• Debashish Goswami

Articles written in Proceedings – Mathematical Sciences

• Invariants for normal completely positive maps on the hyperfinite II1 factor

We investigate certain classes of normal completely positive (CP) maps on the hyperfinite II1 factorA. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps.

• On Equivariant Embedding of Hilbert $C^\ast$ Modules

We prove that an arbitrary (not necessarily countably generated) Hilbert $G-\mathcal{A}$ module on a $G-C^∗$ algebra $\mathcal{A}$ admits an equivariant embedding into a trivial $G-\mathcal{A}$ module, provided 𝐺 is a compact Lie group and its action on $\mathcal{A}$ is ergodic.

• Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups

Let 𝐺 be one of the classical compact, simple, centre-less, connected Lie groups of rank 𝑛 with a maximal torus 𝑇, the Lie algebra $\mathcal{G}$ and let $\{E_{i},F_{i},H_{i},i=1,\ldots,n\}$ be tha standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{\text{Ad}_{g}(H_{1}), g\in G\}$, identified with the homogeneous space $G/L$ where $L=\{g\in G : \text{Ad}_{g}(H_{1})=H_{1}\}$. We prove that the coordinate functions $f_{i}(g):=\gamma_{i}(\text{Ad}_{g}(H_{1}))$, $i=1,\ldots,n$, where $\{\gamma_{1},\ldots,\gamma_{n}\}$ is basis of $\mathcal{G}'$ are quadratically independent' in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithfully on $C(M)$ such that the action leaves invariant the linear span of the above coordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of 𝑀 satisfying a similar linearity' condition must be a Rieffel-Wang type deformation of some compact group.

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019