Lie multiplicative isomorphisms, standard operator algebras. For instance, the space of twotall column vectors and the space of twowide row vectors are not equal because their elementscolumn vectors and row vectorsare not equal, but we have the idea that these spaces differ only in how their elements appear. A subalgebra of alg l is called a standard subalgebra if it contains all finite rank operators in alg l. Bulletin of the london mathematical society this book is extremely clear and well. Before answering these questions, let me rst describe the target audience of this text. In each case, the clifford algebra is algebra isomorphic to a full matrix ring over r, c, or h the quaternions, or to a direct sum of two copies of such an algebra, though not in a canonical way. The theory of clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator. May 17, 2000 the aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to jordan triple isomorphisms of associative rings is automatically additive. An elementary proof of the characterization of isomorphisms of standard operator algebras article pdf available in proceedings of the american mathematical society 411.
Note that every operator of rank one can be written in this form. Moreover, for rational real number 1, 1, lie multiplicative isomorphisms are ring isomorphisms. Irving kaplansky, ring isomorphisms of banach algebras, canadian j. Recall that a standard operator algebra on a banach space is a subalgebra of the whole operator algebra b containing the identity operator and the ideal of all finite rank operators. Particularly, multiplicative isomorphisms are ring isomorphisms and the jordan multiplicative isomorphisms are jordan ring isomorphisms. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the socalled jordan product. The blending of algebra, topology, measure theory, and analysis to study operators has resulting in breathtaking advances, and this trend continues. Ring isomorphisms and complete preservers of fixed points. Ring isomorphisms and complete preservers of fixed points for. In bijective maps which leave the schatten pnorm 1. Qi and hou in 10 gave a characterization of all lie multiplicative isomorphisms between standard operator algebras. Applications to several kinds of operator algebras such as banach algebras, nest algebras, matrix algebras and standard operator algebras. Let a be a standard operator subalgebra of nest algebra which does not contain the identity operator, acting on a hilbert space of dimension greater than one.
In section 2 we deal with additive bijective mappings between standard operator algebras preserving zero products. Additive maps on standard operator algebras preserving. Isomorphisms of algebras from directed graphs 3 paths w 1 and w 2 are composable precisely when sw 2 rw 1 and, in that case, the composition w 2w 1 is just the concatenation of w 1 and w 2. Order isomorphisms and triple isomorphisms of operator. Algebra isomorphisms between standard operator algebras. Tonev um standard operator algebras bedlewo, 2009 1 18. Davidson, nest algebras, pitman research notes in mathematics series, vol. Lu linear algebra and its applications 347 2002 283291 it follows from theorem m that every multiplicative isomorphism of a standard operator algebra on a banach space of dimension at least 2 i. Ams proceedings of the american mathematical society. Pdf an elementary proof of the characterization of. In this paper we give the general form of order isomorphisms and that of triple isomorphisms of standard operator algebras. Below it is shown that distinct clifford algebras may be algebra isomorphic, as is the case of cl 2,0 r and cl 1,1 r, which are both. A certain method of investigation of derivations was suggested in 3, where it was proved that any derivation of a standard algebra of bounded operators on a normed space. On isomorphisms of standard operator algebras core.
The use of c algebras in operator theory is known as a soft technique, in contrast to the hard techniques that use deep results from analysis. In this paper, we show that if an injective map on symmetric matrices snc satisfies then for all, where f is an injective homomorphism on c, s is a complex orthogonal matrix and af is the image of a under f applied entrywise. Derivations on the algebras of bounded operators are rather wellinvestigated 1. Browse other questions tagged functionalanalysis operator algebras banach algebras abstract algebra vonneumann algebras or ask your own question. B y, not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on x. Jordan isomorphisms of rings are defined by two equations.
We refer the reader to these books for more details and for the proofs. Frederic schullers lectures on the geometric anatomy of. Irving kaplansky, algebraic and analytic aspects of operator algebras, american mathematical society, providence, r. It supposed to be a rst linear algebra course for mathematically advanced students. Jordan semitriple multiplicative maps on the symmetric. Effros mathematics department, university of pennsylvania, philadelphia, pennsylvania 19174 and e.
Linear algebradefinition and examples of isomorphisms. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. An elementary proof of the characterization of isomorphisms of standard operator algebras authors. Order isomorphisms and triple isomorphisms of operator ideals.
Linear algebra is one of the most applicable areas of mathematics. In the examples following the definition of a vector space we developed the intuition that some spaces are the same as others. Box 12, 4010 debrecen, hungary b faculty of mechanical engineering, university of. Additivity of jordan maps on standard operator algebras core. The theory of operator algebras tends to be rather technical. Remember that an algebra or a ring a is prime if xay 0 implies x 0 or y 0. As k algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
Applications to several kinds of operator algebras such as banach algebras, nest algebras, matrix algebras and standard operator algebras are presented. Still very useful, in particular on applications to representation. No book can start from scratch, and this book presupposes a level of knowl. Concerning more general operator algebras, in theorem 2. Jordan semitriple multiplicative maps on the symmetric matrices. Recall that a standard operator algebra on a banach space is a subalgebra of the whole operator algebra containing the identity operator and the ideal of all finite rank operators. From vertex operator algebras to conformal nets and back arxiv. This book appeared as lecture notes for the course \honors linear algebra.
Standard operator algebra, jordan derivation, local jordan derivation. In mathematics, a clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. This result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras. For tensor algebras associated with graphs having no. Tensor products of operator algebras sciencedirect. Semrl, elementary operators on standard operator algebras, linear and multilinear algebra 50 2002, 315319. A linear combination of vectors adds to the zero vector and so lemma 1. Isomorphisms and derivations in lie c algebras article pdf available in abstract and applied analysis 20072 august 2007 with 49 reads how we measure reads. Banach space of dimension 1, onto a ring that satisfies. Still very useful, in particular on applications to representation theory. Conference board of the mathematical sciences regional conference series in mathematics, no. Let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y, respectively, and let. In theorem 1 in 23 we described all bijective maps between the related positive semidefinite cones which preserve a schatten p norm 1 standard operator subalgebra of nest algebra which does not contain the identity operator, acting on a hilbert space of dimension greater than one.
In this paper, a characterization of jordan isomorphisms between standard subalgebras of jsubspace lattice algebras is given. An elementary proof of the characterization of isomorphisms of standard operator algebras. Mathematics, and to serve as an introduction and standard reference for the. Jordan maps on standard operator algebras springerlink. Triangular forms for operator algebras on hilbert space. Linear algebra done wrong mathematical and statistical. Box 12, 4010 debrecen, hungary b faculty of mechanical engineering. In this paper we present results showing that on standard operator algebras. Reduced operator algebras of tracepreserving quantum. Local automorphisms of operator algebras on banach spaces, proc.
Linear rank and corank preserving maps on bh and an. The operator x f is a projection if and only if fx 1. The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to jordan triple isomorphisms of associative rings is automatically additive. In each case, the clifford algebra is algebra isomorphic to a full matrix ring over r, c, or h the quaternions, or to a direct sum of two copies of such. This result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively. Additivity of lie triple isomorphisms on standard operator. Liu, additivity of jordan maps on standard jordan operator algebras, linear algebra appl. Research article characterizing lie multiplicative. Tonev the university of montana, missoula, usa bedlewo, poland, 2009 t. Advances in mathematics 25, 4 1977 tensor products of operator algebras edward g. Linear algebraisomorphisms wikibooks, open books for an. Algebra isomorphisms between standard operator algebras t.