arXiv subjects

Tattwamasi Amrutam

Publications and source records attributed to Tattwamasi Amrutam.

At least 19 recordsLinked to original sources

arXiv2026

Invariant von Neumann subalgebras of crossed products and tensor products

Let $Γ$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,τ_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q\rtimesΓ$ and of $L(Γ)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto $Q$ is scalar is necessarily contained in $L(Γ)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\mathbb{F}_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors.

math.OA↗

arXiv2026

Unique equivariant pseudo expectation and generalized Powers averaging for crossed product

Let $Γ$ be a countable discrete group and let $A$ be a unital $Γ$-simple $Γ$-$C^*$-algebra. We prove that $A\rtimes_rΓ$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_rΓ\to I_Γ(A)$ is the unique $Γ$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_Γ(A)\rtimes_rΓ$, with the averaging coefficients still belonging to $A$.

math.OA↗

arXiv2026

Invariant random subgroups of $C^*$-simple groups

Let $Γ$ be a countable discrete $C^*$-simple group. Let $μ$ be a $Γ$-invariant Borel probability measure on $\text{Sub}(Γ)$, i.e., an IRS. We show that $μ$-almost every subgroup $H$ is $C^*$-simple.

math.OA↗

arXiv2026

Boundary Dynamics, Cubical Actions, and Infinite Girth

We develop two methods for proving infinite girth from boundary dynamics. First, we use topologically free extreme boundary actions to give a boundary-dynamical proof of infinite girth for finitely generated acylindrically hyperbolic groups. The second combines Nakamura's criterion with, respectively, flag-space and Roller-boundary dynamics, yielding infinite-girth results for semisimple $S$-algebraic groups and for essential non-elementary actions on finite-dimensional, second countable, non-Euclidean CAT(0) cube complexes. We also prove that every finitely generated large group has infinite girth, thereby answering a question of Akhmedov and Mishra.

math.GR↗

arXiv2026

The $C^*$--ISR Property for $\text{PSL}_n(\mathbb{Z})$

A discrete group $Γ$ has the \emph{$C^*$--ISR property} if every unital $Γ$--invariant $C^*$--subalgebra $A\subseteq C_r^*(Γ)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleftΓ$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$--ISR property for every $n\geq3$.

math.OA↗

arXiv2026

Uniformly recurrent subalgebras in finite von Neumann algebras

We introduce the notion of a uniformly recurrent subalgebra (URA) for a trace-preserving action of a countable discrete group $Γ$ on a finite von Neumann algebra $M$, providing an operator-algebraic counterpart to the theory of uniformly recurrent subgroups (URS). We also show that the Effros-Maréchal space $\text{Sub}(M)$ is compact if and only if $M$ lacks a diffuse direct summand. Leveraging this, we show that URAs can exhibit arbitrary topological complexity and construct exotic URAs homeomorphic to any prescribed minimal Polish space. In the context of crossed products $M \rtimes Γ$ with amenable coefficients, we utilize URAs to formulate a new characterization of C*-simplicity, proving that $Γ$ is C*-simple if and only if the only amenable URA of the crossed product containing $M$ is $\{M\}$. Finally, to bypass the failure of compactness in $\text{Sub}(M)$, we develop a generalized state-space machinery using Baire-category methods on the weak-* compact space of trace-extending states. This construction captures compact, discrete, and exotic URAs, while recovering the classical URS framework as a special case.

math.OA↗

arXiv2026

Relative stationary dynamical systems

Let $G$ be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure $μ$. We generalize the notion of $μ$-stationary systems to $μ$-stationary $G$-factor maps $π: (X,ν)\to (Y,η)$. For these stationary relations between dynamical systems, we provide a structure theorem, which generalizes the structure theorem of Furstenberg-Glasner. Furthermore, we show the existence and uniqueness of a relative version of the Poisson boundary in this setup.

math.DS↗

arXiv2026

Simplicity of q-Gaussian C*-algebras

We show that q-Gaussian $C^*$-algebras for $q \in (-1, 1)$ have the Dixmier averaging property, and hence are simple with a unique trace. We argue by combining rapid decay and spectral gap estimates for the commutators with the generators, which are obtained from free probability.

math.OA↗

arXiv2026

The Bishop--Phelps--Bollobás Property for Extremally Disconnected Ranges: Separable and Low-Density Domains

We prove a Bishop--Phelps--Bollobás theorem for operators into spaces of continuous scalar-valued functions on extremally disconnected compact Hausdorff spaces over both the real and complex scalar fields. The main result applies whenever the density character of the domain is strictly smaller than the Baire number of the underlying compact space. The proof also yields an explicit quadratic Bishop--Phelps--Bollobás modulus. In particular, every separable Banach space paired with such a function space has the Bishop--Phelps--Bollobás property for operators.

math.FA↗

arXiv2026

Relative invariant subalgebra rigidity for Thompson's group $F$

We prove that Thompson's group $F$ satisfies the relative invariant subalgebra rigidity property with respect to its commutator subgroup: every von Neumann subalgebra of $L(F)$ that is invariant under conjugation by $[F,F]$ is of the form $L(N)$ for some normal subgroup $N \trianglelefteq F$. Along the way, we establish a general factoriality criterion for invariant subalgebras whose hypotheses are met whenever the ambient group is i.c.c., simple, and every faithful ergodic measure-preserving action of it on a probability space is essentially free.

