Banach space of dimension 1, onto a ring that satisfies. Seminar on c algebras and applications to physics, 1974. The theory of clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Box 12, 4010 debrecen, hungary b faculty of mechanical engineering.
Reduced operator algebras of tracepreserving quantum. 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. 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. Local automorphisms of operator algebras on banach spaces, proc. Moreover, for rational real number 1, 1, lie multiplicative isomorphisms are ring isomorphisms. 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. On isomorphisms of standard operator algebras core. 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. This book appeared as lecture notes for the course \honors linear algebra. In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finitedimensional real and complex clifford algebras for a nondegenerate quadratic form have been completely classified. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines.
Box 12, 4010 debrecen, hungary b faculty of mechanical engineering, university of. This result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras. 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. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the socalled jordan product. In this paper we present results showing that on standard operator algebras over spaces with dimension at least 2, the bijective solutions of that second equation are automatically additive. Jordan semitriple multiplicative maps on the symmetric.
Let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y, respectively, and let. Still very useful, in particular on applications to representation theory. As k algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. A linear combination of vectors adds to the zero vector and so lemma 1. Applications to several kinds of operator algebras such as banach algebras, nest algebras, matrix algebras and standard operator algebras. Additive maps on standard operator algebras preserving. Linear rank and corank preserving maps on bh and an. No book can start from scratch, and this book presupposes a level of knowl. From vertex operator algebras to conformal nets and back arxiv. In this paper we present results showing that on standard operator algebras. 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.
Irving kaplansky, algebraic and analytic aspects of operator algebras, american mathematical society, providence, r. Triangular forms for operator algebras on hilbert space. Qi and hou in 10 gave a characterization of all lie multiplicative isomorphisms between standard operator algebras. Still very useful, in particular on applications to representation. Effros mathematics department, university of pennsylvania, philadelphia, pennsylvania 19174 and e. Ams proceedings of the american mathematical society. Algebra isomorphisms between standard operator algebras.
Derivations on the algebras of bounded operators are rather wellinvestigated 1. The operator x f is a projection if and only if fx 1. Linear algebraisomorphisms wikibooks, open books for an. Liu, additivity of jordan maps on standard jordan operator algebras, linear algebra appl. Tonev um standard operator algebras bedlewo, 2009 1 18. Frederic schullers lectures on the geometric anatomy of. Before answering these questions, let me rst describe the target audience of this text. 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. Concerning more general operator algebras, in theorem 2. 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. Lie multiplicative isomorphisms, standard operator algebras. Algebra isomorphisms between standard operator algebras article pdf available in studia mathematica 1912.
An elementary proof of the characterization of isomorphisms of standard operator algebras authors. Bulletin of the london mathematical society this book is extremely clear and well. An elementary proof of the characterization of isomorphisms of standard operator algebras article pdf available in proceedings of the american mathematical society 411. For tensor algebras associated with graphs having no. 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. 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. Particularly, multiplicative isomorphisms are ring isomorphisms and the jordan multiplicative isomorphisms are jordan ring isomorphisms. Order isomorphisms and triple isomorphisms of operator. An elementary proof of the characterization of isomorphisms of standard operator algebras. Namely, those algebras are socalled standard operator algebras over a hilbert space, i. On axiomatic approaches to intertwining operator algebras. Additivity of lie triple isomorphisms on standard operator. Applications to several kinds of operator algebras such as banach algebras, nest algebras, matrix algebras and standard operator algebras are presented.
Conference board of the mathematical sciences regional conference series in mathematics, no. Mathematics, and to serve as an introduction and standard reference for the. In bijective maps which leave the schatten pnorm 1. Standard operator algebra, jordan derivation, local jordan derivation. 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. 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. Pdf algebra isomorphisms between standard operator algebras. Browse other questions tagged functionalanalysis operator algebras banach algebras abstract algebra vonneumann algebras or ask your own question. Jordan semitriple multiplicative maps on the symmetric matrices. 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. Note that every operator of rank one can be written in this form. Advances in mathematics 25, 4 1977 tensor products of operator algebras edward g.
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 ideals. A subalgebra of alg l is called a standard subalgebra if it contains all finite rank operators in alg l. Jordan maps on standard operator algebras springerlink. Jordan isomorphisms of rings are defined by two equations.
Research article characterizing lie multiplicative. 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 this paper we give the general form of order isomorphisms and that of triple isomorphisms of standard operator algebras. 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. 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. Linear algebra is one of the most applicable areas of mathematics. Additivity of jordan maps on standard operator algebras core. In this paper, a characterization of jordan isomorphisms between standard subalgebras of jsubspace lattice algebras is given.
Algebra isomorphisms between standard operator algebras t. It supposed to be a rst linear algebra course for mathematically advanced students. Irving kaplansky, ring isomorphisms of banach algebras, canadian j. B y, not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on x. This result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively. In the examples following the definition of a vector space we developed the intuition that some spaces are the same as others. Linear algebradefinition and examples of isomorphisms. 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. 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. Ring isomorphisms and complete preservers of fixed points for. In section 2 we deal with additive bijective mappings between standard operator algebras preserving zero products.
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