogous decomposition fulfilling Conditions over as well relative to which is meta-invariant.

9) Theorem. Let be a bounded graded resolution of the identity on a Hilbert space X over the Cayley-Dickson algebra, , , for each and for each. Then the integral converges to a self-adjoint operator T on X so that and is the spectral resolution of such operator T.

Proof. Take any partitions and of the segment of diameters

and

and denote by their common refinement. Then one gets

and

where

consequently1)

.

Consider the family of partitions of the segment ordered by inclusion. The algebra is complete relative to the operator norm topology. Therefore, a Cauchy net of finite integral sums converges in,

andsince for each and.

Theorem 7 states that is the graded spectral resolution of.

In the space of continuous functions on a compact Hausdorff topological space with values in a family of polynomials is everywhere dense in accordance with the Stone-Weierstrass theorem (see and I.2.7

[13]), since is everywhere dense in

, where is a family of all possible embeddings of into, denotes the first countable ordinal, (see [14]). The real field is the center of the Cayley-Dickson algebra so that each polynomial of a variable with values in has the form

, where are constants. In this calculus we have

for each. Particularly, if f is a nonnegative function on, then is a positive operator. Therefore, is the difference of two positive operators, where and

. These operators _{1}T and _{2}T commute, since they are presented by the corresponding integrals, which have integration over non-intersecting sets so that the graded projectors and

commute (see and). Their sum is the positive operator, consequently, meta-invariant for (see and Lemma 8).

Consider the restricted algebra and isomorphic to it algebra and with, where is a compact Hausdorff extremely disconnected topological space,. The conditions, for each and for each imply that a function w in representing the operator T has range in, consequently,.

10) Theorem. Suppose that T is a unitary metainvariant operator on an Hilbert space X, , , while A is the quasi-commutative von Neumann algebra generated by T and.

1) Then T is a norm limit of finite linear combinations of mutually orthogonal projections in A with coefficients in.

2) If additionally and T is either right or left linear operator on X, then a positive operator K and a purely imaginary Cayley-Dickson number, , , exist such that and .

Proof. 1) Each unitary operator is normal. It preserves an scalar product on X: for each.

Consider the von Neumann algebra over generated by and. In view of Theorem 2.22 [12] and Lemma 3 the restricted algebra is isomorphic with, where is a compact extremely disconnected topological Hausdorff space. The algebra is isomorphic with for each (see also).

Recall the following. A closed subset Q of a topological space P is called functionally closed if for some continuous function. A subset U of a topological space P is called functionally open, if it is the complement of some functionally closed subset Q, i.e.. A covering of a topological space P by functionally open subsets is called a functionally open covering. It is said that a topological space P is strongly zero-dimensional, if P is a non-void Tychonoff space (completely regular), i.e., and each functionally open covering has a finite disjoint open covering refining, that is, for each, for every

an open subset with exists such that. This means that this covering consists of clopen subsets in P.

Theorem 6.2.25 [14] says that each non-void extremely disconnected Tychonoff space is strongly zerodimensional.

Therefore, to the restricted operator up to an isomorphism of algebras a function corresponds such that 3)where and

for each, (see §3 in [16] or §I.3 [13]). The Hilbert subspace

is isomorphic with for each. Then. Generally to a function corresponds with

, where is a topological space homeomorphic with. If is such homeomorphism, then for each.

Certainly it is sufficient to take a function g with values in [0,1], since the exponential function is periodic: for each real number and an integer and a purely imaginary number in the unit sphere. Therefore, a positive operator K on X corresponds to this function g.

The operator K defines an graded resolution of the identity (see Theorem 4) and 4).

Therefore,. We put for b < 0, for, let be the largest clopen subset in on which the function g has values not exceeding.

Let be zero for for, and let be the characteristic function of for. Each clopen subset V in is contained in, since takes values on in for each, consequently,. This implies that is the infimum of the family. This induces a bounded graded resolution of the identity, where corresponds to for each b.

A function g can be chosen such that it does not take the unit value on a non-void clopen subset W in. Indeed, otherwise and W would be disjoint from and the function g could be redefined due to the periodicity of the exponential function. Then a point would exist so that.

For each a positive number exists such that for every partition of the segment [0,1] of diameter and points the inequality is satisfied:

5), consequently6)where and the sum is by for which the subset is non-void, since the exponential function on is continuous. Certainly coefficients are in and hence in.

Since by the conditions of this theorem, the purely imaginary unit sphere is compact in the Cayley-Dickson algebra. Then

, consequently, for each

a finite -net of points in exists. Therefore, for each a finite family of points

exists so that

since.

2) In accordance with the conditions of Theorem 9, an operator T is an meta-invariant either right or left linear operator, consequently, graded resolution of the identity can be chosen either right or left linear for all b (see Theorem 4). Then T due to its either right or left linearity is completely characterized by its restriction on (up to continuous with its inverse isomorphism of Hilbert spaces as two-sided modules). Therefore, to the operator T up to an isomorphism of algebras a function corresponds such that , where for each,. Equivalently up to an isomorphism one can take a purely imaginary (constant) number M instead of i_{1}, that is, , , where is isomorphic with for every (see §2.9). Thus generally. This operator K can be chosen meta-invariant on X, since T is meta-invariant on X.

