Linear maps on operator algebras that preserve elements annihilated by a polynomial

In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.     

Cylindric algebras and finite polyadic algebras

In this survey paper the short history of cylindric and finitary polyadic algebras (term-definitionally equivalent to quasi-polyadic algebras) is sketched, and the two concepts are compared. Roughly speaking, finitary polyadic algebras constitute a subclass of cylindric algebras that include a transposition operator being strong enough. We discuss the following question: should the definition of cylindric algebras include a transposition operator? Results confirm that the existence of a transposition operator ensures representability (by relativised set algebras). The different variants of cylindric algebras including a transposition operator play an important role in the theory of cylindric-like algebras.     

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra V with conformal vector ω,we consider a class of vertex operator subalgebras and their conformal vectors.They are called semi-conformal vertex operator subalgebras and semiconformal vectors of(V,ω),respectively,and were used to study duality theory of vertex operator algebras via coset constructions.Using these objects attached to(V,ω),we shall understand the structure of the vertex operator algebra(V,ω).At first,we define the set Sc(V,ω)of semi-conformal vectors of V, then we prove that Sc(V,ω)is an affine algebraic variety with a partial ordering and an involution map.Corresponding to each semi-conformal vector,there is a unique maximal semi-conformal vertex operator subalgebra containing it.The properties of these subalgebras are invariants of vertex operator algebras.As an example,we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras.As an application,we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.     

环同构及完全保持乘子不动点的映射

令${\mathcal R}_1$, ${\mathcal R}_2$为两个含单位元$I$的环. 在${\mathcal R}_1$某些温和的假设条件下, 本文基于对完全保持乘子不动点映射的分类研究, 给出${\mathcal R}_1$到${\mathcal R}_2$的环同态和环同构的刻画. 然后给出在Banach代数、套代数、矩阵代数以及标准算子代数等算子代数方面的应用.     

Beyond hyperinvariance for compact operators

We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator is compact this leads to a generalization of Lomonosov's theorem.     

Operator algebras of functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Logmodularity and isometries of operator algebras

We generalize some facts about function algebras to operator algebras, using the ``noncommutative Shilov boundary' or ``-envelope' first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided.

     

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.     

C~*标准代数中效应的序列同构

主要借助于C~*标准算子代数中的有限秩算子对一般的效应元进行刻画.证明了C~*标准算子代数A的效应代数E(A)上的每个序列自同构和自同构ψ都具有形式ψ(A)=UAU~*,其中A∈E(A),U为酉算子或反酉算子.     

Simple and pseudosimple algebras with operators

In this work, algebras with operators, i.e., algebras with an additional system of unary operations acting as endomorphisms with respect to the basic operations, are considered. We find necessary conditions for the simplicity and pseudosimplicity of an arbitrary universal algebra with a fixed operator and explore how several properties of congruences of this algebra depend on the structure of the unary reduct of a given algebra. For algebras with one ternary operation, defined in the standard way and satisfying the Mal’cev identities, and one operator (i.e., unars with Mal’cev operation), necessary and sufficient conditions for their simplicity and pseudosimplicity are obtained.     

Roots in Operator and Banach Algebras

We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the ‘sign’ and ‘geometric mean’ of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.     

The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras

Löwner’s operator in Euclidean Jordan algebras, defined via the spectral decomposition of the elements of a scalar function, has been widely used in various optimization problems over Euclidean Jordan algebras. In this note, we shall show that Löwner’s operator in Euclidean Jordan algebras is Hölder continuous if and only if the underlying scalar function is Hölder continuous. Such a property will be useful in designing solution methods for symmetric cone programming and symmetric cone complementarity problem.     

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

强Ockham代数与剩余格

首先讨论了Ockham代数与剩余格的关系,引入了强Ockham代数的概念,并讨论了它的基本性质．然后,将著名的风蕴涵和风算子推广到Ockham代数上,证明了添加广义R0蕴涵和广义风算子后的Ockham代数L成为剩余格的充要条件是L为强Ockham代数．最后给出若干重要例子,以此来说明强Ockham代数的条件是独立的．     

算子代数的斜积

引入了算子代数的一种新运算"斜积",证明了在这个新定义的斜积运算下算子代数的自反性保持不变.研究发现,斜积运算对应的子空间格是拓扑意义下的格的直积关系.这个新发现的重要意义在于由此可从已知的自反子空间格生成更多更复杂的新自反格,从而得到新的自反代数.在此基础上,本文对KS-代数保持性等其他非自伴代数类的性质也作了相应研究.     

Extensions of operator algebras I

We transcribe a portion of the theory of extensions of C^∗-algebras to general operator algebras. We also include several new general facts about approximately unital ideals in operator algebras and the C^∗-algebras which they generate.