From graphs to free products BASU, MADHUSHREE; KODIYALAM, VIJAY; SUNDER, V S
Proceedings of the Indian Academy of Sciences. Mathematical sciences,
11/2012, Letnik:
122, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We investigate a construction (from Kodiyalam Vijay and Sunder V S,
J. Funct. Anal.
260
(
2011
) 2635–2673) which associates a finite von Neumann algebra
M
(Γ,
μ
) to a finite weighted graph (Γ,
μ
). ...Pleasantly, but not surprisingly, the von Neumann algebra associated to a ‘flower with
n
petals’ is the group on Neumann algebra of the free group on
n
generators. In general, the algebra
M
(Γ,
μ
) is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs ‘with one edge’ (or actually a pair of dual edges). This also yields ‘natural’ examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii)
-valued circular and semi-circular operators.
In this note, we first work out some `bare hands' computations of the most elementary possible free products involving \(\mathbb{C}^2 ~(=\mathbb{C} \oplus \mathbb{C} \)) and \(M_2 ~(= ...M_2(\mathbb{C}))\). Using these, we identify all free products \(C \ast D\), where \(C,D\) are of the form \(A_1 \oplus A_2\) or \(M_2(B)\); \(A_1,A_2,B\) are finite von Neumann algebras, as is \(A_1 \oplus A_2\) with the 'uniform trace' given by \(tr(a_1, a_2) = 1/2 (tr(a_1) + tr(a_2))\}\) and \(M_2(B)\) with the normalized trace given by \(tr((b_{i,j}))=1/2(tr(b_{1,1}) + tr(b_{2,2}))\). Those results are then used to compute various possible free products involving certain finite dimensional von-Neumann algebras, the free-group von-Neumann algebras and the hyperfinite \(II_1\) factor. In the process, we reprove Dykema's result `\(R \ast R \cong LF_2\)'.
Continuous minimax theorems Basu, Madhushree; Sunder, V S
arXiv (Cornell University),
11/2013
Paper, Journal Article
Odprti dostop
In classical matrix theory, there exist useful extremal characterizations of eigenvalues and their sums for Hermitian matrices (due to Ky Fan, Courant-Fischer-Weyl and Wielandt) and some consequences ...such as the majorization assertion in Lidskii's theorem. In this paper, we extend these results to the context of self adjoint elements of finite von Neumann algebras, and their distribution and quantile functions. This work was motivated by a lemma in a paper by Voiculescu and Bercovici, that described such an extremal characterization of the distribution of a self-adjoint operator affiliated to a finite von Neumann algebra - suggesting a possible analogue of the classical Courant-Fischer-Weyl minmax theorem, for a self adjoint operator in a finite von Neumann algebra. It is to be noted that the only von Neumann algebras considered here have separable pre-duals.
From graphs to free products Basu, Madhushree; Kodiyalam, Vijay; Sunder, V S
arXiv.org,
02/2011
Paper, Journal Article
Odprti dostop
We investigate a construction which associates a finite von Neumann algebra \(M(\Gamma,\mu)\) to a finite weighted graph \((\Gamma,\mu)\). Pleasantly, but not surprisingly, the von Neumann algebra ...associated to to a `flower with \(n\) petals' is the group von Neumann algebra of the free group on \(n\) generators. In general, the algebra \(M(\Gamma,\mu)\) is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs `with one edge' (or actually a pair of dual edges). This also yields `natural' examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) \(\C \oplus \C\)-valued circular and semi-circular operators.
Aphasia following an acquired neurological insult necessitates an in-depth evaluation of the primary and secondary language symptoms. Of all the tools available for aphasia diagnosis, the Western ...Aphasia Battery (WAB; Kertesz, 1982) has proved to be one of the most comprehensive test batteries for describing the aphasia symptom complex. Several authors have pointed out the need for language-specific tools for the assessment of aphasia. But in Bengali, the most prevalent language in eastern India, no formal language assessment tool was available to date. The present study adapted the original WAB in Bengali to give the Bengali WAB (B-WAB). The study was completed in three phases: development, standardization and validation of the B-WAB. The test material was developed preserving the total number of items, however minor changes were made wherever necessary so that it matched the sociolinguistic norms in this part of the country. It was standardized in a group of 150 normal individuals in five different age groups ranging from 18-70 years, and normative values were provided for each subtest for each group. For establishing validity, it was administered to 30 aphasic subjects and the results indicated that the B-WAB was a valid tool for testing individuals with aphasia.
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