Nonlinear maps preserving the mixed product [A ● B,C]* on von Neumann algebras

Let A and B be two von Neumann algebras. For A,B ? A, define by [A,B]* = AB-BA* and A ? B = AB + BA* the new products of A and B. Suppose that a bijective map ? : A ? B satisfies ?([A ? B,C]*) = [?(A)? ?(B),?(C)]* for all A,B,C ? A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map ?(I)? is a sum of a linear *-isomorphism and a conjugate linear +-isomorphism, where ?(I) is a self-adjoint central element in B with ?(I)2 = I. If A and B are two factor von Neumann algebras, then ? is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

Frobenius W-algebras and traces of Frobenius Heisenberg categories

To each symmetric graded Frobenius superalgebra we associate a W-algebra. We then define a linear isomorphism between the trace of the Frobenius Heisenberg category and a central reduction of this W-algebra. We conjecture that this is an isomorphism of graded superalgebras.

Maps preserving triple product A∗B + BA∗ on ∗-algebras

Let [Formula: see text] and [Formula: see text] be two prime ∗-algebras. Let [Formula: see text] be a bijective and satisfies [Formula: see text] for all [Formula: see text] where [Formula: see text]. Then, [Formula: see text] is additive. Moreover, if [Formula: see text] is idempotent then we show that [Formula: see text] is [Formula: see text]-linear ∗-isomorphism.

Constructions with regular degenerate quadratic forms

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG = 1 and the Cartan k-subgroups of G are tori. It begins with an overview of the lemma and propositions for regular degenerate quadratic forms, coupled with two examples. It then considers the conformal isometry between quadratic spaces over a field, which is a linear isomorphism that respects the quadratic forms up to a nonzero scaling factor. It also introduces a proposition that provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO(q) for a regular quadratic form q. Finally, it describes all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k.

On the extension of unitary group isomorphisms of unital UHF-algebras

H. Dye showed that an isomorphism between the (discrete) unitary groups in two factors not of type In is implemented by a linear (or a conjugate linear) *-isomorphism of the factors. If φ is an isomorphism between the unitary groups of two unital C*-algebras, it induces a bijective map θφ between the sets of projections. For certain UHF-algebras, we construct an automorphism φ of their unitary group, such that θφ does not preserve the orthogonality of projections. For a large class of unital finite C*-algebras, we show that θφ is always an orthoisomorphism. If φ is a continuous automorphism of the unitary group of a UHF-algebra A, we show that φ is implemented by a linear or a conjugate linear *-automorphism of A.

Computations of Turaev-Viro-Ocneanu Invariants of 3-Manifolds from Subfactors

In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2, Z). Via this correspondence and the Dehn surgery formula, we compute Turaev-Viro-Ocneanu invariants from several subfactors for basic 3-manifolds including lens spaces and Brieskorn 3-manifolds by using Izumi's data written in terms of sectors.

REGULAR REPRESENTATIONS OF VERTEX OPERATOR ALGEBRAS

This paper is to establish a theory of regular representations for vertex operator algebras. In the paper, for a vertex operator algebra V and a V-module W, we construct, out of the dual space W*, a family of canonical weak V ⊗ V-modules [Formula: see text] with a nonzero complex number z as the parameter. We prove that for V-modules W, W1 and W2, a P(z)-intertwining map of type [Formula: see text] in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V-homomorphism from W1 ⊗ W2 into [Formula: see text]. Combining this with Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of P(z)-intertwining maps of the same type we obtain a canonical linear isomorphism fromthe space [Formula: see text] of intertwining operators of the indicated type to [Formula: see text]. Denote by RP(z)(W) the sum of all (ordinary) V ⊗ V-submodules of [Formula: see text]. Assuming that V satisfies certain suitable conditions, we obtain a canonical decomposition of RP(z)(W) into irreducible V ⊗ V-modules. In particular, we obtain a decomposition of Peter–Weyl type for RP(z)(V). Denote by ℱP(z) the functor from the category of V-modules to the category of weak V ⊗ V-modules such that ℱP(z)(W)=RP(z)(W'). We prove that for V-modules W1, W2, a P(z)-tensor product of W1 and W2 in the sense of Huang and Lepowsky exactly amounts to a universal from W1 ⊗ W2 to the functor ℱP(z). This implies that the functor ℱP(z) is essentially a right adjoint of the Huang–Lepowsky's P(z)-tensor product functor. It is also proved that RP(z)(W) for [Formula: see text] are canonically isomorphic V ⊗ V-modules.

A Decision Algorithm for Linear Isomorphism of Types with Complexity Cn(log2(n))

It is known that ordinary isomorphisms (associativity and commutativity<br />of "times", isomorphisms for "times" unit and currying)<br />provide a complete axiomatisation for linear isomorphism of types.<br />One of the reasons to consider linear isomorphism of types instead of<br />ordinary isomorphism was that better complexity could be expected.<br />Meanwhile, no upper bounds reasonably close to linear were obtained.<br />We describe an algorithm deciding if two types are linearly isomorphic<br />with complexity Cn(log2(n)).