Angel Rodriguez Palacios

University of Granada | UGR · Department of Mathematical Analysis

In this paper, we prove that the multiplication algebra of a nondegenerate non-commutative Jordan algebra is semiprime as a consequence of the multiplicative primeness of strongly prime non-commutative Jordan algebras, obtained previously by the two first named authors. For that we prove first the coincidence of the McCrimmon radicals for a non-com...

Let X be a real or complex Banach space, let \({\mathcal {A}}\) be a standard operator algebra on X, let n be a positive integer, and let \(D:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\) be a linear mapping such that the equality \(D(A^{2n})=D(A^n)A^n+A^nD(A^n)\) holds for every \(A\in {\mathcal {A}}\). We prove that D can be written in a unique...

Let A be a complete normed complex algebra, let π:A→A be a nonzero contractive linear projection, and consider π(A) as a complete normed complex algebra under the product (x,y)→π(xy). We prove the following results. - If A is unital and associative (respectively, alternative) and satisfies the von Neumann inequality, and if π satisfies the weak con...

This chapter deals with the different approaches to non‐associative models trying to generalize C*‐algebras and discusses how these notions are related. An attractive approach to the non‐associative generalizations of C*‐algebras consists of removing associativity in the abstract characterizations of unital C*‐algebras given either by the Gelfand–N...

We prove that, if A is a (possibly non-unital) non-commutative JB⁎-algebra, and if π:A→A is a positive contractive linear projection, then π(A), endowed with the product (x,y)→π(xy), becomes naturally a non-commutative JB⁎-algebra, and moreover the equality($)π(π(a)•π(b))=π(a•π(b)) holds for all a,b∈A. The appropriate variant of this result, with ‘...

Applications of multiplication algebras to the algebraic and analytic strengthenings of primeness and semiprimeness of (possibly non-associative) algebras are fully surveyed, and complete normed complex algebras whose closed multiplication algebras satisfy the von Neumann inequality are studied in detail.

By deeply refining previous results of Dineen [8], Barton and Timoney [2] proved that the bidual of any ‐triple J becomes a ‐triple under a triple product which extends that of J and is separately ‐continuous. In this note we give a simpler new proof of the above theorem.

Transitivity, almost transitivity, and convex-transitivity are classical notions in the isometric theory of Banach spaces, all of them related to Mazur's problem asking whether transitive separable Banach spaces are Hilbert spaces [24,33]. In this paper we study ‘near’ versions of the above notions. Roughly speaking, ‘nearness’ means that contracti...

Cambridge Core - Algebra - Non-Associative Normed Algebras - by Miguel Cabrera García

In [1], Arazy proves that a complete normed unital associative complex algebra A satisfies the von Neumann inequality if and only if the holomorphic vector field a → 1 – a2 is complete on the open unit ball ΔA of A. We define Arazy algebras as those complete normed unital (possibly non-associative) complex algebras A such that the holomorphic vecto...

We prove that normed unital complex (possibly non-associative) algebras with no non-zero left topological divisor of zero are isomorphic to the field \(\mathbb {C} \) of complex numbers. We also show the existence of a complete normed unital infinite-dimensional complex algebra with no non-zero two-sided topological divisor of zero.

This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-...

π-complemented algebras are defined as those algebras (not necessarily associative or unital) such that each annihilator ideal is complemented by other annihilator ideal. Let A be a semiprime algebra. We prove that A is π-complemented if, and only if, every idempotent in the extended centroid of A lies in the centroid of A. We also show the existen...

Let A be a (possibly non-associative) complex Banach algebra and let * be a conjugate-linear vector space involution on A such that ‖ a*a‖=‖ a*‖ ‖ a ‖ for every a∈A. We prove the following facts: If A has an approximate unit bounded by 1 and consisting of *-invariant elements, and if dim (A)≠2, then A is alternative, and * is an algebra involution...

