0
votes
2answers
28 views

Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
3
votes
1answer
34 views

A detail in Rădulescu's Theorem proof

I've been following one Rădulescu's Theorem proof ($G$ is hyperlinear if and only if $\mathcal{L}_{G}$ can be embedded in a ultrapower of the hyperfinite type-$II_{1}$ factor $\mathcal{R}$, where ...
1
vote
2answers
33 views

Strong closure of a C*-algebra of operators.

In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator ...
2
votes
0answers
56 views

Dual subfactor and commutant

Let $(N \subset M)$ be a subfactor and $N \subset M \subset M_1$ the basic construction. Question: Is $(M \subset M_1) \simeq (M' \subset N')$? Else in which generic case it's true? What's the ...
3
votes
2answers
31 views

Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
1
vote
1answer
32 views

Is $vN(M_1,M_2) \cap M_3= vN(M_1,M3) \cap vN(M_2,M3)$?

Let $M_1,M_2,M_3$ be von Neumann algebras (i.e. weakly closed subalgebras of $B(H)$ where $H$ is a Hilbert space). Let $vN(M_1,M_2)$ denote the von Neumann algebra generated by $M_1$ and $M_2$ inside ...
1
vote
1answer
38 views

Abelian projection in the von-Neumann algebras

As we know, a minimal projection must be Abelian, An Abelian projection must be finite. A minimal projection correspond to a rank one operator, a finite projection correspond to a finite rank ...
2
votes
1answer
38 views

Do two II$_1$-factors with trivial intersection generate $B(H)$?

Let $H$ be an infinite dim. separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $A$, $B \subset B(H)$ be II$_1$-factors such that $A \cap B = \mathbb{C}I$. Examples: (1) Take ...
1
vote
0answers
81 views

Question on amenable direct summand

Given a finite Von Neumann algebra $(N,\tau)$, and Von Neumann subalgebras $A\subset B$ with the same identity, I came across the fact saying that $B$ has an amenable direct summand implies $A$ has an ...
1
vote
1answer
32 views

find element in relative commutant of a matrix subalgebra

Let $M=A*P$ to be a free product von Neumann algebra, and $A$ is a finite dimensional subalgebra, for simplicity, we may assume $A=M_2(\mathbb{C})$, and $P\neq \mathbb{C}$. A standard fact is that ...
3
votes
0answers
30 views

Understanding minimal projections

This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition. Let $A$ be a C*-algebra and let ...
2
votes
0answers
49 views

von Neumann Algebras and measures

I read that any abelian von Neumann algebra is isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-commutative measurable ...
2
votes
1answer
80 views

No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
1
vote
0answers
53 views

Ultra weakly continuous trace on a von Neumann Algebra

Let $M$ be a infinite dimensional von Neumann Algebra with a positive, faithful, ultra weakly continuous trace $tr:M\rightarrow \mathbb{C}$. Is it possible to show that $tr$ is strongly continuous?
3
votes
1answer
29 views

A question on a lemma about the product map

Here is a Lemma in the book “C*-algebras and Finite-Dimensional Approximations”: Lemma 3.8.4. Let $A$ be a C*-algebra, $M\subset B(H)$ be a con Neumann algebra and $\phi: A\rightarrow M$ be a ...
0
votes
1answer
24 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
0
votes
1answer
27 views

Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
2
votes
2answers
34 views

Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
1
vote
1answer
36 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
1
vote
1answer
26 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
0
votes
1answer
30 views

$E(a)=0\Longrightarrow E(a^{n})=0$?

Let $(M; \tau)$ be the hyperfinite $II_{1}$-Factor and consider a W${}^{\ast}$-subalgebra, $N$. Is there a (trace-preserving) conditional expectation, $E:M\to N$? Considering, now, a more general ...
0
votes
1answer
15 views

A symbol of commuting ranges in tensor product

Here is a proposition of tensor product: ($A,~B,~C$ are C*-algebras) Proposition 3.1.17 Given two *-homomorphisms $\pi_{A}: A\rightarrow C$ and $\pi: B\rightarrow C$ with commuting ranges (i.e., ...
0
votes
1answer
38 views

The multiplication of tensor product

Proposition 3.1.15 (Multiplication). Let $A$, $B$ be C*-algebra, the tensor product $A\odot B$ (denotes the algebraic tensor product) has a multiplication defined by $$(\sum\limits_{i}a_{i}\otimes ...
1
vote
1answer
33 views

The involution of tensor product

Proposition 3.1.8 (Linear independence). If $\{x_{1},...,x_{n}\}\subset X$ are linearly independent, $\{y_{1},...,y_{n}\}\subset Y$ are arbitrary and $$0=\sum\limits_{i=1}^{n}x_{i}\otimes y_{i}\in ...
2
votes
1answer
35 views

Zhou operator theory book, Kaplanskys formula

In Zhou's operator theory book, Kaplanskys formula has stated that if $P$ and $Q$ are projection in a von neumann algebra $A$ acting on $H$, then $P\vee Q-Q\sim P-P\wedge Q$. In the proof, it says ...
0
votes
1answer
42 views

A representation of von Neumann algebra of type I

I am reading a book "C*-algebras and Finite-Dimensional Approximations". There is a quotation below: For infinite-dimensional Hilbert space $H$ and a abelian von Neumann algebra $A$, we can represent ...
0
votes
1answer
23 views

A inequality about pointwise absolute value vectors

Let $\Gamma$ be a discrete group and $\xi\in l^{2}(\Gamma)$ be a unit vector. If $|\xi|$ be the pointwise absolute value of $\xi$, then how to verify: ($S$ is a linear bounded operator on ...
1
vote
1answer
33 views

