Inner product on B -algebras of operators on a free Banach space over the Levi-Civita field

Available online at wwwsciencedirectcom ScienceDirect Indagationes Mathematicae 26 (215) 191 25 wwwelseviercom/locate/indag Inner product on B -algebras of operators on a free Banach space over the Levi-Civita field José Aguayo a, Miguel Nova b, Khodr Shamseddine c, a Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepcion, Casilla 16-C, Concepción, Chile b Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile c Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada Received 24 April 214; received in revised form 11 September 214; accepted 16 September 214 Communicated by B de Pagter Abstract Let C be the complex Levi-Civita field and let c (C) or, simply, c denote the space of all null sequences z = (z n ) n N of elements of C The natural inner product on c induces the sup-norm of c In a previous paper Aguayo et al (213), we presented characterizations of normal projections, adjoint operators and compact operators on c In this paper, we work on some B -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators We finish the paper with a characterization of closed subspaces of the B -algebra of all adjoint and compact operators on c which admit normal complements c 214 Royal Dutch Mathematical Society (KWG) Published by Elsevier BV All rights reserved Keywords: Banach spaces over non-archimedean fields; Inner products; Compact operators; Self-adjoint operators; Positive operators; B -algebras Corresponding author Tel: +1 24 474 627 E-mail addresses: jaguayo@udeccl (J Aguayo), mnova@ucsccl (M Nova), khodrshamseddine@umanitobaca (K Shamseddine) http://dxdoiorg/1116/jindag21496 19-3577/ c 214 Royal Dutch Mathematical Society (KWG) Published by Elsevier BV All rights reserved

192 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 1 Introduction Two of the most useful and beautiful mathematical theories in real or complex functional analysis have been Hilbert spaces and continuous linear operators These theories have exactly matched the needs of many branches of physics, biology, and other fields of science The importance of Hilbert spaces over the real or complex fields has led many researchers to try and extend the concept to non-archimedean fields One of the first attempts to define an appropriate non-archimedean inner product was made by G K Kalisch [3] Two of the most recent papers about non-archimedean Hilbert spaces are those of L Narici and E Beckenstein [4] and the authors [1] They define a non-archimedean inner product on a vector space E over a complete non-archimedean and non-trivially valued field K as a non-degenerated K-function in E E, which is linear in the first variable and satisfies what they call the Cauchy Schwarz type inequality Recall that a vector space E is said to be orthomodular if for every closed subspace M of E, we have that E is the direct sum of M and its normal complement The existence of infinite-dimensional non-classical orthomodular spaces was an open question until the following interesting theorem was proved by M P Solèr [8]: Let X be an orthomodular space and suppose it contains an orthonormal sequence e 1, e 2, (in the sense of the inner product) Then the base field is R or C Based on the result of Solèr, if K is a non-archimedean, complete valued field and L (c ) is the space of all continuous linear operators on c, then there exist T L (c ) which does not have an adjoint For example, T (x) = x i e1 is such a linear operator; on the other hand, the normal projections (see the definition below) admit adjoints In this paper, we consider the complex Levi-Civita field C as K; in C, we take the natural involution z z Recall that a free Banach space E is a non-archimedean Banach space for which there exists a family (e i ) i I in E \ {θ} such that any element x E can be written in the form of a convergent sum x = i I x ie i, x i K, ie, lim i I x i e i = (the limit is with respect to the Fréchet filter on I ) and x = sup i I x i e i The family (e i ) i I is called an orthogonal basis Now, if E is a free Banach space of countable type over C, then it is known that E is isometrically isomorphic to c (N, C, s) := (x n ) n N : x n C; lim x n s(n) =, n where s : N (, ) Of course, it could be that, for some i N, s (i) C \ {} But, if the range of s is contained in C \ {}, it is enough to study c (N, C) (taking s to be the constant function 1), which will be denoted by c (C) or, simply, c We already know that c is not orthomodular In a previous paper [1], we characterized closed subspaces of c with a normal complement; that is, we characterized those non-trivial closed subspaces M which admit a non-trivial closed subspace N such that a c = M N, and b for x M and y N, x, y = N is actually the subspace M p = {y c : x, y = for all x M} and then c = M M p Such a subspace, together with its normal complement, defines a special kind of projection, the so-called normal projection; that is, a linear operator P : c c such that i P is continuous; ii P 2 = P; iii z, w =, for all z N (P) and for all w R (P)