We introduce in a nonassociative setting the notions of spectral radius of a bounded subset of a normed algebra, as well as that of topologically nilpotent normed algebra. We generalize and refine most known results on topologically nilpotent associative algebras to the nonassociative context, and prove some new results both in the associative and...

We prove that semisimple algebras containing some algebraic element whose centralizer is semiperfect are artinian. As a consequence, semisimple complex Banach algebras containing some element whose centralizer is algebraic are finite-dimensional. This answers affirmatively a question raised in Burgos et al. (2006) [4], and is applied to show that a...

We describe the non-associative products on a C⁎-algebra A which convert the Banach space of A into a Banach algebra having an approximate unit bounded by 1, and determine among them those which are associative. As a consequence, if such a product p satisfies p(a,b)□=p(b□,a□) and ‖p(a□,a)‖=‖a‖2, for all a,b∈A and some conjugate-linear vector space...

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X⊗ˆπY if the centralizer of X is infi...

We survey Banach space characterizations of unitary elements of C∗-algebras, JB∗-triples, and JB-algebras. In the case of the existence of a pre-dual, appropriate specializations of these characterizations are also reviewed.

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ1, all Hilbert spaces,...

It is well known that, if S is a bounded and multiplicatively closed subset of an associative normed algebra (A,‖⋅‖), then there exists an equivalent algebra norm |||⋅||| on A such that |||s|||⩽1 for every s∈S. Although associativity is not an essential requirement in this result, it is easy to find examples of nonassociative normed algebras A wher...

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity,...

Almost transitive superreflexive Banach spaces have been considered in [C. Finet, Uniform convexity properties of norms on superreflexive Banach spaces, Israel J. Math. 53 (1986) 81–92], where it is shown that they are uniformly convex and uniformly smooth. We characterize such spaces as those convex transitive Banach spaces satisfying conditions m...

We prove that, if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the closed unit ball of X has diameter equal to 2. This result, together with a suitable refinement also proven in the paper, contains (and improves in some cases) previously known facts for C *-algebras, JB *-triples,...

We study absolute valued algebras with involution, as defined in Urbanik (19619. Urbanik , K. ( 1961 ). Absolute valued algebras with an involution . Fundamenta Math. 49 : 247 – 258 . View all references). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x 2, x) = 0, where (·, ·, ·) means associator. We sh...

We generalize the theory of associative unitary normed algebras to the setting of noncommutative Jordan algebras. Special attention is devoted to the case of alternative algebras.

We prove that, in Urbanik's (Urbanik 196110. Urbanik , K. ( 1961 ). Absolute valued algebras with an involution . Fundamenta Math. 49 : 247 – 258 . View all references) definition of absolute valued algebras with involution, the axiom ‖ a*‖ = ‖ a‖ (the unique one which relates the absolute value ‖·‖ and the involution *) is redundant.

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:(1)Every member of R has the Daugavet property.(2)It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product X⊗ˆϵY lie in R.(...

We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Mo...

We prove that, if a JB*-algebra contains a non self-adjoint idempotent, then it also contains a nonzero self-adjoint idempotent. This is achieved through an “almost description” of C*- and JB*-algebras generated by a non self-adjoint idempotent.

Let A be a C∗-algebra generated by a nonself-adjoint idempotent e, and put K:=sp(e∗e)∖{0}. It is known that K is a compact subset of [1,∞[ whose maximum element is greater than 1, and that, in general, no more can be said about K. We prove that, if 1 does not belong to K, then A is ∗-isomorphic to the C∗-algebra C(K,M2(C)) of all continuous functio...

The centroid of an algebra $A$ is the largest ring over which $A$ can be regarded as an algebra. In case $A$ is a $C^\ast$-algebra, the centroid of $A$ also has a natural structure of $C^\ast$-algebra and, for $f$ in the centroid of $A$ with $0\leq f\leq 1$, the $f$-mutation of $A$ (denoted $A^{(f)}$) with the same norm as $A$ is a (complete) norme...

