arXiv subjects

Issan Patri

Publications and source records attributed to Issan Patri.

7 recordsLinked to original sources

arXiv2026

Spaces of UCP maps and subalgebras of von Neumann algebras

In this paper, we establish that several natural topologies on the space of state-preserving unital completely positive maps coincide and that make it a Polish space. We then focus on the subspace of state-preserving conditional expectations and analyse its topology in detail, recovering the Haagerup-Winslow result that it aligns with the Effros-Mar\'echal topology on the space of von Neumann subalgebras. This correspondence is then applied to structural classes of subalgebras, including amenable, Haagerup and weakly amenable subalgebras. Among other consequences, we demonstrate the closedness of amenable subalgebras admitting state-preserving conditional expectations and analyze the semicontinuity and failure of continuity of the Cowling-Haagerup constant as a function on subalgebras. Finally, we investigate the space of von Neumann subalgebras that are not the image of state preserving conditional expectations for a fixed faithful normal state. For several important classes of von Neumann algebras, such as type ${\rm III}_\lambda$ factors with $0 \le \lambda < 1$ and type ${\rm III}_1$ factors with a state whose centraliser is infinite dimensional, we show that the subalgebras lacking state-preserving conditional expectations form an open and dense subset. Thus, in these settings, the generic subalgebra is not the range of state preserving conditional expectation.

math.OA↗

arXiv2024

Operator Algebras of Universal Quantum Homomorphisms

Given two unital C*-algebras $A$ and $B$, we study, when it exists, the universal unital $C^*$-algebra $\mathcal{U}(A,B)$ generated by the coefficients of a unital $*$-homomorphism $\rho\,:\, A\rightarrow B\otimes\mathcal{U}(A,B)$. When $B$ is finite dimensional, it is well known that $\mathcal{U}(A,B)$ exists and we study in this case properties LP, RFD, primitiveness and the UCT as well as $K$-theory. We also construct a reduced version of $\mathcal{U}(A,B)$ for which we study exactness, nuclearity, simplicity, absence of non-trivial projection and $K$-theory. Then, we consider the von Neumann algebra generated by the reduced version and study factoriality, amenability, fullness, primeness, absence of Cartan, Connes' invariants, Haagerup property and Connes' embeddability. Next, we consider the case when $B$ is infinite dimensional: we show that for any non-trivial separable unital $C^*$-algebra $A$, $\mathcal{U}(A,B)$ exists if and only if $B$ is finite dimensional. Nevertheless, we show that there exists a unique unital locally $C^*$-algebra generated by the coefficients of a unital continuous $*$-homomorphism $\rho\,:\, A\rightarrow B\otimes\mathcal{U}(A,B)$. Finally, we study a natural quantum semigroup structure on $\mathcal{U}(A,A)$.

math.OA↗

arXiv2024

Michael's selection theorem and applications to the Mar\'echal topology

The Mar\'echal topology, also called the Effros-Mar\'echal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Mar\'echal from 1973 that this topology is Polish as soon as the ambient algebra has separable predual, but the sketch of proof in her research announcement appears to have a small gap. Our main goal in this paper is to fill this gap by a careful look at the topologies one can put on the space of weak-$*$ closed subspaces of a dual space. We also indicate how Michael's selection theorem can be used as a step towards Mar\'echal's theorem, and how it simplifies the proof of an important selection result of Haagerup and Winsl{\o}w for the Mar\'echal topology. As an application, we show that the space of finite von Neumann algebras is $\mathbf\Pi^0_3$-complete.

math.OA↗

arXiv2017

Topological automorphism groups of compact quantum groups

We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the "inner" automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end with the construction of a compact matrix quantum group whose fusion ring is not finitely generated, unlike the classical case.

math.OA↗

arXiv2016

Maximal torus theory for compact quantum groups

Associated to any compact quantum group $G\subset U_N^+$ is a canonical family of group dual subgroups $\widehat{\Gamma}_Q\subset G$, parametrized by unitaries $Q\in U_N$, playing the role of "maximal tori" for $G$. We present here a series of conjectures, relating the various algebraic and analytic properties of $G$ to those of the family $\{\widehat{\Gamma}_Q|Q\in U_N\}$.

math.QA↗

arXiv2015

On compact bicrossed products

We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and and produce an easy construction of the associated bicrossed product $\mathbb{G}$. We investigate the relative co-property $(T)$ and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property $(T)$ and Haagerup property of the discrete dual $\widehat{\mathbb{G}}$, and review co-amenability of $\mathbb{G}$ as well. We distinguish two such non-trivial compact bicrossed products with relative co-property $(T)$ and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property $(T)$, the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.

math.OA↗

arXiv2013

Normal Subgroups, Center and Inner Automorphisms of Compact Quantum Groups

We introduce a class of automorphisms of compact quantum groups which may be thought of as inner automorphisms and explore the behaviour of normal subgroups of compact quantum groups under these automorphisms. We also define the notion of center of a compact quantum group and compute the center for several examples. We briefly touch upon the commutator subgroup of a compact quantum group and discuss how its relation with the center can be different from the classical case.

math.OA↗