arXiv subjects

Bharat Talwar

Publications and source records attributed to Bharat Talwar.

10 recordsLinked to original sources

arXiv2026

Weak Centrality of unital C(X)-algebras

This paper establishes a fibrewise characterization of weak centrality for unital $C(X)$-algebras whose defining homomorphism maps $C(X)$ onto the center: such an algebra is weakly central if and only if each of its nonzero fibres has a unique maximal ideal. This yields a corresponding characterization for arbitrary unital $C^*$-algebras through their canonical fibres over the spectra of their centers. The resulting criterion explains Vesterstr{\o}m's AF-algebra counterexample, whose obstruction is also interpreted through its Bratteli diagram. A parallel criterion characterizes centrality by simplicity of the fibres. Applications to full group $C^*$-algebras give an alternative proof for the discrete Heisenberg group and show that the full group $C^*$-algebra of every countable non-abelian torsion-free nilpotent group is not weakly central.

math.OA↗

arXiv2024

Peripheral Poisson Boundary

It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a $C^*$-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.

math.OA↗

arXiv2023

Local and Global Analysis of Semilinear Heat Equations with Hardy Potential on Stratified Lie Groups

On stratified Lie groups we study a semilinear heat equation with the Hardy potential, a power non-linearity and a forcing term which depends only upon the spacial variable. The analysis of an equivalent formulation to the problem and an application of a decade old result of Avelin et al. facilitates the management of the singularity in the Hardy potential, thereby yielding results pertaining to both local and global nonexistence. In addition, local existence is verified when the gradient term appearing in the Hardy potential is unimodular almost everywhere. The global existence is also proved under an additional assumption that the forcing term depends on the time variable as well. Through these results this paper sheds light on the possible pivotal exponents for the existence of both local and global solutions to the equation, offering a deeper understanding of the interplay between the model's parameters and the underlying stratified Lie group structure.

math.AP↗

arXiv2022

Peripherally automorphic unital completely positive maps

We identify and characterize unital completely positive (UCP) maps on finite dimensional $C^*$-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We analyze a decomposition of general UCP maps in finite dimensions into persistent and transient parts. It is shown that UCP maps on finite dimensional $C^*$-algebras with spectrum contained in the unit circle are $\ast$-automorphisms.

math.OA↗

arXiv2022

Fujita exponent on stratified Lie groups

We prove that $\frac{Q}{Q-2}$ is the Fujita exponent for a semilinear heat equation on an arbitrary stratified Lie group with homogeneous dimension $Q$. This covers the Euclidean case and gives new insight into proof techniques on nilpotent Lie groups. The equation we study has a forcing term which depends only upon a group element and has positive integral. The stratified Lie group structure plays an important role in our proofs, along with test function method and Banach fixed point theorem.

math.AP↗

arXiv2022

Nonexistence of solutions of certain semilinear heat equations

We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases.

math.AP↗

arXiv2021

Center of Banach algebra valued Beurling algebras

We prove that for a Banach algebra $A$ having a bounded $\mathcal{Z}(A)$-approximate identity and for every $\bf[IN]$ group $G$ with weight $w$ which is either constant on conjugacy classes or $w \geq 1$, $\mathcal{Z}\big(L^1_w(G) \otimes^γA\big) \cong \mathcal{Z}(L^1_w(G)) \otimes^γ\mathcal{Z}(A)$. As an application, we discuss the conditions under which $\mathcal{Z}\big(L^1_w(G,A)\big)$ enjoys certain Banach algebraic properties, for example, weak amenability, semisimplicity etc.

math.FA↗

arXiv2020

Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction of certain inclusions of $C^*$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^*$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^*$-subalgebras of any inclusion of unital $C^*$-algebras admitting a finite index conditional expectation. Then, on the lines of [2], we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^*$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

math.OA↗

arXiv2020

Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras

For a locally compact Hausdorff space $X$ and a $C^*$-algebra $A$ with only finitely many closed ideals, we discuss a characterization of closed ideals of $C_0(X,A) $ in terms of closed ideals of $A$ and certain (compatible) closed subspaces of $X$. We further use this result to prove that a closed ideal of $C_0(X) \otimes^{\min} A$ is a finite sum of product ideals. We also establish that for a unital $C^*$-algebra $A$, $C_0(X,A)$ has centre-quotient property if and only if $A$ has centre-quotient property. As an application, we characterize the closed Lie ideals of $C_0(X,A)$ and identify all closed Lie ideals of $ C_0(X) \otimes^{\min} B(H) $, $H$ being a separable Hilbert space.

math.OA↗

arXiv2018

On closed Lie ideals of certain tensor products of C*-algebras II

We identify all closed Lie ideals of $A \otimes^{\alpha} B$ and $B(H) \otimes^{\alpha} B(H)$, where $\otimes^{\alpha}$ is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, $A$ is any simple C*-algebra, $B$ is any C*-algebra with one of them admitting no tracial states, and $H$ is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of $A\otimes^{\min} B$, where $A$ is a simple C*-algebra with at most one tracial state and $B$ is any commutative C*-algebra.

math.OA↗