Totally multiplicatively prime algebras

Cabrera, M.; Mohammed, Amir A.

doi:10.1017/s0308210500002055

Cited by 7 publications

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References 4 publications

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“…So, for example, for the symmetric algebra, we will prove that: if A is a totally prime algebra, then Q respectively. An analogous result is also obtained for totally multiplicatively prime algebras, which were recently introduced by the authors in [3].…”

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Algebras of quotients with bounded evaluation of a normed semiprime algebra

Cabrera

Mohammed

2003

Studia Math.

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Abstract. We deal with the algebras consisting of the quotients that produce bounded evaluation on suitable ideals of the multiplication algebra of a normed semiprime algebra A. These algebras of quotients, which contain A, are subalgebras of the bounded algebras of quotients of A, and they have an algebra seminorm for which the relevant inclusions are continuous. We compute these algebras of quotients for some norm ideals on a Hilbert space H: 1) the algebras of quotients with bounded evaluation of the ideal of all compact operators on H are equal to the Banach algebra of all bounded linear operators on H, 2) the algebras of quotients with bounded evaluation of the Schatten p-ideal on H (for 1 ≤ p < ∞) are equal to the Schatten p-ideal on H. We also prove that the algebras of quotients with bounded evaluation on the class of totally prime algebras have an analytic behavior similar to the one known for the bounded algebras of quotients on the class of ultraprime algebras.Introduction and preliminaries. Throughout this paper all algebras considered are associative over the field K equal to R or C.The notion of rings of quotients (in which two-sided ideals are used) was introduced by W. S. Martindale for prime rings in [5] and extended to semiprime rings by S. A. Amitsur in [1]. It is usual to define these rings of quotients using partially defined centralizers on essential ideals. However, for our purposes we prefer to give a somewhat more abstract presentation (see for example [9] or [2]). Given a semiprime algebra A, the right algebra of quotients of A, denoted here by Q

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“…Finally, we examine the symmetric algebra of quotients with bounded evaluation of totally multiplicatively prime algebras, which were recently introduced by the authors in [3]. These algebras are totally prime and may not be ultraprime.…”

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Algebras of quotients with bounded evaluation of a normed semiprime algebra

Cabrera

Mohammed

2003

Studia Math.

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Multiplication Algebras: Algebraic and Analytic Aspects

García

Palacios

2020

Associative and Non-Associative Algebras and Applications

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Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

Palacios

García

2023

European Journal of Mathematics

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We provide a proof of Jacobson’s theorem on derivations of primitive rings with nonzero socle. Both Jacobson’s theorem and its formulation (in terms of the so-called differential operators on left vector spaces over a division ring) underlie our paper. We apply Jacobson’s theorem to describe derivations of standard operator rings on real, complex, or quaternionic left normed spaces. Indeed, when the space is infinite-dimensional, every derivation of such a standard operator ring is of the form $$A\rightarrow AB-BA$$ A → A B - B A for some continuous linear operator B on the space. Our quaternionic approach allows us to generalize Rickart’s theorem on representation of primitive complete normed associative complex algebras with nonzero socle to the case of primitive real or complex associative normed Q-algebras with nonzero socle. We prove that additive derivations of the Jordan algebra of a continuous nondegenerate symmetric bilinear form on any infinite-dimensional real or complex Banach space are in a one-to-one natural correspondence with those continuous linear operators on the space which are skew-adjoint relative to the form. Finally we prove that additive derivations of a real or complex (possibly non-associative) $$H^*$$ H ∗ -algebra with no nonzero finite-dimensional direct summand are linear and continuous.

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Totally multiplicatively prime algebras

Cited by 7 publications

References 4 publications

Algebras of quotients with bounded evaluation of a normed semiprime algebra

Algebras of quotients with bounded evaluation of a normed semiprime algebra

Multiplication Algebras: Algebraic and Analytic Aspects

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

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