In view of we get 7)

where the operator is given by Formula (4).

11) Remark. For an infinite dimensional Hilbert space over the Cayley-Dickson algebra one has that and are isomorphic as Hilbert two-sided modules, since and are isomorphic as real Hilbert spaces.

12) Theorem. Let be a compact Hausdorff space, let also be an Hilbert space, and let be a representation of on for each, where, is isomorphic with for each, so that Conditions 2.9 are fulfilled. Then to each Borel subset of a graded projection corresponds such that 1), where denotes the strong operator closure of;

2);

3) for each countable family

of mutually disjoint Borel subsets of; also

for each,;

4) If the span of ranges of those such that and vanishes on is dense in for a Borel subset of, then;

5) The mapping is a regular Borel measure so that

for each continuous function, where denotes a Borel -algebra of all Borel subsets in.

Proof. A multiplication

and an involution are defined pointwise and for each. The C^{*}-algebra is quasi-commutative, consequently, A is a quasi-commutative C^{*}- algebra.

Suppose that U is an open subset in and a continuous function is given such that. Then. Denote by the family of all such functions f directed by the natural order of functions: if for each. Therefore, the family has a supremum in the quasi-commutative C^{*}-algebra A over. This element is a projection, since . For a Borel subset G of we put.

If is a unit vector, then is a state of. Indeed, on is -linear and - additive, for each nonnegative function on, moreover, and for every non-negative continuous function on:

6)particularly for also:

for each due to the identity for each and (see also Formula 4.2 (7) in [13] and §§2.8), since a continuous image of a compact topological space is compact by Theorem 3.1.10 [14] and hence each continuous function on a compact topological space is bounded. From the Riesz representation theorem for continuous R-linear non-negative functionals on the R-linear space we get that 7)

for each, where is a regular Borel -additive non-negative measure on.

Then Formulas (6,7) generate on the space (see §2.23 [12]). The family generate the family due to the bi-- additivity

for all and bi--linearity

for each provided by the corresponding properties of. This induces and hence on due to Conditions 2.9.

The measure is inner regular, that is for each and an open subset in a compact subset B in U exists such that. The topological space is normal, consequently, for each closed subset B in an open subset U in a continuous function exists so that and. For such function we have

consequently, , since f is arbitrary with such properties. In accordance with the definition of we have, hence

. The outer regularity of

implies that

for each Borel subset W in. For a disjoint sequence of Borel subset in we get

If for some and, then

This implies that for eachsince for each. Thus

for each, consequently,

If a set is as in and is an open subset containing and with, a function f is non-zero identically, then a range graded projection of is a graded sub-projection of, where. On the other hand, the range graded projection of is the same as that of, since and and each operator in is -linear, where and for each. Thus contains the range graded projection of the image of each function so that vanishes on, since the span of functions from is everywhere dense in

.

Hence due to Property.

13) Corollary. Suppose that is an essential representation of (see §6 and Theorem 12), then a possibly unbounded graded resolution of the identity exists such that 1)

for each.

Proof. Consider a one-point (Alexandroff) compactification of the real field. Then is isomorphic with (see §6). The latter is a quasi-commutative subalgebra in.

In accordance with Theorem 12 there exists a graded projection valued measure on such that

for each, moreover,

for each Borel subset in

. Put, then is a graded resolution of the identity (possibly unbounded). Indeed, for each, consequently, . Statement of Theorem 12 implies that.

Then we have for each integer and. Therefore,

On the other hand, one has

consequently,

and

.

Consider now the sequence for. Then

such that

Then we get and is a graded resolution of the identity, since

and is the graded projection valued measure.

Certainly one has the identities

also. Together with Statement of Theorem 12 this means that 2)

for each due to the gradation of.

14) Remark. If T is a bounded normal metainvariant operator on an Hilbert space and is a quasi-commutative -algebra over the CayleyDickson algebra generated by T and and

, then in accordance with Theorem 4 and Lemma 3 there exists an graded projection valued measure on either the one-point compactification of for a self-adjoint operator or on the one-point compactification of for a normal operator such that 1)

for each and, where for every Borel subset in. A spectrum is embedded into. If is an open subset of disjoint from and (see), then. Therefore, and we write 2)

and is the spectral measure of T on the Borel - algebra. Then for a self-adjoint operator T, one gets that is a bounded graded resolution of the identity such that for and for, consequently3)

for each vector (see Corollary 13). Using the polarization as in we get 4)

for each.

Formula (1) extends by continuity of the functional on the vector space of all bounded Borel measurable functions, since

.

For a normal bounded operator such polarization gives:

5)

for every and each. Then the identities are satisfied:

6)

for each, consequently7)

for every. If, then one certainly obtains 8)

for all, consequently9)

for each real number. Particularly, for the characteristic function of a Borel subset contained in we get the equalities:

consequently10) and.

If a Borel subset C is disjoint form B, then, hence. Recall that a step function by the definition is a finite vector combination of characteristic functions. For any two step Functions and with real coefficients a_{l} and b_{k} and mutually disjoint sets and for each this implies the identities

and

consequently,. Each bounded Borel function is a limit of step functions, where a limit is taken relative to the family of all semi-norms 11)

. Therefore, for each bounded real-valued Borel functions the formula 12)

is valid. But then for each and, consequently,

and hence

for each. If and are polynomials in variables this implies 13).