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 193 Actually these concepts are not exclusive to c ; if E is a vector space with an inner product, then normal complements and normal projections have similar meaning In this paper we will study some B -subalgebras of L (c ) and we will define an inner product such that the usual norm of linear operators will be induced by this inner product 2 Preliminaries and notations Throughout this paper R (resp C) will denote the real (resp complex) Levi-Civita field; for a detailed study of R (and C), we refer the reader to [6,7] and the references therein Any z C (resp R) is a function from Q into C (resp R) with left-finite support For w R (resp C), we will denote by λ (w) = min (supp (w)), for w, and λ () = + On the other hand, since each z C can be written as z = x +iy, where x, y R, we have that λ (z) = min {λ (x), λ (y)} If we define e λ(z) z = if z if z =, then is a non-archimedean absolute value in C It is not hard to prove that (C, ), where is the metric induced by, is a complete metric space Now let z = x + iy in C be given If x y then z = e λ(z) = e min{λ(x),λ(y)} = max e λ(x), e λ(y) = max { x, y } We can easily also check that z = max{ x, y } when x = or y = Thus, z = max { x, y } for all z = x + iy C In other words, C is topologically isomorphic to R 2 provided with the product topology induced by in R We denote by c (C), or simply c, the space c = z = (z n ) n N : z n C; lim z n = n A natural non-archimedean norm on c is z = sup { z n : n N} Writing z n = x n + iy n and x = (x n ) n N, y = (y n ) n N, we also have the equality z = max x, y It follows that (c, ) is a Banach space For a detailed study of non-archimedean Banach spaces, in general, we refer the reader to [9] Recall that a topological space is called separable if it has a countable dense subset In the class of real or complex Hilbert spaces, we can distinguish two types: those spaces which are separable and those which are not separable If E is a separable normed space over K, then each one-dimensional subspace is homeomorphic to K, so K must be separable too Nevertheless, we know that there exist non-archimedean fields which are not separable, for example, the Levi- Civita fields R and C Thus, for non-archimedean normed spaces the concept of separability cannot be used if K is not separable However, by linearizing the notion of separability, we obtain a generalization, useful for each non-archimedean valued field K A normed space E over K is said to be of countable type if it contains a countable subset whose linear hull is dense in E An example of a normed space of countable type is (c (K), ), for any non-archimedean valued field K, in particular, when K is the complex Levi-Civita field C

194 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 Let us consider the following form:, : c c C; z, w = z n w n n=1 This form is well-defined since lim n z n w n = and, at the same time,, satisfies Definition 241, p 38, in [5] Let z := z, z Then, since 2 = 1, is a non-archimedean norm on c (Theorem 242 (ii) in [5]) It follows easily that x, y =, y c x = θ which is referred to as the non-degeneracy condition The next theorem was proved in [4] and tells us when the non-archimedean norm in a Banach space is induced by an inner product Theorem 1 Let (E, ) be a K-Banach space Then, if E K 1/2 and if every onedimensional subspace of E admits a normal complement, then E has, at least, an inner product that induces the norm If E = c and K = C, then the conditions of the theorem above are satisfied In fact, if z c, z θ, then lim n z n =, which implies that there exists j o N such that z = max z j : j N = z j C Now, since C C 1/2, c C 1/2 The other condition is guaranteed by Lemma 2319, p 34 in [5] It was proved in [1] that, is one of the inner products that induce the norm on c Such a result was guaranteed thanks to the following lemma which will be useful also in this paper Lemma 1 If {z 1, z 2,, z n } C, then z 1 z 1 + z 2 z 2 + + z n z n = max { z 1 z 1, z 2 z 2,, z n z n } Definition 1 A subset D of c such that for all x, y D, x y x, y =, is called a normal family A countable normal family {x n : n N} of unit vectors is called an orthonormal sequence If A c, then [A] and cl [A] will denote the linear and the closed linear span of A, respectively If M is a subspace of c, then M p will denote the subspace of all y c such that y, x =, for all x M Since the definition of the inner product given in [5, p 38], coincides with the definition of inner product given here, the Gram Schmidt procedure can be used

