Werner Timmermann
The paper contains results on the structure of Jordan maps and several kinds of triple maps on standard algebras of unbounded operators in Hilbert spaces. These results are unbounded counterparts to results on algebras of bounded operators obtained by Lu, Molnár and others ([3,4,6 -- 10]).
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R.V. Kadison defined the notion of local derivation on an algebra and proved that every continuous local derivation on a von Neumann algebra is a derivation (Kadison 1990). We provide the analogous result in the setting of Jordan triples.
Let $\mathcal{A}$ be a prime $\ast$-algebra and $Φ$ preserves triple $\ast$-Jordan derivation on $\mathcal{A}$, that is, for every $A,B \in \mathcal{A}$, $$Φ(A\diamond B \diamond C)=Φ(A)\diamond B\diamond C+A\diamond Φ(B)\diamond C+A\diamond B\diamond Φ(C)$$ where $A\diamond B = AB + BA^{\ast}$ then $Φ$ is additive. Moreover, if $Φ(αI)$ is self-adjoint for $α\in\{1,i\}$ then $Φ$ is a $\ast$-derivation.
We prove that Jordan triple elementary surjective maps on unital rings containing a nontrivial idempotent are automatically additive.
The main goal of this paper is to introduce and explore an appropriate notion of weakly Rickart JB$^*$-triples. We introduce weakly order Rickart JB$^*$-triples, and we show that a C$^*$-algebra $A$ is a weakly (order) Rickart JB$^*$-triple precisely when it is a weakly Rickart C$^*$-algebra. We also prove that the Peirce-2 subspace associated with a tripotent in a weakly order Rickart JB$^*$-triple is a Rickart JB$^*$-algebra in the sense of Ayupov and Arzikulov. By extending a classical property of Rickart C$^*$-algebras, we prove that every weakly order Rickart JB$^*$-triple is generated by…
Let $\mathcal{A}$ be a factor with dim$\mathcal{A}\geq2$. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, it is proved that a map $Φ: \mathcal {A}\rightarrow \mathcal {A}$ satisfies $Φ([A, B]_{*}\bullet C)=[Φ(A), B]_{*}\bullet C+[A, Φ(B)]_{*}\bullet C+[A, B]_{*}\bullet Φ(C)$ for all $A, B,C\in\mathcal {A}$ if and only if $Φ$ is an additive $*-$derivation.
We introduce the notion of a Jordan triple module and determine the precise conditions under which every derivation from a JB*-triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or complex JB*-triple into its dual space is automatically continuous. Among the consequences, we prove that every triple derivation from a C*-algebra A to a Banach triple A-module is continuous. In particular, every Jordan derivation from A to a Banach A-bimodule is a derivation, a result which complements a classical theorem due to B.E. Johnson and solves a prob…
We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.
Maps preserving ascent/descent of triple Jordan product
本文研究维数至少为 3 的实或复 Banach 空间上有界算子代数，完整刻画了保持算子 Jordan 三重积升标或降标的满射映射。
Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and $η$ be a non-zero complex number. A nonlinear bijective map $ϕ:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$ϕ([A,B]_{*}^η\diamond_η C)=[ϕ(A),ϕ(B)]_{*}^η\diamond_ηϕ(C)$$ for all $A,B,C\in\mathcal A.$ If $η=1,$ then $ϕ$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $η\neq 1$ and satisfies $ϕ(I)=1,$ then $ϕ$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.
This paper is an elaborated version of the material presented by the author in a three hour minicourse at "V International Course of Mathematical Analysis in Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation. Part II discusses cohomology theory of algebras and triple systems, in both finite and infinite dimensions. Althou…
Let $\mathcal {A}$ be a unital $\ast$-algebra. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, under some mild conditions on $\mathcal {A}$, it is shown that a map $Φ:\mathcal {A}\rightarrow \mathcal {A}$ satisfies $Φ([A\bullet B, C]_{*})=[Φ(A)\bullet B, C]_{*}+[A\bullet Φ(B), C]_{*}+[A\bullet B, Φ(C)]_{*}$ for all $A, B,C\in\mathcal {A}$ if and only if $Φ$ is an additive $*-$derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of t…
Jordan triple product homomorphisms on Hermitian matrices to and from dimension one
本文完整刻画了n×n复厄米矩阵与一维数域（复数、实数、非负实数）之间两个方向上的所有Jordan三重积同态映射。
A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan Banach triples, focussing on commutative JB*-triples and some Cartan factors.
Generalised triple homomorphisms and Derivations
该研究引入Jordan Banach三重系统上的广义三重同态与广义三重导子概念，证明了JB$^*$-三重上两类映射的自动连续性，推广了已有Banach代数相关结果。
We study pointwise-generalized-inverses of linear maps between C$^*$-algebras. Let $Φ$ and $Ψ$ be linear maps between complex Banach algebras $A$ and $B$. We say that $Ψ$ is a pointwise-generalized-inverse of $Φ$ if $Φ(aba)=Φ(a)Ψ(b)Φ(a),$ for every $a,b\in A$. The pair $(Φ,Ψ)$ is Jordan-triple multiplicative if $Φ$ is a pointwise-generalized-inverse of $Ψ$ and the latter is a pointwise-generalized-inverse of $Φ$. We study the basic properties of this maps in connection with Jordan homomorphism, triple homomorphisms and strongly preservers. We also determine conditions to guarantee the automati…
On the Jordan-Chevalley-Dunford decomposition of operators in type $I$ Murray-von Neumann algebras
本文研究类型 $I$ Murray-von Neumann 代数中算子的 Jordan-Chevalley-Dunford 分解，证明矩阵分解映射的无界性，并给出稠密定义闭算子的典范分解，表明无界算子的必要性。
We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $θ$-deformations. In particular we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around…
By considering the general properties of approximate units in differentiable algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular, by strengthening Kasparov's technical theorem, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.
