Miguel Cabrera Garcia

New associative and nonassociative Gelfand-Naimark theorems

We prove that, ifA is a complex Banach algebra with a unit 1 and a conjugate-linear vector space involution* such that 1*=1 and‖a*a‖=‖a*‖ ‖a‖ for alla inA, and ifdim(A)≥3, thenA is a C*-algebra. The two-dimensional case is also considered and described.

On Primitive Jordan Banach Algebras

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 Qs(A) of A containing A as an ideal. Precisely, we prove that there exists a Banach space X and a one-to-one homomorphism o from Qs(A) into the Banach algebra BL(X) of all bounded linear operator on X such that o(A) acts irreducibly on X and the restriction of o to J is continuous.

Zel’Manov’s Theorem for Nondegenerately Ultraprime Jordan-Banach Algebras

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 quotients of A (denoted by Q b (A))is provided. The main result asserts that, up to bicontinuous isomorphisms, the non-degenerately ultraprime Jordan-Banach algebras are the following: the only simple exceptional 27-dimensional Jordan complex algebra, the Jordan-Banach algebras of a regular continuous non-degenerate symmetric bilinear form on a complex Banach space, and the closed Jordan subalgebras of Q b (A) contained in H(Q b (A), *) and containing H(A, *) as an ideal, where A is an ultra-*-prime associative complex Banach algebra such that H(A, *) generates A as a Banach algebra.