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 195 Theorem 2 If (z n ) n N is a sequence of linearly independent vectors in c, then there exists an orthonormal sequence (y n ) n N such that [{z 1,, z n }] = [{y 1,, y n }] for every n N Definition 2 A sequence (z n ) n N of non-null vectors of c has the Riemann Lebesgue Property (RLP) if for all z c, lim z n, z = n Obviously, any orthonormal basis of c has this property The following theorem was proved in [4] Theorem 3 If S c, is a finite orthonormal subset, say {z 1,, z n }, or is an orthonormal sequence (z n ) n N which satisfies the RLP, then S can be extended to an orthonormal basis for c ; that is, there exists a countable orthonormal sequence (w n ) n N (possibly finite) such that S {w n : n N} is an orthonormal basis for c Lemma 2 If (z n ) n N is an orthonormal sequence in c, then (z n ) n N is orthogonal in the van Rooij s sense (see [9, p 57]) If E and F are normed spaces over K, then L (E, F) will be the normed space consisting of all continuous linear maps from E into F L (E, K) will be denoted by E and L (E, E) will be denoted by L (E) For a T L (E, F), N(T ) and R(T ) will denote the kernel and the range of T, respectively It is well-known that the dual of c is c = l A linear map T from E into F is said to be compact if, for each ϵ >, there exists a continuous linear map of finite-dimensional range S such that T S ϵ Any continuous linear operator u L (c ) can be identified with the following matrix whose columns converge to : α 11 α 12 α 13 α 1 j α 21 α 22 α 23 α 2 j α 31 α 32 α 33 α 3 j [u] = α i1 α i2 α i3 α i j Definition 3 A linear operator v : c c is said to be an adjoint of a given linear operator u L (c ) if u (x), y = x, v (y), for all x, y c In that case, we will say that u admits an adjoint v We will also say that u is self-adjoint if v = u In [1] we showed that if a continuous linear operator u has an adjoint, then the adjoint is unique and continuous Lemma 3 Let u L (c ) with associated matrix α i, j Then, u admits an adjoint operator v if and only if lim j α i j =, for each i N In terms of matrices, this means

196 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 that α 11 α 12 α 13 α 1 j α 21 α 22 α 23 α 2 j α 31 α 32 α 33 α 3 j [u] = α i1 α i2 α i3 α i j In the classical Hilbert space theory, any continuous linear operator admits an adjoint This is not true in the non-archimedean case For example, the operator u L(c ) given by the matrix: b b 2 b 3 b j, with 1 < b, does not admit an adjoint, by Lemma 3 The following two theorems provide characterizations for normal projections (see the proofs in [1]) Theorem 4 Let P L(c ) If P is a normal projection, then P is self-adjoint Conversely, if P is self-adjoint and P 2 = P then it is a normal projection Theorem 5 If P : c c is a normal projection with R (P) = [{y 1, y 2, }], where y1, y 2, is an orthonormal finite subset of c or an orthonormal sequence with the Riemann Lebesgue Property, then Px = x,y i y i,y i y i The following theorem (proved in [1]) provides a way to construct compact and self-adjoint operators starting from an orthonormal sequence Theorem 6 Let (y i ) i N be an orthonormal sequence in c Then, for any λ = (λ i ) i N in c such that λ i R, the map T : c c defined by T ( ) = λ i P i ( ), where P i ( ) =,y i y i,y i y i, is a compact and self-adjoint operator The converse is also true, as the following theorem shows Theorem 7 Let T : c c be a compact, self-adjoint linear operator of infinite dimensional range Then there exists an element λ = (λ n ) n N c (R) and an orthonormal sequence (y n ) n N