In view of the Stone-Weierstrass theorem the set of -valued polynomials on the ball

is dense in. Thus Formula (13) is spread on, since for and due to Formula (2). Together with (12) this induces the formula.

14)

for each real-valued continuous function and every bounded real-valued Borel function , since is a self-adjoint operator (see also §9). In the latter situation

and if, then. Therefore, for any increasing sequence of continuous real-valued functions a sequence is increasing and composed of self-adjoint operators. If in addition g_{n} converges point-wise to a bounded Borel function g, then converges point-wise to.

There is said that the mapping

with the monotone sequential convergence property is -normal.

15) Theorem. Suppose that T is a bounded normal meta-invariant operator on an Hilbert space X and A is a quasi-commutative C^{*}-algebra over the Cayley-Dickson algebra generated by T and T^{*} and with. Then the-homomorphism extends to a -normal-homomorphism

, where C denotes a quasicommutative von Neumann algebra over consisting of operators quasi-commuting with each operator quasicommuting with A such that 1)

2)

where for each

. If, then, moreover, if, then. For each

the composition is. The mapping is an graded projection valued measure. If an operator T is self-adjoint, then it possesses an graded spectral resolution of the identity with and, where.

Proof. Part of the proof is given in Section 16.

In view of Theorem V.4.4 [17] a point-wise of a sequence of -measurable functions converging at each point of a set V with a -algebra of its subsets is an -measurable function f on V, that is, where denotes the Borel - algebra of the real field R.

Take any vector and an increasing sequence tending point-wise to a bounded Borel function. Then 3)

from the monotone convergence theorem. By the decomposition properties of the valued scalar product and the gradation of we get this property on, since it is accomplished on and each has the form 4) with real-valued functions.

Let be an open subset in. It can be written as a countable union of bounded open subsets in

of radius, since the topological space

is of countable topological weight when. The topological space is normal, consequently, there exist open subsets of radius contained in for which continuous functions exist such that and such that for each and

for every. Take their combination

. From the construction of this sequence we get for each, consequently, for each.

Denote by the family of all Borel subsets of whose characteristic function satisfies Equation 16(14). This family contains characteristic functions of each closed and every open subset of. The -normality property (see Formula) implies that is a -algebra, consequently, contains the Borel -algebra. Thus Formula 16(14) is valid for a characteristic function for each Borel subset B of. Using norm limits one gets this formula for any bounded real-valued Borel functions g and h.

Mention that the decompositions 5) and 6)

Are fulfilled so that the-homomorphism extends to a -normalhomomorphism, where C_{0} denotes a commutative von Neumann algebra over consisting of operators commuting with each operator commuting with. Algebras are isomorphic with and each algebra is isomorphic with. Then is provided for each (see Conditions 2.9), where C^{*}-algebras and are considered relative to the point-wise addition and multiplication of functions from the left and right on Cayley-Dickson numbers and point-wise conjugation as involution and with the norm:

.

This implies for each j due to the -gradation of the projection valued measure and due to the commutativity of the complex field for each. Therefore, decompositions extend the -normal-homomorphism from

up to

.

For a self-adjoint operator its spectrum is contained in, while the real field is separable as the normed space.

16) Theorem. Let A and B be -algebras over the Cayley-Dickson algebra, , let also be a-homomorphism. Then 1) and for each;

2) if K is a self-adjoint element in A and and and for each, then ;

3) if is a-isomorphism, then and for each and is a -subalgebra in.

Proof. 1). Suppose that and, then has an inverse in. This implies , since and has an inverse in, hence, where I denotes the unit element in and in. Then

and

(see [18]). But, since

, consequently, (see also Lemma 2.26 [18]). That is the-homomorphism is continuous.

2) If an element K is self-adjoint, then. At the same time one has for each, since for real. If is a sequence of polynomials tending to f uniformly on, then

and

since the restriction induces a-homomorphism from into.

3) If is a-isomorphism and is a self-adjoint element in A, then by (1) we have. If this inclusion would be strict, then there would exist a non-zero element such that

. But (2) means that and

, contrary to the assumption that is one-to-one. Thus and for each self-adjoint element. Particularly, this is accomplished for and from (1) it follows that

consequently, , where is an arbitrary element. On the other hand, is a complete normed space andis the-isometry, consequently, is closed in B and contains I. That is is a C^{*}-subalgebra in B.

If V is an element in B, it induces a quasi-commutative C^{*}-sub-algebra over generated by V, V^{*} and I. This sub-algebra has the decomposition, where are isomorphic algebras over R. Now take the restricted subalgebra, where

. More generally take sub-algebras, where is isomorphic with and impose Conditions 2.9. To the latter C^{*}-algebras Theorem 2.24 [18] is applicable. In view of this theorem

, consequently,

, since is the-isomorphism, where denotes the spectrum of an element P in B.

