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Questions tagged [operator-algebras]

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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unital and completely isometric imply completely positive?

Let $B, C$ be unital $C^*$-algebras, $S\subset C$ an operator system, $f:S\to C$ a unital linear map. Then, $f$ is completely isometric if and only if $f$ is isometric and both $f$ and $f^{-1}:f(S)\...
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27 views

Can infinite matrices represent nonlinear operators?

I read this reddit post and this SE thread discussing how to represent nonlinear/linear transforms in matrix notations but they were not sufficient. In quantum mechanics, scientists use infinite ...
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1answer
32 views

Conditional expectation in free probability

Given is a unital $C^*$-algebra $A$ with state $\mathbb E$ and closed sub-$*$-algebra $B$ (unital, with the same identity). Think of $(A,\mathbb E)$ as a (possibly) non-commutative probability space. ...
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1answer
18 views

Enough existence of faithful normal states on von Neumann algebra acting on separable Hilbert space

Under which condition , given a von Neumann algebra $M$ acting on separable Hilbert space $\mathcal{H}$ have uncountable number of faithful normal states?
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25 views

How can I represent functionals in tensor-notation?

Recently I read this post: can-non-linear-transformations-be-represented-as-transformation-matrices So I concluded that normal matrices cannot represent every non-linear transformations. But I know ...
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2answers
53 views

How to calculate the $K_0$ and $K_1$ groups for $A$

Let $A=\{f\in C([0,1],M_n)\mid f(0)$ is scalar matrix $\}$. Then find the $K_0(A)$ and $K_1(A)$. I am trying to use the SES $J \rightarrow A \rightarrow A/J$ where $J$ can be taken as some closed ...
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0answers
29 views

Separable Commutative $C*$ algebra [duplicate]

Consider the algebra $C(X)$ of continuous complex functions over a compact space $X$. On what conditions this algebra is separable? What if $X$ is a compact subset of $\mathbb{R}^n$?
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1answer
15 views

On clarification on representation of operators on direct sum of Hilbert spaces as matrices

Suppose $G$ is a countable group, consider $\mathcal{H}=\oplus\{\mathcal{H}_{g}:g\in G\}$, Suppose $T$ is an operator in $\mathcal{H}$, then how to give a isomorphism from $B(\mathcal{H})$ into ...
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1answer
16 views

Two ordered relations on projections.

Let $A$ be a vn-algebra. Let $p$ and $q$ be two projections. In the literature, we say $p$ is majorised by $q$ if $pq=p$. Q. Suppose that $q-p$ is a positive element in $A$ (meaning $q-p=x^*x$ for ...
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1answer
33 views

A relation between projections

Let $A$ be a vn-algebra. Let $e,p$ and $q$ be projections in $A$. Suppose that $p\leq q$. True or false? $$q\wedge e-p\wedge e\leq q-p$$
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Can a projection in factor $R$'s commutant fail to meet a maximal projection in a factor that includes $R$?

If $\mathcal{R} \subseteq \mathcal{S}$ are factors acting on Hilbert space $H$, with $\mathcal{S}$ type I and $\mathcal{R}$ not type I, and $P$ is a maximal projection in $\mathcal{S}$ (meaning the ...
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32 views

Conditional Expectation for von Neumann algebra.

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
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1answer
36 views

Pure states on subalgebras of $\mathcal{B}(\mathcal{H})$ in finite dimensions.

I consider only finite-dimensional Hilbert spaces. We know that pure states on $\mathcal{B}(\mathcal{H})$ are exactly the vector states or in terms on density matrices, the rank one projections. My ...
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1answer
28 views

the induced trivial $*$ homomorphism

Let $A=c_{0}\oplus \mathbb{K}$,$I=c_{0}$ is the closed ideal of $A$,there is an induced $*$ homomorphism $\phi:A/I\rightarrow M(I)/I$,where $M(I)$ is the multiplier algebra of $I$.$\phi(a+I)=(L_{a},R_{...
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1answer
45 views

Matrix notations of binary operators (Multi-input operators)

In quantum mechanics, we can represent functions as vectors, and unitary operators as matrices, with given orthogonal function system (basis). I wonder if we could notate binary operators like + ,- ,×...
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28 views

On direct integral of states of von Neumann algebras

Suppose we consider a direct integral of GNS states of a measure space in von Neumann algebra, get the new state by direct integral. Does the GNS represenation of the state breaks down into direct ...
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0answers
36 views

Union of $C^*$-algebras generated by one element also generated by one element?

