Operator algebras in India in the past decade. (English) Zbl 1444.46051
Indian J. Pure Appl. Math. 50, No. 3, 801-834 (2019); erratum ibid. 50, No. 4, 1147-1151 (2019).
Summary: Operator algebras come in many flavours. For the purpose of this article, however, the term is only used for one of two kinds of self-adjoint algebras of operators on Hilbert space, viz., \(C^\ast\)-algebras (which are norm-closed) or von Neumann algebras (which are closed in the topology of pointwise strong convergence or, equivalently, in the weak\(*\)- topology it inherits as a result of being a Banach dual space). To be fair, there are a number of people in India (e.g., Gadadhar Misra, Tirthankar Bhattacharyya, Jaydeb Sarkar, Santanu Dey, etc.) who work on non-selfadjoint algebras, mostly from the point of view of connections with complex function theory; but in the interest of restricting the size of this paper, I confine myself here to selfadjoint algebras. I apologise for ways in which my own personal taste and limitations colour this depiction of operator algebras. Another instance of this arbitrary personal taste is a decision to concentrate on the work of younger people. Thus, the work of the more senior people who have worked in operator algebras is only seen via their collaborations with younger people: e.g., KRP via Srinivasan and Rajarama Bhat, Kalyan Sinha via Debashish, Partha, Arup, Raja, etc. and me via Vijay, Srinivasan and Panchugopal.
Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU, \(\dots\). My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process.) My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the \(C^\ast\)-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay.
What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) \(E_0\) semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) masas in \(II_1\) von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).
Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU, \(\dots\). My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process.) My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the \(C^\ast\)-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay.
What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) \(E_0\) semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) masas in \(II_1\) von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).
MSC:
| 46L99 | Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.) |
| 46-03 | History of functional analysis |
| 01A32 | History of Indian mathematics |
| 01A61 | History of mathematics in the 21st century |
Keywords:
von Neumann algebras; subfactors; planar algebras; free probability; TQFTs, Hopf (and Kac) algebras; Skein theories; quantum dynamical systems; complete positivity; (sub- and super-)product systems; \(E_0\) semigroups on \(B(H)\), resp., on general factors; types I–III of \(E_0\) semigroups; extendability of \(E_0\) semigroups; masas; \(q\)-deformed Araki-Woods von Neumann algebras; ergodic theory; \(C^\ast\)-algebras; semigroup \(C^\ast\)-algebras; operator-algebraic quantum groups; Elliott’s classification program; noncommutative geometry (NCG); spectral triples for quantum groups; dimensional invariants; quantum isometry groups; local index formula in NCG; metric properties; Dirac differential graded algebra; Yang Mills functionals; compact quantum metric spaces; rapid decay propertyReferences:
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