in c such that T = λ n P n, where n=1 P n =, y n y n, y n y n is a normal projection defined by y n J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 197 The uniqueness of the element (λ n ) n N of c (R) in Theorem 7 is shown by the following proposition, also proved in [1] Proposition 1 Let T = n=1 λ n,y n y n,y n y n be a compact and self-adjoint operator and let µ in C be an eigenvalue of T Then µ = λ n for some n 3 B -algebras Some of the results of this section have already been developed in [2] and our goal here is to connect those results with linear compact operators and linear normal projections Recall that each T L (c ) has an associated matrix α i j such that (1) sup αi j < ; (2) lim i α i j =, for any j N; (3) T = α i je j e i, where e j e i : c c is given by e j e i(z) = z, e j ei ; (4) T = sup αi j = supn N T e n If we let αi M (c ) = j : sup α i j < ; j N, lim α i j =, i equipped with the natural supremum norm, then L (c ) and M (c ) are isometrically isomorphic A Riesz functional is a continuous linear functional of the form x x, y for some y c If we take T L (c ) and y c, then the functional x T x, y belongs to c ; but, in general, this functional is not necessarily a Riesz functional As an example, take T x = x i e1 On the other hand, if T x = I d(x) = x, then x T x, y is of the Riesz type for any y c Lemma 4 Let T L(c ) Then, for a given y c, the following conditions are equivalent: (1) There exists y c such that T x, y = x, y, x c (2) lim n T e n, y = Proof (1) (2): Assume that there exists y c such that T x, y = x, y, x c Then T e n, y = e n, y = y n n (2) (1): Assume that lim n T e n, y = Then ( T e n, y ) c If we define y = n=1 T e n, y e n, then for x c x, y = x, T e n, y e n = T e n, y x, e n n=1 n=1

198 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 and T x, y = T x, e n e n, y = x, e n T e n, y n=1 n=1 Remark 1 (1) Given T L (c ), we denote by D (T ) the following set: D T = y c : lim T e n, y = n = y c : y c, T x, y = x, y for all x c Clearly, D (T ) is a closed subspace of c Moreover, y, the corresponding element associated to y D (T ), is unique (2) If T satisfies condition (1) (or (2)) of Lemma 4, then for any arbitrary fixed y c, the functional x T x, y is a Riesz functional Let us denote by A (c ), or simply A, the collection of all T L (c ) such that D (T ) = c Clearly, I d A and T A, where T x = x i e1 ; therefore A L (c ) Also, A is a non-commutative Banach algebra with unity Since for T A and for any y D (T ) = c, y is unique, we can define T : c c by T (y) = y Note that T x, y = x, y = x, T (y), for any x, y c Even more, if T (y) θ, then T (y) 2 = T (y), T (y) = T T (y), y from which it follows that T (y) T y T T (y) y T T (y) y ; Therefore, T L (c ) and it is called the adjoint operator of T It is not hard to show that T A if and only if lim j α i j =, for all i N, where [T ] = α i j is the associated matrix of T L (c ) with respect to the canonical base {e n : n N} In other words, A is the collection of all continuous linear operators which admit adjoints In particular, A contains normal projections By Lemma 4, A can be rewritten as A = T L (c ) : y c, lim T e j, y = j Now, if T A, T = α i je j e i, then the sequence α i j j N = T e j, e i j N c, for each i N In terms of matrices, T admits an adjoint if and only if α 11 α 12 α 13 α 1 j α 21 α 22 α 23 α 2 j α 31 α 32 α 33 α 3 j [T ] = ; α i1 α i2 α i3 α i j

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 199 in such a case the matrix associated with the adjoint T of T is the complex conjugate of the transpose of the matrix associated with T Thus, if αi M (c ) = j M (c ) : i N, lim α i j =, j then A and M (c ) are isometrically isomorphic Also, T x, y = y, T x = T y, x = x, T y ; that is, T A and, by uniqueness, (T ) = T = T Therefore, the map : A A ; T T, is an involution on A Altogether, we say that A is a non-archimedean B -algebra For each a c, the linear operator M a, defined by M a x = a i x, e i e i, belongs to A ; moreover, lim M ae n = lim n n meanwhile, I d is also an element of A, but lim n I d (e n) = lim n e n = 1 a i e n, e i e i = lim n a n =, Let us denote by A 1 (c ), or simply A 1, the collection of all T L (c ) such that lim n T e n = θ, ie, A 1 = T L (c ) : lim n = θ n From the fact that T e n, y T e n y, we have that A 1 A ; but A 1 A since I d A 1 The next theorem will identify linear compact operators in A In order to do that, we will use the following result given in [2] Proposition 2 A continuous linear operator T = α i je j e i is compact if and only if lim i sup j N αi j = Theorem 8 T A 1 if and only if T is compact and T A Proof Suppose first T A 1 Now, for each j N, we define T j : c c by T j x = j x i T e i Clearly, T j is a continuous linear operator and its range is finite-dimensional; hence it is compact Moreover, if y c, then Tj e k, y = T ek, y if k j if k j + 1, therefore T j A