17) Lemma. Let be a-homomorphism of C^{*}-algebras over the Cayley-Dickson algebra,. Then the testriction induces a-iosmorphism of into

Proof. A-homomorphism is R-linear and -additive and so the restricrion also has there properties. This provides a-homomorphism from into. Since

the image of is,. Due to the -linearity this means that for each real number, i.e.. For each purely imaginary Cayley-Dickson number, , the identities are satisfied, consequently,

and.

If M and K are two Cayley-Dickson numbers which are orthogonal, then

Therefore, and are orthogonal for each

and and for every real number, also for each. Moreover, we infer that

for each. That is,

is-isomorphic with fore each,. Using embedded sub-algebras satisfying Condition 2.9into we get that is-isomorphic with, since

and hence for each Cayley-Dickson number.

18) Remark. Lemma 17 means that up to a-isomorphic of the Cayley-Dickson algebra one can considerhomomorphisms satisfying additional restriction:, that is for each. This will simplify notations, for example, , also in Theorem 16, i.e.

.

19) Theorem. If T is a bounded normal meta-invariant operator on an Hilbert space X, ,. Suppose that is a -normal homomorphism into a quasi-commutative von Neumann algebra over such that and, where for each.

Then, where

is the quasi-commutative von Neumann algebra over generated by T and T^{*} and I so that for every.

Proof. As in §15 is supplied with the structure of a quasi-commutative C^{*}-algebra with instead of. On the other hand, the inclusion is valid and 1,. For each and there exists such that, consequently, and hence . If, then has an inverse h in for each, consequently, is self-adjoint in A. The homomorphism is order preserving and does not increase norm by Theorem 18.

Next we apply Theorems 2.24 and 3.22 [17] and §2.9. Consider restricted to. Then

for each and

. The mappings and

are -normal, as in §§14, 15 we get that, when is the characteristic function of an open subset in.

We consider the family of all Borel subsets with their characteristic functions satisfying the equality. By the -normality this family contains the union of each countable subfamily. It contains also the complement of each set, since. Thus contains the family of all Borel subsets of and for each

. The mapping is R-linear and - additive and norm continuous so that. Due to the meta-invariance of the operator the set of all step functions is dense in relative to the norm topology on it, since is compact for each, while and the Cayley-Dickson algebra is separable as the normed space for. Therefore for each.

20) Theorem. Let be a symmetrical metainvariant operator, and, then there exists its resolvent function and for each. Let be a closed operator, then the sets, , and are not intersecting and their union is the entire Cayley-Dickson algebra,

.

Moreover, if additionally an operator T is self-adjoint and quasi-linear, then the inclusion is satisfied and.

Proof. In the general case for a quasi-linear operator (not necessarily symmetrical) for the decomposition of the components of the projection valued measure defined (see also [18]) as the sum of the point, absolutely continuous

and continuous singular measures in accordance with the Lebesgue theorem cited in gives, where

,

and

.

At the same time for a symmetrical operator, the inclusion is satisfied and due to the relations given in [18] for components of the projection valued measure the supports of all these measures for different are consistent and are contained in.

21) Theorem. For a self-adjoint quasi-linear - meta-invariant operator in a Hilbert space over the Cayley-Dickson algebra, , , there exists a uniquely defined regular countably additive self-adjoint spectral measure on, such that 1)

2)

for each.

Proof. Due to Proposition 2.22.9, §2.29 and Theorem 2.30 [18] the space is -vector. Use Lemma 2.26 [18] and take a marked element, then we consider from into. It also is the homomorphism of the unit sphere

into and for

for each is accomplished the identity in accordance with Lemma 2.5 [18]. If, then, consequently, .

Let, then there exists, where, consequently, is bijective, and

, that is,. For

the operator

is bounded and it is defined everywhere and this case was considered in Theorem 2.24 [18]. Therefore, the mapping is the homeomorphism from onto, consequently, also from onto.

If and, then is associative and is alternative, since, while a purely imaginary quaternion or octonion number has the decomposition, where, ,. If, then is alternative as the subalgebra contained in,. If and, then anti-commutes with and, also is orthogonal to q and, consequently, is alternative in this case also. Therefore, the subalgebra is alternative for each and.

For each put, where is the -graded decomposition of the unity for a normal operator. On the other hand, each vector has the decomposition and projection valued measure is graded, where for each l (see §2.8).

Mention that, since in the contrary case

for each such that contradicting the invertibility of. Therefore, , consequently, is an graded spectral measure. This spectral measure is self-adjoint, countably additive and regular due to such properties of (see Theorem 15). Then and for, where. We now put

Evidently, for each bounded Borel subset contained in, since

and

Thus the equality 1)

is fulfilled.

We next verify, that. Take an arbitrary vector. From the definition we have the formula, consequently, a vector exists such that and

Making the change of measure we infer the relations:

2)

consequently,

This demonstrates that and hence.

In view of Formula (2) and Theorem 15 we have

consequently,. At the same time one has:

and this together with (1) implies that the sequence converges for each. The operator is closed and, consequently, and. Thus

and together with the opposite inclusion demonstrated above this gives the equality.

Suppose that another graded measure exists with the same properties as. In accordance with statement of this theorem we have

and together with this implies that

In accordance with Theorems 15 above and 2.24, Propositions 2.22.20, 2.22.12, 2.22.18 [18] one has

and

consequently,

Taking the limit with tending to the infinity and substituting the measure one gets:

consequently, and hence for each Borel subset contained in. Thus the graded spectral measure is unique.