We study unitary Banach algebras, as defined by M. L. Hansen and R. V. Kadison in 1996, as well as some related concepts like maximal or uniquely maximal Banach algebras. We show that a norm-unital Banach algebra is uniquely maximal if and only if it is unitary and has minimality of the equivalent norm. We prove that every unitary semisimple commut...

We prove that, if A is an absolute-valued ∗-algebra in the sense of [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247–258], then the normed space of A becomes a trigonometric algebra (in the meaning of [P.A. Terekhin, Trigonometric algebras, J. Math. Sci. (New York) 95 (1999) 2156–2160]) under the product ∧ defined...

Let X be a Banach space. For a norm-one element u in X we put σ( X , u ):=sup{|| ψ -Π( ψ )||: ψ ∈ D ( X **, u )}, where D ( X **, ·) denotes the duality mapping of X **, and Π : X ***→ X * stands for the Dixmier projection. The element u is said to be a big point of X if the closed convex hull of the orbit of u under the group of all surjective iso...

Absolute-valuable Banach spaces are introduced as those Banach spaces which underlie complete absolute-valued algebras. Examples and counterexamples are given. It is proved that every Banach space can be isometrically enlarged to an absolute-valuable Banach space, which has the same density character as the given Banach space, and whose dual space...

We prove that, for a compact metric space X not reduced to a point, the existence of a bilinear mapping ⋄:C(X)×C(X)→C(X) satisfying ∥f⋄g∥=∥f∥∥g∥ for all f,g∈C(X) is equivalent to the uncountability of X. This is derived from a bilinear version of W. Holsztyński’s theorem [Stud. Math. 26, 133–136 (1966; Zbl 0156.36903)] on isometries of C(X)-spaces,...

We prove that, given a real JB *-triple X, there exists a nonempty relatively weakly open subset of the closed unit ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB *-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB *-trip...

In a Banach algebra an invertible element which has norm one and whose inverse has norm one is called unitary. The algebra is unitary if the closed convex hull of the unitary elements is the closed unit ball. The main examples are the C*-algebras and the l 1 group algebra of a group. In this paper, different characterizations of unitary algebras ar...

We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X0 and nonzero complex spaces X1,..., Xn in such a way that the equality ∥ x0 + eiq1ρ x1 + ⋯ + e iqnρ xn ∥ = ∥ x0 + ∥ + x n ∥ holds for suitable positive integers q1,..., qn, and every ρ ∈ ℝ and every Xj ∈ Xj (j = 0, 1,..., n)...

Let A be an infinite-dimensional C*-algebra. It is proved that every nonempty relatively weakly open subset of the closed unit ball BA of A has diameter equal to 2. This implies that BA is not dentable, and that there is not any point of continuity for the identity mapping (B A, weak.)→(BA,norm).

We prove that the norm of a J B *‐triple X is strongly subdifferentiable at a norm‐one element x if and only if 1 is an isolated point of the triple spectrum of x , and this is the case if and only if the support of x in the bidual of X lies in X . Moreover we show that the J B *‐triples whose norms are strongly subdifferentiab...

The aim of this paper is to discuss the non‐associative side of a celebrated theorem of C. E. Rickart asserting the automatic continuity of dense range homomorphisms from complete normed associative algebras to complete normed strongly semisimple associative algebras. As the main result, we prove that associativity can be removed in Rickart's theor...

The aim of this paper is to discuss the non‐associative side of a celebrated theorem of C. E. Rickart asserting the automatic continuity of dense range homomorphisms from complete normed associative algebras to complete normed strongly semisimple associative algebras. As the main result, we prove that associativity can be removed in Rickart'...

We prove that, if A and B are complete normed non-associative algebras, and if B is strongly semisimple and algebraic, then dense range homomorphisms from A to B are continuous. As a consequence, we obtain that homomorphisms from complete normed algebras into a complete normed power-associative algebraic algebra B are continuous if and only if B ha...