2 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 Now, since T x Tj x = i= j+1 x i T e i max i j+1 x i T e i = max i j+1 x i T e i x max i j+1 T e i and since lim j T e j = θ, we have that T j converges uniformly to T which implies that T is compact Conversely, we already know that if T A, then T A On the other hand, since T = α i je j e i and T is compact, we have that lim i sup αi j N j = Now, since sup αi j N j = T e i, we get that T A 1 Applying the first part of the proof, we conclude that T is also compact Thus, lim i sup βi j N j =, where T = β i je j e i, with β i j = α ji, which is equivalent to lim i sup α j N ji = Using the fact that sup α j N ji = T e i, we prove that T A 1 Remark 2 If T = α i je j e i A 1, then (1) lim i sup j N αi j =, since T is compact (2) lim j α i j =, i N, since T A Therefore, combining (1) and (2), we have that the double limit of the sequence α i j (i, j) N N is, that is, lim α i j = (i, j) N N Conversely, if lim (i, j) N N α i j =, then T = α i je j e i A 1 A linear operator T = α i je j e i A is said to be self-adjoint if T = T or, equivalently, if α i j = α ji As a corollary of the previous result we have that any compact and self-adjoint operator T belongs to A 1 Proposition 3 A 1 is a closed subalgebra of A Proof Let T A 1 and let ϵ > in R be given Then there exists S A 1 such that T S < ϵ Since S A 1, there exists N N such that Se n < ϵ for n N Therefore, for n N, we have that T e n max { (T S)e n, Se n } max { T S, Se n } < ϵ We will denote by A 2 the collection of all self-adjoint and compact operators; that is, A 2 = T A 1 : T = T Note that the operator S ( ) = a i, e i e i A 2 (a i ) i N c (R) Therefore, if α C \ R, then αs A 2, which implies A 2 is a proper subset of A 1 Nevertheless, we have the following:

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 21 Proposition 4 A 2 is a closed subset of A 1 Proof Let T A 2 ; hence there exists a sequence {T n } in A 2 such that lim n T n = T Since T x, y = lim n T nx, y = lim n x, T n y = x, T y, T is self-adjoint That T is compact follows from the fact that the space of compact operators is closed in L(c ) 4 Inner product in A 1 Now, we are ready to define an inner product in A 1 Since lim n Se n lim n T e n = θ for S, T A 1, the mapping, : A 1 A 1 C; (S, T ) S, T = = θ and since Se i, T e i, (41) is well-defined, linear in the first variable and linear conjugate in the second variable Note that S, T = T, S for any S, T A 1 It is clear that if w c, w, w R and then there exists a z C such that w, w = zz Therefore, if w 1, w 2,, w n c, then by Lemma 1 w 1, w 1 + w 2, w 2 + + w n, w n = max { w i, w i : i = 1,, n} From this lemma, we can prove that if S, S =, then S θ and then, is an inner product according to Definition 241, p 38, [5] On the other hand, since 2 = 1, we can conclude that S, S is a norm on A1 and S, T 2 S, S T, T The next proposition shows that the norm in A 1 is induced by the above inner product Proposition 5 Let T A 1, T θ Then, T, T = T 2 Proof Since T A 1, there exists N N such that i N T e i, T e i = T e i 2 < T 2 It follows that T e i, T e i max { T e i, T e i : i N} i=n < T 2 = max { T e i, T e i : i N} = max { T e i, T e i : i = 1,, N 1} N 1 = T e i, T e i Thus, N 1 T, T = T e i, T e i = T e i, T e i + T e i, T e i i=n N 1 = T e i, T e i = T 2

