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

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.1
• /
• pp.113-119
• /
• 2011

Let T be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. An operator T is called class A operator if ${\left|{T^2}\right|}{\geq}{\left|{T^2}\right|}$ and is called class A(k) operator if $({T^*\left|T\right|^{2k}T})^{\frac{1}{k+1}}{\geq}{\left|T\right|}^2$. In this paper, we show that ${\sigma}$ is continuous when restricted to the set of class A operators and consider the tensor products of class A(k) operators.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.997-1000
• /
• 2010

We prove that the reduced free product of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras is not the minimal tensor product of reduced free products of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras. It is shown that the reduced group $C^*$-algebra associated with a group having the property T of Kazhdan is not isomorphic to a reduced free product of abelian $C^*$-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian $C^*$-algebras is not isomorphic to the tensor product of a nuclear $C^*$-algebra and a reduced free product of abelian $C^*$-algebra. We discuss the freeness of free product $II_1$-factors and solidity of free product $II_1$-factors weaker than that of Ozawa. We show that the freeness in a free product is related to the existence of Cartan subalgebras in free product $II_1$-factors. Finally, we give a free product factor which is not solid in the weak sense.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.751-763
• /
• 2016

The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-p-group, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.

• Journal of the Korean Data and Information Science Society
• /
• v.18 no.3
• /
• pp.803-808
• /
• 2007

In present article, we consider the embeddability problems for finite dimensional irreducible modules over a complex simple Lie algebra L. For s1(n,C) modules, we determine when one can be embedded into the other if s1(n,C) modules are tensor products of fundamental modules.

• Journal of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.193-207
• /
• 2004

In this expository article we survey some recent developments on the construction of a holomorphic functional calculus for an infinite number of elements in a commutative unital Banach algebra.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.269-277
• /
• 2005

The tensor product $A{\bigotimes}B$ of Hilbert space operators A and B will be shown to be log-hyponormal if and only if A and Bare log-hyponormal. Some general comments about generalized hyponormality are also made.

• The Pure and Applied Mathematics
• /
• v.11 no.2
• /
• pp.149-154
• /
• 2004

We show that the Choquet boundary of the tensor product of two real function algebras is the product of their Choquet boundaries.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.3
• /
• pp.653-659
• /
• 2005

In this paper it is proved that quasisimilar n-tuples of tensor products of injective p-quasihyponormal operators have the same spectra, essential spectra and indices, respectively. And it is also proved that a Weyl n-tuple of tensor products of injective p-quasihyponormal operators can be perturbed by an n-tuple of compact operators to an invertible n-tuple.

• Journal of the Chungcheong Mathematical Society
• /
• v.10 no.1
• /
• pp.127-139
• /
• 1997

The spherical non-commutative $\mathbb{S}_{\omega}$ were defined in [2,3]. Assume that no non-trivial matrix algebra can be factored out of the $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_k(\mathbb{C})$. It is shown that the tensor product of the spherical non-commutative torus $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has a trivial bundle structure if and only if k and 2d - 1 are relatively prime, and that the tensor product of the spherical non-commutative torus $S_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when k > 1.