We study morphisms of the generalized quantum logic of tripotents in JBW*-triples and von Neumann algebras. Especially, we establish generalization of celebrated Dye's theorem on orthoisomorphisms between von Neumann lattices to this new context. We show one-to-one correspondence between maps on tripotents preserving orthogonality, orthogonal suprema, and reflection $u\to -u$, on one side, and their extensions to maps that are real linear on sets of elements with bounded range tripotents on the other side. In a more general description we show that quantum logic morphisms on tripotent structur…
In this note, we will discuss what kind of operators between C*-algebras preserves Jordan triple products {a,b,c}= (ab*c + cb*a)/2. These include especially isometries and disjointness preserving operators.
Continuing our study of spectral triples on quantum domains, we look at unbounded invariant and covariant derivations in the quantum annulus. In particular, we investigate whether such derivations can be implemented by operators with compact parametrices, a necessary condition in the definition of a spectral triple.
We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.
We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive.
We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.
We prove that every surjective isometry between the unit spheres of two atomic JBW$^*$-triples $E$ and $B$ admits a unit extension to a surjective real linear isometry from $E$ into $B$. This result constitutes a new positive answer to Tignley's problem in the Jordan setting.
We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we …
In a first result we prove that every continuous local triple derivation on a JB$^*$-triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB$^*$-triple are continuous even if not assumed a priori to be so. In particular every local triple derivation on a C$^*$-algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalised (Jordan) derivations on a C$^*$-algebra.
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonalit…
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 this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illu…
We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles …
Let $X_1$ and $X_2$ be complex Banach spaces with dimension at least three, $\mathcal{A}_1$ and $\mathcal{A}_2$ be standard operator algebras on $X_1$ and $X_2$, respectively. For $k\geq2$, let $(i_1,...,i_m)$ be a sequence with terms chosen from $\{1,\ldots,k\}$ and assume that at least one of the terms in $(i_1,\ldots,i_m)$ appears exactly once. Define the generalized Jordan product $T_1\circ T_2\circ\cdots\circ T_k=T_{i_1} T_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2} T_{i_1}$ on elements in $\mathcal{A}_i$. This includes the usual Jordan product $A_1A_2+A_2A_1$, and the Jordan triple $A_1A_2…
We study new classes of linear preservers between C$^*$-algebras and JB$^*$-triples. Let $E$ and $F$ be JB$^*$-triples with $\partial_{e} (E_1)$. We prove that every linear map $T:E\to F$ strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C$^*$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan $^*$-homomorphism $S: A\to B$ satisfying $T(x) = T(1) S(x)$, for every $x\in A$. We also study the connections between linea…
We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a …
A Kaplansky Theorem for JB*-triples
该研究针对从复JB*-三元系或实J*B-三元系到赋范Jordan三元系的未必连续三元同态，证明其连续时具有闭值域，推广了多类代数框架下的经典Kaplansky类定理。
In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods…
We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components---a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the doma…
Parabolic noncommutative geometry
本文提出 tangled 谱三元组概念，用于研究抛物几何中的各向异性及非交换几何中的共形等价性，通过构造次椭圆谱三元组和交叉乘积谱三元组等方法实现。
In the paper "Maps preserving the spectrum of generalized Jordan product of operators", we define a generalized Jordan products on standard operator algebras $A_1, A_2$ on complex Banach spaces $X_1, X_2$, respectively. This includes the usual Jordan product $A_1 \circ A_2 = A_1 A_2 + A_2 A_1$, and the triple $\{A_1,A_2,A_3\} = A_1 A_2 A_3 + A_3 A_2 A_1$. Let a map $Φ: A_1 \to A_2$ prserving the spectra of the products $$ σ(Φ(A_1) \circ ... \circ Φ(A_k)) = σ(A_1\circ ... \circ A_k) $$ whenever any one of $A_1, ..., A_k$ has rank at most one. It is shown in this paper that if the range of $Φ$ c…
In this paper we introduce the curvature of densely defined universal connections on Hilbert $C^{*}$-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorisations of unbounded Kasparov modules. Our refined definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a $KK$-factorisation.
M-ideals in real operator algebras
本文将实 JB*-三元组中 M-理想与范数闭三元理想的对应关系推广到可能非自伴的实算子代数及 Jordan 算子代数，并刻画单侧 M-理想。
A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on a JB$^*$-algebra $A$ the following assertions are equivalent: $(a)$ $T$ is triple derivable at zero; $(b)$ $T$ is triple derivable at orthogonal elements; $(c)$ There exists a Jordan $^*$-derivation $D:A\to A^{**}$, a central element $ξ\in A^{**}_{sa},$ and an anti-symmetric element $η$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + ξ\circ a +…
We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger alge…
We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples Z-actions on Cantor sets. The C*-algebra of this dynamical system is generated by functions in C(X) and a unitary element u implementing the action. Building on ideas of Christensen and Ivan, we give a construction of a family of spectral triples on the commutative algebra C(X). There is a canonical choice of eigenvalues for the Dirac operator D which ensures that [D,u] is bounded, so that it extends to a spectral triple on the crossed product. We study the s…
This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $θ$-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a r…
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