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Ilwoo Cho

Publications and source records attributed to Ilwoo Cho.

At least 37 records · Page 2Linked to original sources
arXiv2006

Measure Theory on Graphs

In this paper, we defined three kinds of measures depending on the given finite directed graphs. For the given finite directed graph, we can construct the free semigroupoid, the diagram set and the reduced diagram set, as algebraic structures determined by the admissibility on the graph. The power sets of such structures are sigma algebras of them, respectively. On them, we can define a measure. The pain purpose of this paper is to introduce those three measures and observe some properties of them. Moreover, we consider Measure Theory on them

math.CO↗
arXiv2005

Boundary Qoutinet Graphs and Graph Index

In this paper, we define the quotinet graphs. In particular, we introduce the boundary quotient graphs, admissible boundary quotient graphs and subgraph boundary qoutient graphs. By the property of the quotient spaces, the boundary quotients need not be invariants on graphs. But, in certain cases, the admissible boundary quotient can be an invariant. The graph index is defined and observed. The concept is somewhat artificial, but we can construct the group-theory-like index on graphs, with respect to the subgraph qoutient graphs.

math.CO↗
arXiv2005

Free Probability on a Direct Product of Noncommutative Probability Spaces

In this paper, we observevd the amalgamated free probability of direct product of noncommutative probability spaces. We defined the amalgamated R-transforms, amalgamated moment series and the amalgamated boxed convolution. They maks us to do the amalgamated R-transform calculus, like the scalar-valued case.

math.OA↗
arXiv2005

Graph W*-Probability on the Free Group Factor L(F_N)

In this paper, we will use the graph W*-probability technique to re-compute the moments and cumulants of the operator which is the N-free sum of semicircular elements. This computation is well-known, but I used the graph probability technique to resolve them. By considering this idea, we will consider the embedding of the free group factor L(F_N) into a graph W*-algebra.

math.OA↗
arXiv2005

Free Product Structure of a Graph W*-probability space

In this paper, we observe the amalgamated free product structure of a Graph W*-probability space. In [16] and [17], we already observed the operator-valued freeness conditions on a graph W*-algebra. By using the conditions, we will consider the amalgamated free product structure of a graph W*-algebra. In particular, we can make the algebra be blocked, by so-called the loop free blocks (D_G-semicircular blocks) and non-loop blocks (R-diagonal blocks). Finally, we can see the the graph W*-algebra is the free product of edge blocks of the algebra.

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arXiv2005

Compressed Random Variables in a Graph W*-Probability Space

In this paper, we will consider the free probabilistic information about compressed random variables in a graph W*-Probability space. Recall the diagonal compressed random variables in a graph W*-probability space. In particular, we can see that the free moments and cumulants of the fixed compressed random variable of a random variable x are exactly same as the diagonal compressed part of x.

math.OA↗
arXiv2005

Random Variables in Graph W*-Probability Spaces

In [16], we observed the graph W*-probability theory. In this paper, we will review [16] and introduce special amalgamated random variables in this amalgamated W*-probability space. In particular, we will observe the amalgamated semicircularity, amalgamated evenness and amalgamated R-diagonality. As an example, we will compute the trivial moments and trivial cumulants of the generating operator of the graph W*-algebra.

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arXiv2005

The Moment of an Operator in the Free Group Factor L(F_N)

In this paper, we will compuet the moment of the operator which is the product of the generating operator and the word operator. The word operator means the operator which is the total sum of the word with length n. We constructed a new recurrence relation and by using this recurrence relation, we can compute the first moment of the operator.

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arXiv2005

Diagonal Compressed Random Variables in a Graph W*-probability Spaces

In [15] and [16], we constructed a graph W*-probability space over its diagonal subalgebra. To study this strudture is a good example of Speicher's amalgamated W*-probability spaces. In this paper, we will observe the diagonal compressed random variables. we will compute the moments and cumulants of them and we will show that the freeness on diagonal compressed random variables in the graph W*-probability space is preserved by that of (uncompressed) random variables.

math.OA↗
arXiv2005

Operator-Valued Moment Series of the Generating Operator of L(F_2) Over the Commutator Group von Neumann algebra L(K)

In this paper, we will consider the generating operator of the free group factor L(F_2). Then we can construct the group von Neumann algebra L(K), where K is the commutator group of F_2 and the conditional expectation E. Then (L(F_2), E) is the W*-probability space with amalgamation over L(K). In this paper, we will compute the trivial operator-valued moment series of the generating operator of L(F_2) over L(K). This computation is the good example for studying the operator-valued distribution, since the operator-valued moment series of operator-valued random variables contain algebraic and combinatorial free probability information about the opeartor-valued distributions.

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arXiv2004

The Trivial L(F_1)-Valued moment Series of the Generating Operator of L(F_2)

In [13] and [14], we found so-called the reccurence diagram. This paper is the application of such recurrence diagram. As a first application for the reccurence diagram, we will re-compute the well-known moment series of the generating opeator of the free group factor L(F_N). Also, we will compute the trivial L(F_1)-valued moment series of the generating operator of L(F_2). This result can be extended for the general case when we have L(F_N), instead of L(F_2). To study operator-valued moment series is to study the operator-valued distribution. By doing that, we can get some algebraic and combinatorial free probability information about the operator-valued distribution of certain random variables. So, this paper provides a good example for studying the operator-valued distribution.

math.OA↗
arXiv2004

Amalgamated R-diagonal Pairs

In this paper, we will consider the properties of amalgamated R-diagonal pairs. We characterize the amalgamated R-diagonality of pairs of amalgamated random variables by certain cumulant-relation.

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arXiv2004

Moments of the Generating Operator of Two Copies of L(F_2)'s over L(F_1)

064 In this paper, we will consider an example of a (scalar-valued) moment series, under the compatibility. Suppose that we have an amalgamated free product of free group algebras, $L(F_{2})*_{L(F_{1})}L(F_{2})=L( )*_{L( )}L( )$ 064 We will provide the method how to find the moment series of $a+b+a^{-1}+b^{-1}+c+d+c^{-1}+d^{-1}.$ Amalgamated freeness of $a+b+a^{-1}+b^{-1}$ and $c+d+c^{-1}+d^{-1}$ over $L(F_{1})$ is used and some combinatorial functions (to explain the recurrence relations) are used to figure out the $n$-th moment of this element.

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arXiv2004

Compatibility of a noncommutative probability space and a noncommutative probability space with amalgamation

In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued R-transforms and operator-valued moment series, under the compatibility. Also, we can see that there is a big gap between freeness and amalgamated freeness.

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arXiv2004

R-transform Theory On a Tower of Amalgamated Noncommutative Probability Spaces

In [9], we observed Amalgamated R-transform Theory. Different from the original definition of Voiculescu and Speicher, we define R-transforms of operator-valued random variable(s) by operator-valued formal series. By doing that we can establish Amalgamated R-transform calculus. In this paper, we will concentrate on observing R-transforms on a tower of amalgamated noncommutative probability spaces. We will consider the R-transform calculus and relation between R-transforms on the tower and distributions with respect to the tower. As application, we will observe the compatibility of the given tower and commuting ladders of amalgamated noncommutative probability spaces

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