math.OA↗

arXiv2026

$C^*$-simplicity, confined subalgebras, and operator algebraic uniform recurrence

We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that a countable discrete group is $C^*$-simple if and only if it admits no non-trivial amenable confined subalgebras. This generalizes the well-known result of Kennedy that characterizes $C^*$-simplicity in terms of trivial amenable uniformly recurrent subgroups.

math.OA↗

arXiv2026

On Relative Invariant Subalgebra Rigidity Property

A countable discrete group $Γ$ is said to have the relative ISR-property if for every non-trivial normal subgroup $N\trianglelefteqΓ$ and every von Neumann subalgebra $\mathcal{M}\subseteq L(Γ)$ invariant under conjugation by $N$, one has $\mathcal{M}=L(K)$ for some subgroup $K\leΓ$. Similarly, $Γ$ has the relative $C^*$-ISR-property if every $N$-invariant unital $C^*$-subalgebra $\mathcal{A} \subseteq C_r^*(Γ)$ is of the form $C_r^*(K)$. We show that every torsion-free acylindrically hyperbolic group with trivial amenable radical satisfies the relative ISR property. Moreover, we also show that all torsion-free hyperbolic groups have the relative $C^*$-ISR property. Furthermore, we establish an analogous relative ISR-property for irreducible lattices in higher-rank semisimple Lie groups, such as $\mathrm{SL}_d(\mathbb{Z})$ ($d \geq 3$), with trivial center.

math.OA↗

arXiv2026

On the Convergence of Numerical Index via Operator Openings and Ultraproducts

The numerical index of a Banach space is a geometric constant relating the numerical radius of bounded linear operators to their standard operator norm. In this paper, we study the continuity of the numerical index under two distinct notions of subspace convergence. First, we establish a full limit theorem in the operator opening topology: if $\{X_n\}_{n \in \mathbb{N}}$ and $X$ are closed subspaces of a Banach space $Y$ with $X_n \to X$ in the operator opening, then $\lim_{n \to \infty} n(X_n) = n(X)$. Second, we develop ultraproduct methods for the numerical index, proving that the numerical radius is exactly preserved by ultraproduct operators, i.e., $v((T_n)_{\mathcal{U}}) = \lim_{\mathcal{U}} v(T_n)$. As a consequence, we show that $n(X_{\mathcal{U}}) \le n(X)$ for every ultrapower $\mathcal{U}$.

math.FA↗

arXiv2026

Invariant $C^$-subalgebras of the reduced group $C^$-algebra

Let $Γ$ be a countable discrete group. We say that $Γ$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $Γ$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(Γ)$ is of the form $C_r^*(N)$ for some normal subgroup $N\triangleleftΓ$. We show that all torsion-free, non-amenable (cylindrically) hyperbolic groups with property-AP and a finite direct product of such groups have this property. We also prove that an infinite group $Γ$ has the C$^*$-ISR property only if $Γ$ is simple amenable or $C^*$-simple.

math.OA↗

arXiv2026

Non-commutative Factor theorem for tensor products of lattices in product groups

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $Γ< G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, τ)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimesΓ$ and $(L^\infty(G/P,ν_P)\overline{\otimes}\mathcal{N})\rtimesΓ$ is again a crossed product of the form $(L^\infty(G/Q,ν_Q)\overline{\otimes}\mathcal{N})\rtimesΓ$.

math.OA↗

arXiv2026

Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $Γ$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $Γ$-system and strongly proximal with respect to $H$. When $H\le_cΓ$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $Γ$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an $X$-plump subgroup given a $Γ$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product.

math.OA↗

arXiv2025

Fejér property and Galois correspondence for groupoid $C^*$-algebras

We introduce a notion of the Fejér property for topological étale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal étale second countable groupoid satisfying the Fejér property, every closed $C_0(\mathcal{G}^0)$-bimodule $M\subset C_r^*(\mathcal{G})$ is of the form $\overline{C_c(U)}^r$ for some open set $U$. Moreover, we get a Galois correspondence in the sense that every intermediate $C^*$-algebra $\mathcal{B}$ with $C_0(\mathcal{G}^0)\subseteq \mathcal{B}\subseteq C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\leq \mathcal{G}$.

math.OA↗

arXiv2025

Strict comparison in reduced group $C^*$-algebras

We prove that for every $n\geq 2$, the reduced group $C^*$-algebras of the countable free groups $C^*_r(\mathbb{F}_n)$ have strict comparison. Our method works in a general setting: for $G$ in a large family of non-amenable groups, including hyperbolic groups, free products, mapping class groups, right-angled Artin groups etc., we have $C^*_r(G)$ have strict comparison. This work also has several applications in the theory of $C^*$-algebras including: resolving Leonel Robert's selflessness problem for $C^*_r(G)$; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C^*_r(G)$; full computations of the Cuntz semigroup of $C^*_r(G)$ and future directions in the $C^*$-classification program.

math.OA↗