Let $A$ be a unital $C^*$-algebra and $a_1, a_2, ... \in A$ be elements such that $C^*(a_1, 1) \subseteq C^*(a_2, 1)\subseteq ...$ where $C^*(., 1)$ denotes the generated unital $C^*$-algebra. The ...
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45 views

$G$-invariant operators on $L^2(G)$

Let $G$ be a Lie group equipped with a left-invariant Haar measure. Then elements of $G$ act as bounded operators on $L^2(G)$, with action given by translation: $$(g\cdot f)(h):=f(g^{-1}h),$$ for ...
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1answer
35 views

Upgrading Injectivity of a *-homomorphism From a Dense Subalgebra

In the proof of Lemma A.4 of this document, the author proves that a C$^*$-algebra is simple by showing that every one of its representations is faithful. They proceed by taking a representation and ...
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38 views

Norm inequality for linear combination of non-commuting unitaries

Let $u, v$ be unitaries in a unital $C^*$-algebra satisfying $uv=e^{2\pi i \theta}vu$ where $\theta$ is irrational (so $\{e^{2 \pi i n \theta} : n \in \mathbb{Z} \}$ is dense in $\mathbb{T}$). For ...
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1answer
33 views

Does the algebra of real valued functions in Stone-Weierstrass Thm. equal to the set of polynomials?

I am studying Stone-Weierstrass Theorem. I wonder whether A is equal to the set of polynomials? If so, how can I proof this? And the statement is as follows: Let $S$ be a compact set, and let $A$ ...
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1answer
48 views

An statement concerning finite rank projections

Let $H$ be a Hilbert space. Let $T$ be an isometric operator on $H$. Suppose that $P$ is a finite rank projection with $PT^nP=PT^n$ for every $n\geq1$. Q. Can we conclude that $P(H)\subseteq \...
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1answer
42 views

Numerical range of selfadjoint elements in non-unital C*-algebras

If $a$ is an element of a C*-algebra $A$ then $V(a)=\{\varphi(a): \varphi\text{ is a state of }A\}$ is the numerical range of $a$. If $a$ is selfadjoint and $A$ is unital then it is known that $V(a)=[\...
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3answers
76 views

What's wrong with my proof that $\sigma(a)\subseteq[-\|a\|, \|a\|]$ for $a$ self-adjoint?

Let $U$ be a $C^*$-algebra and $a\in U$ be self-adjoint. I have a simple proof that $\sigma(a)\subseteq [-\|a\|,\|a\|]$, where $\sigma(a)$ is the spectrum of $a$. It goes as follows (the facts used ...
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1answer
46 views

About unitary group of a von Neumann algebra

Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?
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1answer
46 views

the uniform norm of a positive linear map on von Neumann algebras

Let $H$ be a Hilbert space. Let $\phi:B(H)\to B(H)$ be a positive linear map. Q. Do we have $\|\phi\|=\sup\{\|\phi(x)\| : 0\leq x\leq1\}$?
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1answer
36 views

Finite dimensional von Neumann algebra [closed]

How to prove that finite dimensional von Neumann algebra is direct sum of matrix algebras?
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1answer
45 views

minimal projections in finite dimensional von Neumann algebras

The algebras I'm working with are defined as follows Let $\mathcal{H}$ be a Hilbert space of finite dimension and denote by $\mathcal{B}(\mathcal{H})$ the bounded operators on $\mathcal{H}$. A ...
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1answer
33 views

Monic projections in finite von Neumann algebra

The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation ...
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27 views

Does the following metric metrize the weak*topology on the state space?

Let A be a finite-dimensional C*-algebra. In a paper of Rieffel, it is shown that its state space, S(A), equipped with the energy metric, is isometrically embedded in a Hilbert space. Does this imply ...
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1answer
58 views

Is the double commutant $A''$ commutative if $A$ is commutative?

If $A$ is a commutative C*-subalgebra of linear bounded operator space $B(H)$ on some Hilbert space $H$, so is the double commutant $A''$. It follows from $A$ is dense in $A''$ and the multiplication ...
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2answers
35 views

Banach algebras with trivial center

Let $A$ be a Banach algebra. The center of $A$, denoted by $Z(A)$, is the set of elements of $A$ that commute with all elements of $A$. Please give some examples of Banach algebras with trivial center....
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2answers
28 views

Bounding a positive element from below using a dense subset

Let $A$ be a $C^*$-algebra. Let $a$ be a nonzero positive element of $A$. Suppose that $A$ equals the closed span of a subset $B$ of $A$, where $B$ is closed under the $*$ operation, linear ...
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1answer
15 views

Does there exist a connection between contractive completely positive map and surjective map

If $\psi:A \rightarrow M_n(\mathbb{C})$ is a c.c.p map.What is the relationship between c.c.p maps and surjective maps?Can we deduce that $\psi$ is a surjective map?If not,does there a close ...
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1answer
45 views

State space of finite dimensional, abelian C*-algebra is a simplex.