22 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 Theorem 9 c is isometrically isomorphic to a closed subspace of A 1 Moreover, the restriction of the inner product in A 1 to this closed subspace coincides with the inner product defined in c Proof Let a = (a i ) i N c ; hence M a : c c defined by M a x = a i x, e i e i is an element of A 1, whose corresponding matrix is given by a 1 a 2 a 3 [M a ] = It is easyto verify that M a = a and V = {M a : a c } is a closed subspace in A 1 If we define Ψ : c A 1, Ψ (a) = M a, then Ψ is a linear isometry Finally, if a = (a i ) i N, b = (b i ) i N c, then M a, M b = M a e i, M b e i = a i b i = a, b Remark 3 Observe that if T A 1, then, T : A 1 C, S S, T is a continuous linear functional; that is,, T A 1 Such functionals are called Riesz functionals But not all elements of A 1 are Riesz functionals For example, consider f : A 1 C; T = α i j f (T ) = α i j, then f is well-defined, since lim j α i j = for each i N and lim i α i j = uniformly in j N, (this is true because T is compact and then lim i sup αi j N j = ) f is also linear and continuous, since f (T ) = α i j max αi j = T We show that f, S, for all S A 1 Suppose there exists S A 1 such that f =, S ; since f e j e i = 1, for each i, j N, S should be non-null Now, if S = β i je j e i, then, among other limits, lim j β i j = But, 1 = f e j e i = e j e i, S = β i j, implies that β i j = 1, for any i, j N, which is a contradiction, since S A 1 Proposition 6 Let f be acontinuous linear functional Then f is a Riesz functional if and only if the double sequence f e j e i is convergent to, that is, lim f e j e i = (i, j) N N (i, j) N N

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 23 Proof First assume that f is a Riesz functional Then there exists an S = β i je j e i A 1 such that f =, S Now, for a fixed (i, j) N N f e j e i = e j e i, S = k=1 e j e i (e k ), S (e k ) = e i, Se j = βi j Since S A 1, we have that lim (i, j) N N f e j e i = Conversely, assume that lim (i, j) N N f e j e i = We set f e j e i = β i j, then lim (i, j) N N β i j = lim (i, j) N N f e j e i = We define S = (i, j) N N β i je j e i and we claim that f =, S In fact, if T = (i, j) N N α i je j e i A 1, then f (T ) = α i j f e j e i = α i j β i j = T, S Definition 4 Let M be a closed subspace of A 1 We shall say that M admits a normal complement if A 1 = M M p, where M p = {S A 1 : S, T =, for all T M} Remark 4 (1) Suppose that f is a non-null Riesz functional and S A 1, S θ, are such that f =, S We affirm that if T A 1, then T f (T ) S,S S N ( f ), R ( f ) = [{S}] = N ( f )p, where N ( f ) and R ( f ) are the kernel and the range of f In fact, V, S =, V N ( f ); hence R ( f ) = [{S}] N ( f ) p On the other hand, for U N ( f ) p, U, S U = U S, S S U, S + S, S S N ( f ) N ( f )p = A 1 =V Now, U, S = U, V = V, V + S, S S, V = V, V implies that V = θ and then U = U,S S,S S [{S}] = R ( f ) Summarizing, A 1 = N ( f ) R ( f ) = N ( f ) N ( f ) p (2) As in c, the Riemann Lebesgue Property (RLP) in A 1 can be defined as follows: a sequence (T n ) n N in A 1, of nonzero elements, is said to have the Riemann Lebesgue Property if for any S A 1, S, T n as n (3) Obviously, e e j e i has the Riemann Lebesgue Property Note that j e i = 1 and e j e i, e t e s = if (i, j) (s, t) A sequence like this is called an orthonormal sequence (4) The following results have similar proofs to those of the corresponding ones in [1]: (a) If S A 1 is a finite orthonormal set {T 1, T 2,, T n } or an orthonormal sequence {T n : n N} with the Riemann Lebesgue Property then S can be extended to an orthonormal basis for A 1

