• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.423-430
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• 2002

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{}i$ = $Y_{i}$ for i/ = 1,2,…, n. In this article, we obtained the following : Let X ＝ ($x_{i\sigma(i)}$ and Y ＝ ($y_{ij}$ be operators in B(H) such that $X_{i\sigma(i)}\neq\;0$ for all i. Then the following statements are equivalent. (1) There exists an operator A in Alg L such that AX = Y, every E in L reduces A and A is a self-adjoint operator. (2) sup ${\frac{\parallel{\sum^n}_{i=1}E_iYf_i\parallel}{\parallel{\sum^n}_{i=1}E_iXf_i\parallel}n\;\epsilon\;N,E_i\;\epsilon\;L and f_i\;\epsilon\;H}$ < $\infty$ and $x_{i,\sigma(i)}y_{i,\sigma(i)}$ is real for all i = 1,2, ....

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.699-705
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• 1999

Given a probability space and a subsigma algebra A, each measure equivalent to the probability measure generates a different conditional expectation operator. We characterize those which act boundedly on the original $L^2$ space, and show there are sufficiently many such conditional expectations to generate the commutant of $L^{\infty}$ (A).

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.167-170
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• 1988

Versions of Tannaka duality in operator algebraic context have been obtained in [6], [8] etc. Suppose .sigma.is an automorphism of a von Neumann algebra M, on which there is an action .alpha. of a compact group G such that .sigma. vertical bar $M^{\alpha}$=id, where $M^{\tau}$is the fixed point algebra under the action .alpha.. Then it is shown that if there is an action .tau. of a group H which commutes with .alpha., and which is ergodic in the sense that the fixed point algebra $M^{\tau}$ is trivial, then there exists g.mem.G such that .sigma.=.alpha.(g). Recently Evans and Kishimoto ([4]) showed the versions of Tannaka duality in $C^{*}$-settings under some conditions.s.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.391-397
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• 2017

In this note we extend our previous result about the structure of the dual of a crossed product $C^*$-algebra $A{\rtimes}_{\sigma}G$, when G is a finite group. We consider the space $\tilde{\Gamma}$ which consists of pairs of irreducible representations of A and irreducible projective representations of subgroups of G. Our goal is to endow $\tilde{\Gamma}$ with a topology so that the orbit space e $G{\backslash}{\tilde{\Gamma}}$ is homeomorphic to the dual of $A{\rtimes}_{\sigma}G$. In particular, we will show that if $\widehat{A}$ is Hausdorff then $G{\backslash}{\tilde{\Gamma}}$ is homeomorphic to $\widehat{A{\rtimes}_{\sigma}G}$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1209-1219
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• 2021

Let G be a topological group with a locally compact and Hausdorff topology. Let ω be a diagonally bounded weight on G. In this paper, (σ, σ)-derivation and (σ, 𝜏)-weak amenability of the Beurling algebra L¹_ω(G) are studied, where σ, 𝜏 are isometric automorphisms of L¹_ω(G). We prove that every continuous (σ, σ)-derivation from L¹_ω(G) into measure algebra M_ω(G) is (σ, σ)-inner and the Beurling algebra L¹_ω(G) is (σ, 𝜏)-weakly amenable.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.207-213
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• 1995

Let $(\Omega, F, P)$ be a probability space with F a $\sigma$-algebra of subsets of the measure space $\Omega$ and P a probability measure on $\Omega$. Suppose $a > 0$ and let $(F_t)_{t \in [0,a]}$ be an increasing family of sub-$\sigma$-algebras of F. If $r > 0$, let $J = [-r,0]$ and $C(J, R^n)$ the Banach space of all continuous paths $\gamma : J \to R^n$ with the sup-norm $\Vert \gamma \Vert = sup_{s \in J}$\mid$\gamma(s)$\mid$$ where $$\mid$\cdot$\mid$$ denotes the Euclidean norm on $R^n$. Let E,F be separable real Banach spaces and L(E,F) be the Banach space of all continuous linear maps $T : E \to F$.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.731-737
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• 1994

Let $(\Omega, F, P)$ be a probability space with F a $\sigma$-algebra of subsets of the measure space $\Omega$ and P a probability measures on $\Omega$. Suppose $a > 0$ and let $(F_t)_{t \in [0,a]}$ be an increasing family of sub-$\sigma$- algebras of F. If $r > 0$, let $J = [-r, 0]$ and $C(J, R^n)$ the Banach space of all continuous paths $\gamma : J \to R^n$ with the sup-norm $\Vert \gamma \Vert_C = sup_{s \in J} $\mid$\gamma(x)$\mid$$ where $$\mid$\cdot$\mid$$ denotes the Euclidean norm on $R^n$. Let E and F be separable real Banach spaces and L(E,F) be the Banach space of all continuous linear maps $T : E \to F$ with the norm $\Vert T \Vert = sup {$\mid$T(x)$\mid$_F : x \in E, $\mid$x$\mid$_E \leq 1}$.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.105-121
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• 2020

The main result of this article is to factorize the first (σ, τ)-cohomology group of triangular Banach algebra 𝓣 of order three with coefficients in 𝓣 -bimodule 𝓧 to the first (σ, τ)-cohomology groups of Banach algbras 𝓐, 𝓑 and 𝓒, where σ, τ are continuous homomorphisms on 𝓣. As a direct consequence, we find necessary and sufficient conditions for 𝓣 to be (σ, τ)-weakly amenable.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.125-134
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• 2001

Independent $\sigma$-algebras Α and Β on X, L$^2$(X, Α V Β), L$^2$(X x X, Α x Β), and the Hilbert space tensor product L$^2$(X,Α), (※Equations, See Full-text) L$^2$(X,Β), are isomorphic. In this note, we show that various Hilbert C(sup)*-algebra tensor products provide the analogous roles when independence is weakened to conditional independence.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.37-43
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• 2014

In this paper, we show that for any ${\sigma}$-complete Boolean subalgebra $\mathcal{M}$ of $\mathcal{R}(X)$ containing $Z(X)^{\sharp}$, the Stone-space $S(\mathcal{M})$ of $\mathcal{M}$ is a basically diconnected cover of ${\beta}X$ and that the subspace {${\alpha}{\mid}{\alpha}$ is a fixed $\mathcal{M}$-ultrafilter} of the Stone-space $S(\mathcal{M})$ is the the minimal basically disconnected cover of X if and only if it is a basically disconnected space and $\mathcal{M}{\subseteq}\{\Lambda_X(A){\mid}A{\in}Z({\Lambda}X)^{\sharp}\}$.