I am looking for a proof that the state space of a finite dimensional C*-algebra is a simplex and, vice versa, if the state space is a simplex, the C*-algebra is abelian. I've found one proof, but it ...
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1answer
16 views

limit of sequence of bounded operators

Suppose $\phi_n:A\rightarrow B(H_n)$ is a sequence of nonzero representations, where $A$ is a nonunital $C^*$-algebra,$H_n$ is a Hilbert space, and $P_n$ is a sequence of projections on $H_n$. Does ...
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1answer
42 views

Identifying tensors with functions in $C^*$ -algebras

We know the result $C(Y,C(X)) \cong C(X)\otimes C(Y)$, I don't able to construct the isomorphism mapping that by starting an arbitrary function $f$ from $C(X,Y)$, how to get tensor element in $C(X\...
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1answer
35 views

If an operator $X$ on a Hilbert space satisfies $X^*=-X$, then is $X$ equal to $0$? [closed]

If a linear operator $X$ on a Hilbert space satisfies $X^*=-X$, then is $X$ equal to $0$? $X^*$ is the adjoint.
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0answers
31 views

Structure theory of type 1 von Neumann algebras

Why structure theory of Type 1 von Neumann algebras are coming from spectral theorem? I read Arveson, their heavy technical things are used of multiplicity theory (Hahn-Hellinger Theorem) to get the ...
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1answer
46 views

If $r(a) < 1$, does $\sum_{n=0}^\infty a^{*n}a^n$ necessarily converge?

In Murphy, exercise 2.6: Let $A$ be a unital C$^*$-algebra. If $r(a) < 1$ and $b = (\sum_{n=0}^\infty a^{*n}a^n)^{1/2}$, show that $b \geq 1$ and $\lVert bab^{-1}\rVert < 1$.... where $r(a)$ ...
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0answers
61 views

MASAS in finite von Neumann algebras

While reading the book "Finite vN algebras and Masas", I realized the following facts. A maximal abelian self adjoint algebra $A$ in a type $II_{1}$ factor $N$ is * isomorphic to $L^{\infty}[0,1]$. ...
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1answer
69 views

Convergence of a net in a Hilbert space

Suppose $\phi:A\rightarrow B(H)$ is a nonzero $*$ homomorphism, where $A$ is a nonunital $C^*$ algebra, $H$ is a Hilbert space, $\{x_{i}\}$ is a net of unit vectors in $H$, does there exist $a_0\in A$ ...
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1answer
34 views

MASAS in Type $II_{1}$ factor

Let $N$ be a type $II_{1}$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?
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1answer
52 views

Representations of simple C$^*$-algebras

I am reading from the following document, and am a bit stumped by footnote 4 on page 5: https://arxiv.org/pdf/math-ph/0006011.pdf Actually, I will copy the relevant text because it disappears off ...
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1answer
99 views

Why do we need the spectral theorem? What is its purpose?

One realization of spectral theorem for me that we want to make sense "the object $:f(T)$" in von Neumann algebra $M$ where $f$ is bounded measurable function with respect to some measure. ...
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3answers
46 views

Finding eigenfunction to this operator

I have the operator $$ A : -e^{-2ax} \frac{\partial}{\partial x} \left(e^{2ax}\frac{\partial}{\partial x}\right)\\ D_A = \left( v \in C^2[0,L] \quad | \quad v(0) = v(L) = 0 \right) $$ ...
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0answers
34 views

The Image of Unitary Representation in the Space of Bimodules

TL;DR: Bimodules over a von Neumann algebra are commonly understood as a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $\mathcal{L} G$, the unitary ...
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1answer
32 views

$*$ homomorphism $\phi$ from $A$ to multiplier algebra $M(I)$

If $I$ is a closed ideal in $C^*$ algebra $A$, then there is a unique $*$ homomorphism $\phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $I\to M(I)$, where $M(I)$ is the multiplier algebra ...
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1answer
24 views

infinite representation of a $C^*$ algebra

Suppose $\pi$ is a finite dimensional representation of $C^*$ algebra $A$ on a Hilbert space $H$,then $\pi$ is the direct sum of finite dimensional irreducible subrepresentations. My quesion is :If $\...
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1answer
50 views

Some particular example clarification on algorithm of Hahn Hellinger Theorem

Consider the self adjoint matrix $T$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 &2 \end{bmatrix} The question is the following: I want to understand the ...