22) Definitions. A unique (graded) spectral measure, related with a self-adjoint quasi-linear operator T in X is called a decomposition of the unity for T. For an - valued Borel function f defined -almost everywhere on R a function of a self-adjoint operator T is defined by the relations:

1)where for; while;

2) for each, where.

23) Theorem. Let be a decomposition of the unity for a self-adjoint quasi-linear -meta-invariant operator T in a Hilbert space X over the Cayley-Dickson algebra and let a Borel function f be as in §22, ,. Then is a closed quasilinear operator with an everywhere dense domain of its definition, moreover:

a)

b)

c)

d)

e)

Proof. Take a sequence of functions from and subsets. Then

for each, from this it follows and that is contained in

.

If, then

for each, consequently,

This demonstrates statement a).

To the non-commutative measure given on a - algebra of subsets of the set a quasi-linear operator with values in corresponds and due to Theorem 2.24 and Definitions 2.23 [18] this measure is completely characterized by the family of -valued measures, such that

where with real-valued functions,

for, for each -integrable -valued function with components, where . Then it can be defined the variation of the measure

by all finite disjoint systems of subsets in with.

We have the embedding. Here we consider the Borel -algebra on the CayleyDickson algebra. Mention that each Borel measure on has a natural extension to a Borel measure on so that. If is bounded, then it is the quasi-linear operator of the bounded variation with

moreover, is -additive on, if is -additive, that is in the case considered here with and.

The function we call -measurable, if each its component is -measurable for each and The space of all -measurable - valued functions with

we denote by while, also is the space of all for which there exists

In details we write instead of, where

with non-negative -measurable function , ,

since the function is real-valued. A subset in we call -zero-set, if, where is the extension of the complete variation V by the formula

for. An -valued measure on we call absolutely continuous relative to, if for each subset with. A measure we call positive, if each is non-negative and there exist indices for which is positive (i.e. non-negative on its -algebra and positive on some elements of this -algebra).

For a reference we formulate the following non-commutative variants of Radon-Nikodym’s theorem.

RNCD. Theorem. 1) If is a space with a -finite positive -valued measure, where 2 ≤ v ≤ 3, also is an absolutely continuous relative to bounded -valued measure defined on, then there exists a unique function, such that for each, moreover,

2) If is a space with bounded -valued measure, also is an absolutely continuous relative to -valued measure defined on, where, then there exists a unique function, such that for each.

Proof. This follows from the corresponding RadonNikodym’s theorems for R-valued and C-valued measures.

Recall the classical Radon-Nikodym theorem (III.10.2, 10.7 [19]). Let be a measure space with a -finite positive measure and let be a defined on finite real-valued measure absolutely continuous relative to, then there exists a unique function such that

for each, moreover,. Let be a space with a finite complex-valued measure and let be a defined on finite complex-valued measure absolutely continuous relative to, then there exists a unique function such that

for each.

In the quaternion skew field and the octonion algebra each equation of the form or for nonzero a has the solution or respectively.

On the other hand, a measure is absolutely continuous relative to for each, consequently, with

and hence

, where. Analogously, one gets with. Thus almost everywhere on one obtains

Continuation of the proof of Theorem 23.

The -meta-invariance of T implies, that the families of subalgebras isomorphic with exist, which are defined on vector sub-spaces over satisfying Conditions 2.9. Therefore, we take restrictions and reduce the consideration to the case over up to an isomorphism of the CayleyDickson algebras.

Recall that a mapping of a topological space U into a topological space W is called closed if its graph is a closed subset in, where

. In view of criterion (14.1.1)

[15] a mapping has a closed graph if and only if for each net in U converging to and with the equality is fulfilled.

We now prove that is a closed operator. For each we have and for each, consequently, a domain of definition of is everywhere dense in. Take a sequence so that and. Then for each natural number we get the equalities:

Thus

hence and the operator is closed.

Let,. Due to for the variation of the measure there exists a Borel measurable function, such that for each. From it follows that -almost everywhere. Consider, then due to and

since is the center of the Cayley-Dickson algebra and for each. Therefore,

and from this it follows b).

d) From for each, , , where and for each, it follows, that. Take, then

consequently,. If, then for each and one gets that

consequently, converges to for and inevitably.

e) Due to Theorem 2.24 [18] the statement follows from the fact that is the decomposition of the unity for the bounded restriction, where. On the other hand,

for each and

. Clearly. Therefore,

that finishes the proof.

24) Theorem. A bounded normal -meta-invariant operator on a Hilbert space over the Cayley-Dickson algebra, , , is unitary, Hermitian or positive if and only if is contained in, or respectively.

Proof. Due to Theorem 2.24 [18] the equality is equivalent to for each . If, then

for each. Applying Theorems 4 and 12 we get the statement of this theorem.

25) Definition. A family of bounded quasi-linear operators in a vector space X over the Cayley-Dickson algebra, , is called a strongly continuous semigroup, if a) for each;

b);

c) is a continuous function by for each;

For let an operator be defined by the equality d) for each.

Denote by the set of all vectors for which the limit exists and put e) for each. This operator A is called an infinitesimal generator of the oneparameter semigroup.