The integral about probability measures

Definition For a discrete group $\Gamma$, we let Prob$(\Gamma)$ be the space of all probability measures on $\Gamma$: $$Prob(\Gamma)=\{\mu\in l^{1}(\Gamma): ...
1
vote
1answer
49 views

Universal property about discrete group in C*-algebra

Universal property: Let $u:\Gamma \rightarrow B(H)$ be any unitary representation of $\Gamma$. Then, there is a unique $*-$homomorphism $\pi:C^{*}(\Gamma) \rightarrow B(H)$ such that ...
1
vote
1answer
72 views

Amenable group in C*-algebra

Definition 2.6.1. A group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
0
votes
1answer
47 views

Positive definite function on discrete group in C*-algebra

Recall A function $\phi: \Gamma\rightarrow\mathbb{C}$ is said to be positive definite if the matrix $$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set $F\subset ...
1
vote
1answer
30 views

A simple question about 1-norm

Let $\Gamma$ be a discrete group, if $\mu \in l^{1}(\Gamma)$, then what is the 1-norm of $\mu$, I mean $||\mu||_{1}=?$. As we know, $l^{1}(\Gamma)=\{(\alpha_{x})_{x\in\Gamma}: ...
0
votes
1answer
73 views

A question about reduced C*-algebra of discrete group

There is a quotation below: Let $\Gamma$ be a discrete group and $\Lambda\subset \Gamma$ be a subgroup. The right cosets give a direct sum decomposition $$l^{2}(\Gamma)\cong\bigoplus l^{2}(\Lambda ...
0
votes
1answer
38 views

An easy (I guess) question about vector state in C*-algebra

I meet with some problems when I read a book about C*-algebra. Definition 2.5.10. Let $\phi:\Gamma \rightarrow \mathbb{C}$ be a function ($\Gamma$ is a discrete group here). We define a corresponding ...
0
votes
1answer
14 views

A simple question on infinite dimensional von Neumann algebra

Recall a projection $p\in N$ is called abelian if $pNp$ is an abelian algebra. If $N$ is a von Neumann algebra without abelian projections, then can we conclude that $N$ must be infinite dimensional? ...
1
vote
1answer
37 views

About Fourier transform

The reduced C*-algebra of $\Gamma$, denoted $C^{*}_{\lambda}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm $$\|x\|_{r}=\|\lambda(x)\|_{\mathbb{B}(l^{2}(\Gamma))},$$ The ...
1
vote
1answer
23 views

A question about the positive definite function

Definition 2.5.6. A function $\phi:\Gamma \rightarrow \mathbb{C}$ is said to be positive definite if the matrix$$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set ...
0
votes
1answer
39 views

A question about compact operator

For a discrete group $\Gamma$, $T\in \mathbb{B}(l^{2}(\Gamma))$ is constant down the diagonals-meaning that for every $s, t, x, y\in \Gamma$ such that $ts^{-1}=yx^{-1}$, we have $\langle T\delta_{s}, ...
0
votes
1answer
55 views

A question about full group C*-algebra

There is a quotation below: $\qquad$The $full~group$ C*-algebra of $\Gamma$, denoted $C^{*}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm ...
0
votes
1answer
29 views

The one to one map between two representations

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular ...
0
votes
1answer
35 views

A question on left regular representation of a discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}~$ for all ...
1
vote
1answer
60 views

Full (or universal) group C*-algebra of discrete group $\Gamma$

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums ...
0
votes
2answers
48 views

A question on discrete group

There is a quotation below: For a discrete group $\Gamma$ , $f\in l^{\infty}(\Gamma)$ and $s, t\in \Gamma$. We let $s.f \in l^{\infty}(\Gamma)$ be the function $s.f(t)=f(s^{-1}t)$. My question is ...
0
votes
1answer
66 views

Support and range projections in von Neumann algebra

There is a quotation below: Let $M$ be a von Neumann algebra, take a noncentral projection $p\in M$ and find some $m\in M$ such that $pm(1-p)\neq0$. The partial isometry in the polar decomposition of ...
1
vote
2answers
46 views

A question about discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, ...
1
vote
1answer
20 views

Strong operator sum of corner projections is a normal map

Suppose that we are given a Hilbert space $H$ with an orthogonal basis $(e_i)_{i\in I}$ and let $P_i$ denote the projection of $H$ onto $\mathbb{C}e_i$. Then we can consider the map ...
0
votes
1answer
41 views

The direct sum of two nuclear C*-algebra

Recall: Definition 2.1.2 If $A$ is a C*-algebra and $N$ is a von Neumann algebra, a map $\theta:A \rightarrow N$ is called weakly nuclear if there exist c.c.p. maps $\phi_{n}: A\rightarrow ...
2
votes
1answer
36 views

Are contractive completely positive maps trace decreasing?

Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is ...
0
votes
1answer
30 views

An exercise about nuclear C*-algebra

Definition 2.3.1. A C*-algebra $A$ is nuclear if the identity map id$_{A}:A \rightarrow A$ is nuclear. Exercise 2.3.7. If for each finite set $F\subset A$ and $\epsilon>0$ one can find a nuclear ...
1
vote
1answer
52 views

A question about nuclear C*-algebra

Definition 2.3.1. A C*-algebra $A$ is nuclear if the identity map $id_{A}: A\rightarrow A$ is nuclear. Definition 2.3.2. A C*-algebra $A$ is exact if there exists a faithful representation $\pi:A ...