This paper is a natural continuation of that of Moreno Galindo and Rodríguez-Palacios (2000). The present paper contains nontrivial reformulations of the previous classification theorem for prime complex JB*-triples-, by replacing "matricially decomposed C*-algebras" (a nonfamiliar concept) with "projections in the multiplier algebra of a given C*-...

We prove that, if A is a normed *-algebra of the form B ⊗ C for some central simple finite-dimensional algebra B with involution different from ± IB and some algebra C with involution and a unit, then homomorphisms from A to normed algebras and derivations from A to normed A-bimodules are continuous whenever they are continuous on the hermitian par...

Contents 1. Introduction. 2. Basic notions and results on transitivity of the norm. Transitivity. Almost-transitivity. Transitivity versus almost-transitivity. Convex transitivity. Maximality of norm. Transitivity and isomorphic conditions. 3. Isometric one-dimensional perturbations of the identity. 4. Multiplicative characterization of Hilbert spa...

We review the main results obtained in 2 other papers concerning Grothendieck's inequalities for real and complex JB*-triples. We improve the constants involved in this inequalities. We show that for every complex (respectively, real) JB*-triples (respectively, and every bounded bilinear form U on ε x F, there exist states ΦɛD(BL(ɛ), Iɛ and Ψ∈D(BL(...

We give a detailed survey of some recent developments of non-associative C*-algebras. Moreover, we prove new results concerning multipliers and isometries of non-associative C*-algebras.

Let A be a Banach algebra, and let D : A → A*be a continuous derivation, where A*is the topological dual space of A. The paper discusses the situation when the second transpose D**:A**→ (A**)*is also a derivation in the case where A" has the first Arens product.

We prove that, if $M > 4(1+2\sqrt{3})$ and $\varepsilon > 0$, if $\mathcal{V}$ and $\mathcal{W}$ are complex JBW*-triples (with preduals $\mathcal{V}_{*}$ and $\mathcal{W}_{*}$, respectively), and if $U$ is a separately weak*-continuous bilinear form on $\mathcal{V} \times \mathcal{W}$, then there exist norm-one functionals $\varphi_{1},\varphi_{2}...

Let B be a real or complex complete normed quadratic algebra. All homomorphisms from arbitrary (possibly non associative) complete normed algebras into B are continuous if and only if B has no non-zero element with zero square.

We characterize C *-algebras as those complete normed associative complex algebras having approximate units bounded by one and whose open unit balls are bounded symmetric domains. Such a characterization follows from the more general fact, also proved in the paper, that non-commutative JB *-algebras coincide with complete normed (possibly non-ass...

Almost transitive superreflexive Banach spaces have been considered in (7) (see also (4) and (6)), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach...

We prove that a Banach space X is uniformly smooth if and only if, for every X-valued bounded function f on the unit sphere of X, the intrinsic numerical range of f is equal to the closed convex hull of the spatial numerical range of f.

We prove that if A is a prime non-commutative JB*-algebra which is neither quadratic nor commutative, then there exist a prime C*-algebra B and a real number λ with 1/2 < λ ≤ 1 such that A = B as involutive Banach spaces, and the product of A is related to that of B (denoted by o, say) by means of the equality xy = λx o y + (1 - λ)y o x.

Let E be a real Banach space. A norm-one element e in E is said to be an isometric reflection vector if there exist a maximal subspace M of E and a linear isometry F : E → E fixing the elements of M and satisfying F(e) = -e. We prove that each of the conditions (i) and (ii) below implies that E is a Hilbert space. (i) There exists a nonrare subset...

We study transitivity conditions on the norm of JB *-triples, C *-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW *-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of...

Throughout this paper, X will denote a Banach space, S=S(X) and B=B(X) will be the unit sphere and the closed unit ball of X, respectively, and [script G]=[script G](X) will stand for the group of all surjective linear isometries on X. Unless explicitly stated otherwise, all Banach spaces will be assumed to be real. Nevertheless, by passing to...