26) Theorem. For each strongly continuous semigroup of unitary quasi-linear - meta-invariant operators in a Hilbert space X over the Cayley-Dickson algebra, , , there exists a unique self-adjoint - meta-invariant quasi-linear operator B in X so that 1)

2)

for each, where is an graded projection valued measure, is a Borel function from into.

If additionally and is either left or right -linear operator for each t, then there exists a marked purely imaginary Cayley-Dickson number such that 3) for each.

Proof. For a marked this was partially demonstrated in Section 10. Then we demonstrate it relative to the parameter t. Extent a semigroup for to a group for each putting.

Consider the space of all infinitely differentiable functions with compact support. For each and consider the integral 4)

Since the group is strongly continuous, this integral can be considered as the Riemann integral. Take the -vector space

Then we choose a function with the support

and for each b and on some interval and

where. Then we put for. It can be lightly seen that 5)

The one-parameter group is strongly continuous, consequently, the -vector space is everywhere dense in. For we deduce 6)

consequently,

since the function uniformly converges to on when tends to zero. Then we put. From the definition of we get that for each and, since is the one-parameter group and is the center of the Cayley-Dickson algebra, which implies

.

Then the equalities are satisfied:

since and

and is the one-parameter group, consequently, is the normal -meta-invariant operator. We have that

hence and the operator is skew-adjoint.

Then is self-adjoint on. By Theorem 2.24 [18] it has the graded projection valued measure and

for each, since, consequently,

. Take and

for, where denotes as usually the closed ball in with center at zero and of radius. Then

is the bounded operator on, up to an isomorphism of Hilbert spaces. In view of Theorems 3.4 [18], 4, 9, 12 above the von Neumann algebra over generated by is-isomorphic with for an extremely disconnected compact topological space. Then is a skew-symmetric function

for each and hence has the form with and for each, where and,

denotes a^{*}-isomorphism. We have that for each Borel subsets and contained in. Therefore, up to a^{*}-isomorphism of-algebras the aforementioned functions and algebras can be chosen consistent: and for each.

Then we get is the self-adjoint - meta-invariant operator with an -graded projection valued measure on so that

for each. Therefore,

where is a Borel function with values in (see Theorems 15, 16).

We now take a consistent family of -graded projection valued measures on and put

This operator is self-adjoint with a domain of definition given by Theorem 21. Consider the operator

. Since for each and is the -graded projection valued measure on, operators are well defined. Their family by forms the one-parameter group of normal -meta-invariant unitary operators with a domain given by Theorem 24. Indeed, the projection valued measure is graded and

for each, since for every.

It remains to show that for each. There is the inclusion. Let, then

and. On the other hand,

for each. We consider

and get

consequently,

since the operator Q is skew-symmetric. But, consequently, for each. Thus the equality is fulfilled for each and. The vector subspace Y is everywhere dense in X, consequently, for each.

When the one-parameter group satisfies additional conditions either left or right -linearity and, then there exists a marked purely imaginary Cayley-Dickson number such that for each due to Theorem 10 and the proof above, since in this case and are constants in accordance with Conditions 25.

27) Remark. Another way of the preceding theorem proof in the particular case and either left or right linear operators for each t is the following.

If is a semigroup continuous in the operator norm topology (see also the complex case in Theorem VIII.1.2 [19]), then there exists a bounded -meta-invariant either left or right -linear operator on such that for each. If, then

for. For such due to the Lebesgue theorem:

also by Lemma 2.8 there exists

.

For each let, where, for which there exists, the set of all such we denote by. Evidently, is the - vector subspace in X. Take some infinitesimal quasilinear operator. Considering as the Banach space over, we get the analogs of Lemmas 3, 4, 7, Corollaries 5, 9 and Theorem 10 from VIII.1 [19], moreover, is dense in X, also A is the closed quasi-linear -meta-invariant operator on. Let and with. For each due to Corollary VIII.1.5 [19] a constant exists such that

for each. Then there exists

for each and, consequently,.

Let be a quasi-linear operator corresponding to instead of for, where, moreover,. Then

consequently, for each and. Thus,

for each and.

From [18] it follows, that for the quasi-linear operator A the quasi-linear operator B exists such that, where, ,. In view of the identities we get that A commutes with and. From the equality it follows, that we can choose. If is the decomposition of the unity for B and, then by Theorem 2.30 [18] we have

then due to the Fubini theorem one gets the equality

for each with. Therefore, we infer

for. Due to Lemma the equality

is fulfilled for each and for every with, consequently,.

28) Definitions. A topological -vector space is called locally convex, if it has a base of open neighborhoods of zero consisting of -convex open subsets, that is for each with and every. Let be an -vector locally convex space. Consider left and right and two sided -vector spans of the family of vectors, where

where is a vector prescribing an order of the multiplication in the curled brackets.

29) Lemma. In the notation of

Proof. Due to the continuity of the addition and multiplication on scalars of vectors in X and using the convergence of nets in X, it is sufficient to prove the statement of this lemma for a finite set A. Then the space is finite dimensional over and evidently left and right -vector spans are contained in it. Then in Y it can be chosen a basis over and each vector can be written in the form , where.

