Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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Subalgebras of von Neumann algebras

In the late 70s, Cuntz and Behncke had a paper H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
Andreas Thom's user avatar
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When are two C*-algebras isomorphic as Banach spaces?

We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
Hannes Thiel's user avatar
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Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative ...
Ali Taghavi's user avatar
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Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
Andreas Thom's user avatar
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21 votes
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Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
Suvrit's user avatar
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18 votes
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What is the geometric intuition behind Wilf-Zeilberger theory?

This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
Henry.L's user avatar
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An "exercise" on von Neumann algebra tensor product

The following problem appears to be an easy exercise on von Neumann algebra tensor products, but since I've been failing to find a rigorous proof, I'd like to make sure it's not that trivial. Suppose $...
Narutaka OZAWA's user avatar
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What is operator tmf?

One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, ...
Qiaochu Yuan's user avatar
18 votes
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Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group. General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal coefficient theorem (UCT)? I am mainly ...
Caleb Eckhardt's user avatar
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Can we define spectral triples using the language of rigged Hilbert spaces?

The traditional mathematical approach to quantum mechanics, as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators. Another approach, which more closely resembles ...
Dmitri Pavlov's user avatar
16 votes
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$C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
Yemon Choi's user avatar
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15 votes
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Countably decomposable von Neumann algebras

A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
Nik Weaver's user avatar
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Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
Jonas Meyer's user avatar
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Stable isomorphism of group C$^*$-algebras

For a discrete group $G$, let $C^*_r(G)$ be its reduced group C$^*$-algebra. Question: Do there exist discrete, torsion-free non-isomorphic groups $G,H$ such that $C^*_r(G)$ and $C^*_r(H)$ are stably ...
Caleb Eckhardt's user avatar
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Schwartz kernel theorem for A-linear operators

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
Ulrich Pennig's user avatar
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Unitary group of a von Neumann algebra: is it a retract of $U(H)$?

Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$). Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...
André Henriques's user avatar
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Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard bold.)$\newcommand{\FA}{{...
Yemon Choi's user avatar
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Symmetric (extended) Haagerup tensor product

Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
Matthew Daws's user avatar
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Existence of more than two C*-norms on algebraic tensor product of C*-algebras

Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$. If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
Alcides Buss's user avatar
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Why do some tricks in homological algebra work over the category of C*-algebras?

The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...
xuq01's user avatar
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Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
tattwamasi amrutam's user avatar
12 votes
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Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows: Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of linear ...
yurius's user avatar
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Subfactors of $L(F_{\infty})$

It is a well known result that any subfactor of the hyperfinite $II_{1}$ factor is hyperfinite. I wonder if there is any finite index version of this for free group factors. In particular is it true ...
Mike Hartglass's user avatar
11 votes
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474 views

Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry

In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry. For graphs this had been an open ...
David Roberson's user avatar
11 votes
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408 views

Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...
Jon Bannon's user avatar
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Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
Henrique Tyrrell's user avatar
11 votes
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472 views

Groups with reduced C*-algebras of stable rank 1

Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in $...
Denis Osin's user avatar
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Twisted crossed product von Neumann Algebras

I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I ...
user193319's user avatar
10 votes
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Subobject classifiers with a quantale structure

Quantales were kind of popular a few decades ago, as some sort of "quantum" version of the much more widespread concept of locale (Mulvey introduced them to study $C^*$ algebras). I was ...
Mirco A. Mannucci's user avatar
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Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space: Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
erz's user avatar
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The term "absolute geometry"

My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
santker heboln's user avatar
10 votes
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249 views

What are these matrices called?

A paper I'm writing heavily uses block diagonal matrices with the property that each block is upper triangular and constant along its diagonal. Like this: $$\begin{bmatrix} A_1 \\ & \ddots \\ &...
Nik Weaver's user avatar
10 votes
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183 views

Masas in SAW*-algebras

I asked this question three years ago at MSe but it has no response; let me try here. Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
Tomasz Kania's user avatar
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Are ideals in separable C*-algebras complemented subspaces?

Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a ...
Tobias Fritz's user avatar
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Mackey's Program on Algebraic Ergodic Theory

I knew about Mackey's Program from Arnold's book Random Dynamical Systems and it referred to K. Schmidt's book Algebraic Ideas in Ergodic Theory, which was published in 1990. However, that is the ...
Fan's user avatar
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Commutative spectral triples not coming from manifolds

There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...
truebaran's user avatar
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Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
TrzyTrypy's user avatar
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10 votes
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320 views

Can we find arbitrarily many elements of SU(2) generating a good copy of MAX($\ell_1^n$) inside VN(SU(2))?

In trying to prove that the answer to the title is "no", I was led to the following problem (which I think is equivalent to the question asked in the title, but can be stated independently). If ...
Yemon Choi's user avatar
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10 votes
0 answers
497 views

Lacunary hyperbolic groups and weak amenability

In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
Denis Poulin's user avatar
10 votes
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970 views

Centralizers of group actions

Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...
Steven Deprez's user avatar
10 votes
0 answers
319 views

H-space structure on the Calkin algebra

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
Ulrich Pennig's user avatar
9 votes
0 answers
288 views

Why are projectionless $C^*$-algebras important (Kadison's conjecture)

It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
Alexandar Ruño's user avatar
9 votes
0 answers
201 views

Examples of pure state spaces of C*-algebras

The pure state space of $C(X)$ for compact Hausdorff $X$ is just $X$. The pure state space of $M_n(\mathbb{C})$ is given by the $n$-dimensional row vectors $v$ (normalised and up to a multiple) by $\...
Edwin Beggs's user avatar
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9 votes
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Real Rank of $M_n(A)$

The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest ...
Hannes Thiel's user avatar
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9 votes
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Geometric motivation behind the Fredholm module definition

If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
Max Schattman's user avatar
9 votes
0 answers
108 views

Morita equivalence for graded von Neumann algebras

I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...
Anton Kapustin's user avatar
9 votes
0 answers
267 views

A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this: Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
dimquasar's user avatar
9 votes
0 answers
222 views

Using Property (T) to approximate invertible matrices

In the wikipedia article for Kazhdan's Property (T), there's an intriguing application: Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
Eric Reckwerdt's user avatar
9 votes
0 answers
254 views

Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map. Can $T$ be extended to a `positively linear map' ...
Tomasz Kania's user avatar
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9 votes
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Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
Alexander Chervov's user avatar

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