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Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps

Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
Ran's user avatar
  • 73
1 vote
1 answer
117 views

Lower bound for a commutator trace

I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
Chilperic's user avatar
  • 121
1 vote
0 answers
86 views

Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group

I am posting my question of mathstack exchange here. (see: My post on MSE) Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
Tomás Pacheco's user avatar
9 votes
0 answers
163 views

Moore-Penrose partial isometries and hermitian elements

Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
Hannes Thiel's user avatar
  • 3,497
-6 votes
1 answer
180 views

An analog of Anderson's result in C* algebra setting [closed]

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$. For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$ It's known that $...
SoG's user avatar
  • 307
6 votes
1 answer
170 views

Do projections in an $AW^\ast$-algebra form an orthomodular lattice?

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 2,830
3 votes
1 answer
116 views

Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?

This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t., $\phi$ is sesquilinear,...
David Gao's user avatar
  • 2,830
1 vote
1 answer
122 views

distance in the matrix algebra w.r.t. the nuclear norm

Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
Krzysztof's user avatar
  • 375
1 vote
0 answers
55 views

Characterizing one-sided M-projections on real C*-algebras

Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map $$ v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
Neal B's user avatar
  • 11
4 votes
1 answer
251 views

Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra

Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put $$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
Andromeda's user avatar
  • 175
7 votes
1 answer
280 views

Norm in the minimal tensor product of C*-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
Hannes Thiel's user avatar
  • 3,497
5 votes
1 answer
183 views

Question about modular group (Modular theory in operator algebras, section 2.14)

Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20: I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
Andromeda's user avatar
  • 175
0 votes
0 answers
57 views

Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?

I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
S-F's user avatar
  • 63
2 votes
0 answers
142 views

A $C^*$-algebra with the bidual $B(H).$

Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} \equiv B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$...
Karimbergen Kudaybergenov's user avatar
6 votes
1 answer
255 views

Example/Existence of Positive Linear Functional which is NOT Hermitian

We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
UtsabrajSarkar's user avatar
7 votes
2 answers
349 views

Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?

This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
David Gao's user avatar
  • 2,830
1 vote
0 answers
176 views

Unitary operators in $\mathcal{U}(L^2(X,\mu))$ not coming from $\operatorname{Aut}(X,\mu)$

$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see ...
Kajal Das's user avatar
  • 105
1 vote
1 answer
146 views

Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$

I would like to show that: "every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
VvvV's user avatar
  • 11
3 votes
0 answers
96 views

Excising the trace of a $II_1$-factor

Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
pitariver's user avatar
  • 297
0 votes
1 answer
118 views

Minimal norm problem whose unknown is an operator

Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that $$ f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2 $...
user8469759's user avatar
13 votes
0 answers
573 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
4 votes
0 answers
134 views

Automorphism-invariant positive linear functionals on $C*$-algebras

Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
Bedovlat's user avatar
  • 1,959
1 vote
0 answers
94 views

positive invertible maps which are not *-automorphisms

Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
fidaleo's user avatar
  • 41
3 votes
1 answer
162 views

Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
Logan Hyslop's user avatar
1 vote
0 answers
108 views

Infinite tensor product of Hilbert spaces [duplicate]

Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
Jake's user avatar
  • 11
0 votes
0 answers
157 views

Dependence of functional integral on the function space

In physics, the following functional integral is considered \begin{gather} Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf )) \end{gather} It is usually said that the integration is performed ...
0x11111's user avatar
  • 593
2 votes
0 answers
316 views

What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows: Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
S-F's user avatar
  • 63
6 votes
0 answers
98 views

Conditions for completely positive maps to act homomorphically across multiple subalgebras

For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
BiPolarBear's user avatar
7 votes
0 answers
164 views

Nontrivial examples of locally compact quantum groups

What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
Cameron Zwarich's user avatar
1 vote
1 answer
284 views

A certainty principle?

Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where $$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
JP McCarthy's user avatar
  • 1,027
2 votes
1 answer
237 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
B.Hueber's user avatar
  • 1,161
2 votes
1 answer
264 views

Continuous path of unitary matrices with prescribed first column?

Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$. Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
ccriscitiello's user avatar
1 vote
1 answer
295 views

An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible

Given: $X$ - any Banach space $F : X \to X$ (linear bounded and non-invertible) $P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$ Can you help me come ...
TorteDeline's user avatar
5 votes
2 answers
342 views

Projections in atomless von Neumann algebras

Let $\varepsilon>0$. If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
user92646's user avatar
  • 617
8 votes
1 answer
389 views

Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
Jochen Glueck's user avatar
2 votes
0 answers
232 views

Show that $\mathbb{K}\cong M_{n}(\mathbb{K})$ [closed]

I would like to show the following isomorphy but not sure how to go about this: $\mathbb{K}\cong M_{n}(\mathbb{K})$ Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism ...
craaaft's user avatar
  • 129
1 vote
1 answer
128 views

Compare the weight of $p\vee q$ and that of $p+q$

Let $M$ be a von Neumann algebra. If it has a semifinite faithful normal trace $\tau$, then we have $\tau(p\vee q)\le \tau(p)+\tau(q)$. However, for the weight (even a faithful normal state) $\omega$ ...
user92646's user avatar
  • 617
1 vote
1 answer
211 views

Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful. I also ...
J_P's user avatar
  • 439
1 vote
0 answers
125 views

Transforming nilpotency into diagonalizability [closed]

We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$. We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows: $Te_1=0$ and $...
ABB's user avatar
  • 4,058
4 votes
1 answer
203 views

weights of projections and norms of operators in a von Neumann algebra

Let $M$ be an atomless von Neumann algebra equipped with a (semifinite faithful normal) weight $w$. Let $x\in M$ and let $\varepsilon>0$. Can we find a constant $\delta>0$ such that whenever a ...
user92646's user avatar
  • 617
5 votes
1 answer
221 views

Arens regularity of $\mathrm{BV}(\mathbb{R})$

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
Tobias Fritz's user avatar
  • 6,406
4 votes
0 answers
242 views

On the Dunford-Pettis property and multiplier algebras

I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that: Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
Alexander Dobrick's user avatar
5 votes
1 answer
613 views

Is every character of the algebra of continuous functions on a locally compact space some evaluation?

Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$. Question. Given an arbitrary character (i.e. a non-zero ...
Hua Wang's user avatar
  • 960
2 votes
0 answers
157 views

Why do von Neumann algebras possess identity?

My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
MrPajeet's user avatar
  • 433
6 votes
0 answers
253 views

Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?

QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
Yemon Choi's user avatar
  • 25.8k
-2 votes
1 answer
143 views

Relationship between noncommutative torus for different values of theta [closed]

Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
A beginner mathmatician's user avatar
1 vote
1 answer
207 views

Weak-star convergence implies trace-norm convergence

By definition, if bounded operators $a_i$ converge to $0$ in the weak*-star topology, then $\operatorname{tr} a_it \to 0$ for any trace-class $t$. Does this also hold for the trace-norm instead of the ...
Dominique Unruh's user avatar
4 votes
0 answers
145 views

Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces

Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed? The spaces in question include e.g. \begin{equation} X = (x: x \in l_2: p_i(x) ...
0x11111's user avatar
  • 593
9 votes
0 answers
240 views

What is known about when $vN(G)$ is a factor, for a locally compact group $G$?

When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group. What is known ...
Jared White's user avatar
1 vote
1 answer
180 views

Conditioning a $\mathrm{C}^*$-algebra state with infinite precision

This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success. Let $\mathcal{A}$ be a unital $\...
JP McCarthy's user avatar
  • 1,027

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