Each can be written in the polar form , where, is a real parameter, and M is a purely imaginary Cayley-Dickson number, of unit absolute value and. Hence for each with real parameter, and a Cayley-Dickson number, since the real field is the center of the Cayley-Dickson algebra.

On the other hand, , where are pairwise isomorphic -linear locally convex spaces. Therefore, we have

that together with the inclusion

proved above leads to the statement of this lemma.

30) Lemma. Let X be a Hilbert space over the CayleyDickson algebra, also be the same space considered over the real field R. A vector is orthogonal to an -vector subspace Y in X relative to the -valued scalar product in X if and only if x is orthogonal to relative to the scalar product in. The space X is isomorphic to the standard Hilbert space over of converging relative to the norm sequences or nets with the scalar product, moreover,

, where is the cardinality of a set, , denotes the topological density of X.

Proof. Due to Lemma 29 and by the transfinite induction in Y, an -vector independent system of vectors exists, such that is everywhere dense in Y. In another words in Y, a Hamel basis over exists. A vector x by the definition is orthogonal to Y if and only if for each, that is equivalent to for each. The space X is isomorphic to the direct sum

, where are the pairwise isomorphic Hilbert spaces over, also. The scalar product in then can be written in the form 1)where due to above. Then the scalar product in induces the scalar product.

2).

in. Therefore, the orthogonality of x to the subspace Y relative to is equivalent to for every and each k, n, that implies the orthogonality of x to the subspace relative to. Due to Lemma 31 from it follows, that for each. If, then, where either or. Moreover, by the definition Y is the two-sided module over the CayleyDickson algebra. Then from for each and due to Formula 2.8 (SP) it follows, that for each k, n, since Y is the -vector space, consequently, for each.

Then by the theorem about transfinite induction [14] in X, the orthogonal basis over exists, in which every vector can be presented in the form of the converging series of left (or right) -vector combinations of basic vectors. The real Hilbert space is isomorphic with, consequently, is isomorphic with. The Cayley-Dickson algebra is normed, while the real field R is separable, hence. The space X is normed, consequently, the base of neighborhoods of x is countable for each, hence for the topological density it is accomplished the equality. Particularly, for a finite dimensional algebra, i.e., and one gets.

31) Lemma. For each quasi-linear operator T in a Hilbert space X over the Cayley-Dickson algebra an adjoint operator in X relative to the -scalar product induces an adjoint operator in relative to the -valued scalar product in.

Proof. Let be a domain of the definition of an operator T, which is dense in X. Due to Formulas 30 and the existence of the R-linear automorphisms in as the R-linear space for each

, the continuity of scalar products implies that of by. For these continuities are equivalent. Therefore, due to Lemma 29 the family of all, for which is continuous by forms an -vector subspace in X and this provides a domain of the definition of the operator everywhere dense in. Then the adjoint operator is defined by the equality, also is given by the way of, where, also

. Due to Formula 30 one deduces for each,

and. In view of Propositions 2.22.18 and 2.22.20 [18] and Lemma 29 above and are -vector spaces, then the family of -linear automorphisms of the CayleyDickson algebra as the -linear space given above lead to the conclusion that on induces in.

32) Definition. A bounded quasi-linear operator P in a Hilbert space over the Cayley-Dickson algebra is called a partial - (or -) isometry, if there exists a closed - (or -) vector subspace such that for each and (or respectively), where

.

33) Definition. An operator with an - vector domain in a Hilbert space X over the Cayley-Dickson algebra is called densely defined, if is (every)where dense in X, where Y is a Hilbert space over.

An operator Q extends T (or Q is an extension of T) if and for each. This situation is denoted by.

Denote by a graph of T. If is the graph of a quasi-linear operator, then one says that T is pre-closed (or closable) and refers to as the closure of T.

34) Theorem. Let T be a densely defined quasi-linear operator in a Hilbert space X over the Cayley-Dickson algebra, where Y is a Hilbert space over,. Suppose that T is either left or right - linear. Then 1) If T is pre-closed, then;

2) T is pre-closed if and only if is dense in;

3) If T is pre-closed, then;

4) If T is closed, then the operator is oneto-one with range X and positive inverse of norm not exceeding 1;

5) If T is closed, then the operator is self-adjoint and positive;

6) If T is either left or right -linear on, then is either left or right -linear on respectively.

Proof. If an operator T is either left or right -linear, then is an -vector space due to Lemma 29. Therefore, consider the case of being an - vector space.

6) Let T be right -linear and , , then

in accordance with Formula 2.8, consequently, for each, since

induces the -linear isomorphism of onto, for each. Analogously Statement of this theorem is verified in the case of a left -linear operator T.

1) If Q is densely defined and T is an extension of Q, then for each and, consequently, and

, hence is an extension of. From it follows that. For an arbitrary vector take a sequence in converging to such that converges to. Therefore,

for each, consequently, and. Thus.