24 J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 (b) If M is finite-dimensional or has an orthonormal basis {T n : n N} with the Riemann Lebesgue Property then M admits a normal complement; that is, A 1 = M M p (5) A linear operator P : A 1 A 1 is said to be a Normal Projection if: (a) P 2 = P (which entails that A 1 = N (P) R (P)), (b) P is continuous, and (c) S, T = for S N (P) and T R (P) As a consequence, we have the following results which are similar to those proved in [1] with c instead of A 1 there: (i) Let M be a closed subspace of A 1 which admits a normal complement Then there exists a unique normal projection P such that N ( P) = M (ii) If P is a normal projection and {S n : n N} is an orthonormal basis of N (P), then it has the Riemann Lebesgue Property (6) Summarizing, if M be an infinite dimensional closed subspace of A 1, then, the following statements are equivalent: (a) M has a normal complement (b) M has an orthonormal basis with the Riemann Lebesgue Property (c) There exists a normal projection P such that N( P) = M Remark 5 (1) By Remark 4, the kernel of each Riesz functional admits an orthonormal basis with the Riemann Lebesgue Property (2) (A 1,, ) is not orthomodular For example, for the closed subspace V = T = α i j e j e i A 1 : α i j = V p is the null subspace In fact, suppose that S = ω i je j e i V p, then = S, T = Se i, T e i, T V In particular, for each T i j = e j e i e j+i e i V, we have that = S, T i j = Sek, T i j e k = ωi j ω i j+i, k=1 or equivalently, ω i j = ω i j+i Since S A 1, it then follows that ω i j = i, j N Therefore, V does not admit a normal complement (3) While (A 1,, ) is not orthomodular, there exist closed subspaces of A 1 which admit normal complements In the following, we shall give an example of such a closed subspace S We have already shown above that c can be embedded in A 1 and its image is S = {M a : a c } We claim that S has a normal complement In fact, for each a c, we have that M a = n=1 a n, e n e n ; in particular, M en =, e n e n = e n e n ( ); hence M a = a n e n e n ( ) n=1 Thus, S = cl e n e n : n N Note that since e n e n = 1 and since e n e n, e m e m = Men, M em = en, e m = for m n, it follows that e n e n : n N is an orthonormal basis of S

J Aguayo et al / Indagationes Mathematicae 26 (215) 191 25 25 Now, let T A 1 be given; then lim n T e n = θ and hence T, M en = T en, e n T e n Thus, S admits an orthonormal basis with the Riemann Lebesgue Property It follows that A 1 = S S p Next, we will explicitly find the subspace S p Let T = T = ω i j e j e i A 1 : ω ii =, i N ; then T is clearly a subspace of A 1 and it is closed, since if T T and (T n ) n N is a sequence in T such that T n T then, for each k N, T n e k, e k T e k, e k or, equivalently, ωkk n ω kk which implies that ω kk = On the other hand, for M a S and T T, we have that M a, T = M a e k, T e k = a k e k, T e k = a k ω kk =, k=1 k=1 and therefore T S p The other inclusion follows from the fact that = M ek, T = ω kk for k N Acknowledgments This work was partially supported by Proyecto DIUC, No 291333-1 and by a University of Manitoba start-up fund #316152-3527-2 References [1] J Aguayo, M Nova, K Shamseddine, Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field, J Math Phys 54 (2) (213) [2] B Diarra, Bounded linear operators on ultrametric Hilbert spaces, Afr Diaspora J Math 8 (2) (29) 173 181 [3] GK Kalisch, On p-adic Hilbert spaces, Ann of Math 48 (1) (1947) 18 192 [4] L Narici, E Beckenstein, A non-archimedean inner product, Contemp Math 384 (25) 187 22 [5] C Perez-Garcia, WH Schikhof, Locally Convex Spaces Over Non-Archimedean Valued Fields, Cambridge University Press, Cambridge, 21 [6] K Shamseddine, New elements of analysis on the Levi-Civita field (PhD thesis), Michigan State University, East Lansing, Michigan, USA, 1999, Also Michigan State University report MSUCL-1147 [7] K Shamseddine, M Berz, Analysis on the Levi-Civita field, a brief overview, Contemp Math 58 (21) 215 237 [8] MP Solèr, Characterization of Hilbert spaces by orthomodular spaces, Comm Algebra 23 (1) (1995) 219 243 [9] ACM van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York, 1978 k=1