Jaeseong Heo

On KSGNS representations on Krein C∗-modules

Motivated by the notion of P-functional, we introduce a notion of α-completely positive map between  ∗-algebras which is a Hermitian map satisfying a certain positivity condition, and then a α-completely positive map which is not completely positive is constructed. We establish the Kasparov-Stinespring-Gelfand-Naimark-Segal constructions of C∗-algebra and  ∗-algebra on Krein C∗-modules with α-completely positive maps.

Quantum stochastic processes for maps on Hilbert C*-modules

We discuss pairs (ϕ, Φ) of maps, where ϕ is a map between C*-algebras and Φ is a ϕ-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinesprings theorem for such a pair (ϕ, Φ) is established, and quantum stochastic processes constructed from pairs ({ϕt}, {Φt}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a ϕ-quantum dynamical semigroup Φ = {Φt} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup ϕ = {ϕt}, and that J is covariant if the ϕ-quantum dynamical semigroup Φ is covariant.

Reproducing kernel Hilbert C*-modules and kernels associated with cocycles

In this paper we discuss reproducing kernels whose ranges are contained in a C∗-algebra or a Hilbert C∗-module. Using the construction of a reproducing Hilbert C∗-module associated with a reproducing kernel, we show how such a reproducing kernel can naturally be expressed in terms of operators on a Hilbert C∗-module using representations on Hilbert C∗-modules. We prove that for each positive definite G-kernel associated with a cocycle there is a representation associated with an operator-valued cocycle on the corresponding Hilbert C∗-module. Finally, some examples will be considered.

Radon-Nikodým type theorem for α-completely positive maps

We introduce a new notion of α-completely positive map on a C∗-algebra as a generalization of the notion of completely positive map. Then we study a theorem of the Radon–Nikodým type that there is a one-to-one correspondence between α-completely positive maps and positive operators and, as an application of the Radon–Nikodým type theorem, we give a characterization of pure α-completely positive maps. Finally, we study a covariant version of the Stinespring’s theorem for a covariant α-completely positive map (see Theorem 4.3).

A Radon–Nikodým theorem for completely positive invariant multilinear maps and its applications

Noncommutative Radon–Nikodým theorems have attracted a great deal of attention in the theory of operator algebras. There has been considerable work on non-commutative Radon–Nikodým theorems not only for C *-algebras but also for algebras of unbounded operators [ 3 , 5 , 8 , 9 , 12 , 13 ]. In this paper, we will develop a Radon–Nikodým type theorem for completely bounded and completely positive invariant multilinear maps. The concept of matricial order has turned out to be very important to understand the infinite-dimensional non-commutative structure of operator algebras. As the natural ordering attached to this structure, completely positive maps and completely bounded maps have been studied extensively. Results concerning completely bounded maps have many applications: cohomology of operator algebras, multipliers on group algebras, dilation theory, similarity theory, free product representations, and abstract characterizations of operator algebras.

α-COMPLETELY POSITIVE MAPS ON LOCALLY C * -ALGEBRAS, KREIN MODULES AND RADON-NIKODÝM THEOREM

In this paper, we study -completely positive maps between locally -algebras. As a generalization of a completely positive map, an -completely positive map produces a Krein space with indefinite metric, which is useful for the study of massless or gauge fields. We construct a KSGNS type representation associated to an -completely positive map of a locally -algebra on a Krein locally -module. Using this construction, we establish the Radon-Nikodm type theorem for -completely positive maps on locally -algebras. As an application, we study an extremal problem in the partially ordered cone of -completely positive maps on a locally -algebra.

RECONSTRUCTION THEOREM FOR STATIONARY MONOTONE QUANTUM MARKOV PROCESSES

Based on the Hilbert C � -module structure we study the re- construction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum sto- chastic monotone process constructed from a covariant quantum dynam- ical semigroup is again covariant in the strong sense.

Stationary stochastic processes in a group system

In this paper, we discuss positive definite functions of locally compact groups into the predual of von Neumann algebras or C*-algebras. For a predual-valued or a C*-algebra valued weakly continuous covariant positive definite function, the corresponding covariant representation in a covariant group system is constructed. We consider some generalization of the time domain of stochastic processes to a locally compact group, and we use covariant representations in a group system to obtain invariant stationary stochastic processes induced from predual-valued covariant positive definite functions and covariant stationary stochastic processes induced from C*-algebra valued covariant positive definite functions.

ON OUTER AUTOMORPHISM GROUPS OF FREE PRODUCT FACTORS

In this paper, we answer the Dixmiers question for type II1-factors with property T in the negative, that is, if G is a discrete i.c.c group with property T of Kazhdan, L(G) is not isomorphic to

Representations of invariant multilinear maps on HilbertC*-modules

{\mathcal N} \otimes L ( {\mathbb F}_2 )