2) If is closed in, then it is the Hilbert space over so that. Consider the mapping, consequently, P is bounded and quasi-linear. Thus P has a bounded adjoint mapping Y into. This operator P has null space, since only when y = 0. In view of Theorem 2.30 [18] the range of is dense in as the -vector space. Thus contains a dense -vector subspace the range of consisting of all pairs with

. If a vector y is orthogonal to

, then for eachhence and it is annihilated by T^{*}. Thus contains a dense subset of the -vector range of as well as the orthogonal complement of this range due to Lemma 31. The formula of the scalar product 2.8 on X and Cauchy-Schwartz’ inequality implies that there is the -vector subspace in Y, since is the -vector subspace. Since is an -vector subspace, it is dense in Y.

Suppose now that is dense in Y and is a sequence in converging to zero so that converges to. Then if we get, hence converges to 0 and simultaneously. Therefore, the operator T is pre-closed, since is dense in Y and hence w = 0.

3) At first we demonstrate that if T is densely defined, then is a closed operator. Take arbitrary vectors and. Let T be left -linear, so we get

for each and and

, since the subalgebra is associative for each, consequently, and. Thus is an -vector space by Lemmas 30 and 31 and is a left -linear operator, since each Cayley-Dickson number b has the decomposition with for every l. Analogously it can be demonstrated in the right - linear case (see also (6) above). If is a sequence in converging to z so that converges to, then, hence the sequence converges to and inevitably this gives for each. Thus and. This means that the operator is closed.

Suppose that conditions of (3) are accomplished. The domain of definition is dense in X in accordance with (2), when T is pre-closed, so that T^{*} has an adjoint operator. Then for each and, consequently, and. Therefore, the operator is closed, consequently,. From Section 1 of this proof, it follows that. On the other handwe have, since the operator T^{*} is closed.

Thus.

We have that and. This implies that for each

when is orthogonal to. Particularly it holds for and implies

.

From (2) of this proof we get that. Indeed, if and, then and

for each, consequently, and. Then and

. Thus and and.

4) If, then

Hence, that is, this operator is one-to-one and has a bounded inverse of norm not exceeding one. Each has the form, consequently, the inverse operator

is positive, since

.

5) From 4) we have that is dense in X. For each we get

consequently, and have the same domain and. With

one has the equality such that

and, consequently,

. If a vector y is such that

, then

for each, consequently, , since

.

Thus the operator is bijective and extends the operator and these two operators have the same range. Therefore, one infers that

so that.

Then the inequality for each shows that the operator is positive.

35) Theorem. If T is a closed quasi-linear -metainvariant operator in a Hilbert space X over the CayleyDickson algebra, where, , then T = PA, where P is a partial R-isometry on X_{R} with the initial domain, also A is a self-adjoint quasi-linear operator such that, where denotes a range of T. If additionally T is either left or right -linear, then there are either left or right -linear isometry P and either left or right -linear operator A respectively such that T = PA.

Proof. Due to the spectral theorem 21, a self-adjoint quasi-linear -meta-invariant operator is positive if and only if its spectrum is contained in. In the Cayley-Dickson algebra each polynomial has a root, i.e. zero, (see Theorem 3.17 [16] or [13]). Therefore, if is a positive self-adjoint quasi-linear operator, then there exists a unique positive quasi-linear -meta-invariant operator, such that,

With for and a domain of its definition is given by Theorem 21. Therefore, generally there exists a positive square root of the operator. The space is R-linear, where.

If the operator A is in addition either left or right - linear, then is the -vector subspace due to Lemma 29. In view of Lemma 30, there exists the perpendicular projection from X on, moreover, is right or left -linear, if is the -vector subspace.

Then Q is a correctly defined isometry with the domain of definition, where B is the restriction of A on, , where is considered as the R-linear space embedded into X in general, but for either left or right -linear operator T it is considered as the -vector space, analogously for domains and ranges of the considered here other operators, that is clear from the context. In view of Theorem 34 is dense in, since. Therefore, if , then there exists a sequence in converging to, consequently, and the vector space is dense in . Let P^{1} be an isometric extension of the operator Q on and F be a perpendicular projection in X on. If put, then P is a partial isometry with an initial domain. Moreover, for every.

Let be a sequence in such that and. Then there exists the limit, since is the isometric R-linear mapping. Then and, since the operator is closed. Thus and, consequently, the operator is closed.

It is necessary to verify that. Let K be a restriction of T on. Then by Theorem 34 is dense in. If, there exists a sequence in such that . From and closeness of PA it follows that

consequently,.

We have that is dense in. Therefore, for each a sequence exists so that. Then we deduce that, since

and the operator T is closed. Thus, consequently, together with the opposite inclusion demonstrated above, it implies that.

In view of Theorem 34 one gets

since if and only if

We next demonstrate that this decomposition into the product of operators is unique. In accordance with Theorem 34 AP^{*} = T^{*}, consequently,. On the other hand, is the projector on, hence. A uniqueness of A follows from Theorem 21. Since the mapping A is unique, the operator P is defined on by the equation in the unique manner. A continuous extension of P from on is unique. The restriction of operator P on the orthogonal complement is zero. Thus the operator P is uniquely defined by the operator T.

At the same time A is either right or left -linear, if T is either right or left -linear, since so is for such T and also is either right or left -linear due to Theorem 34(6). Since T and A are either right or left -linear, then P also should be such.

The presented above results of this paper and from works [16,20-30] can be used for further developments of the operator theory over the Cayley-Dickson algebras including that of PDO.

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