Let A be a C -algebra, and B a complex normed non-associative algebra. We prove that, if B has an approximate unit bounded by one, then, for every linear isometry F from B onto A, there exists a Jordan-isomorphism G : B ! A and a unitary element u in the multiplier algebra of A such that F(x) = uG(x) for all x in B. We also prove that, if G is an i...

The following result is well known and easy to prove (see [14, Theorem 2.2.6]). Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A↪BL(X) i...

We show that, if A is a finite-dimensional ∗-simple associative algebra with involution (over the field of real or complex numbers) whose hermitian part H(A, ∗) is of degree ⩾ 3 over its center, if B is a unital algebra with involution over , and if ‖ · ‖ is an algebra norm on H(A ⊗ B, ∗), then there exists an algebra norm on A ⊗ B whose restrictio...

We prove that ifAis a non-associative algebra over the field of real numbers, if there is a norm ‖·‖ onAsatisfying ‖xy‖=‖x‖‖y‖ for allx,yinA, and if every one-generated subalgebra ofAis finite-dimensional, thenAis finite-dimensional.

We prove that, if A is a real C * -algebra having a predual A * , then A * is the unique predual of A and the product of A is σ(A,A * )-continuous.

We prove that there exists a real or complex central simple associative algebra M with minimal one-sided ideals such that, for every non-Jordan associative polynomial p, a Jordan-algebra norm can be given on M in such a way that the action of p on M becomes discontinuous.

For a Banach space X, we show how the existence ob a norín-one element u in X and a norm-one continuous bilinear mapping f : X xX —.~ X satisfying f(xu) = f(u,x) = x for aH x luX, togetber with sorne more intrinsic conditions, can be utiliza! to characterize A? as a member of some relevant subclass of the class ob Banach spaces. O Intraduction Some...

We introduce real JB*-triples as real forms of (complex) JB*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB*-triples is an isometry if and only if it commutes with the cube mappinga→a 3={aaa}. This genera...

We provide an almost purely algebraic proof of Kaplansky’s refinement of the Gelfand-Mazur theorem asserting that the reals, complex, and quaternions are the only associative normed real algebras with no nonzero topological divisors of zero.

We provide an almost purely algebraic proof of Kaplansky’s refinement of the Gelfand-Mazur theorem asserting that the reals, complex, and quaternions are the only associative normed real algebras with no nonzero topological divisors of zero.

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint oper...

We introduce normed Jordan Q-algebras, namely, normed Jordan algebras in which the set of quasi-invertible elements is open, and we prove that a normed Jordan algebra is a Q-algebra if and only if it is a full subalgebra of its completion. Homomorphisms from normed Jordan Q-algebras onto semisimple Jordan-Banach algebras with minimality of norm top...

We introduce normed Jordan Q -algebras, namely, normed Jordan algebras in which the set of quasi-invertible elements is open, and we prove that a normed Jordan algebra is a Q -algebra if and only if it is a full subalgebra of its completion. Homomorphisms from normed Jordan Q -algebras onto semisimple Jordan-Banach algebras with minimality of norm...

We give a description of primitive Jordan Banach algebras J for which there exists an associative primitive algebra A such that J is a Jordan subalgebra of the two-sided Martindale ring of fractions Q S (A) of A containing A as an ideal. Precisely, we prove that there exists a Banach space X and a one-to-one homomorphism ϕ from Q S (A) into the Ban...

Following [2], we will say that a (nonassociative) algebra A is algebraic if, for every x in A, the subalgebra A(x) of A generated by x is finite-dimensional. If in fact dim(A(x)) ≤ m for all x in A and some natural number m only depending on A, then the algebraic algebra A is called of bounded degree, and the smallest such a number m is called the...

This note is a review of the paper of the authors [8]. In this paper, nondegenerately ultraprime normed Jordan algebras are introduced and studied under the light of Zel’manov’s prime theorem. With this aim, ultra-*-prime normed associative algebras are introduced and, for such an algebra A, a normed version of the symmetric Martindale ring of quot...