1
vote
1answer
41 views

Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
0
votes
0answers
23 views

Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
3
votes
2answers
28 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 ...
2
votes
1answer
22 views

How to prove the following map is a c.c.p map

Here is a quotation in a book "C*-algebrass and Finite-Dimensional Approximations" by Nate and Taka (P122). Let $A$ be a C*-algebra, $\Gamma$ be a discrete group and the $\alpha$ is an action of ...
1
vote
1answer
37 views

An exercise about reduced crossed product of a C*-dynamical system

Here is an Exercise in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If ...
1
vote
1answer
25 views

A question about cross product of C*-dynamical system

Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. ...
1
vote
1answer
24 views

$R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$ \exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad $$ implies $A$ is unital. I know this is true if A is a ...
2
votes
0answers
54 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
2
votes
1answer
28 views

A question on nuclearity

Definition 2.1.1. If $A$, $B$ are C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
2
votes
1answer
55 views

A simple lemma in tensor product

Here is a quotation of a book: ($\otimes$ denotes the minimal tensor product) Lemma 3.9.2. Let $A$ be a C*-algebra. If $E\subset A$ is an operator system and $J\triangleleft B$ is an ideal, then ...
2
votes
1answer
72 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
48 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
0answers
19 views

Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
-1
votes
1answer
37 views

Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
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
33 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
25 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentialky selfadjoint if it contains a dense subset of analytic vectors?
0
votes
1answer
61 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
1
vote
1answer
41 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$
1
vote
1answer
40 views

The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
3
votes
0answers
87 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
4
votes
1answer
49 views

*-representations of dense subalgebras

Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a ...
1
vote
0answers
29 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
1
vote
1answer
25 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 $f$ ...
1
vote
1answer
22 views

The exactness of a C*-algebra

Here is a quotation: Corollary 3.7.12 If $\Gamma$ is a non-amenable residually finite group, then $C^{*}(\Gamma)$ is not exact. It follows from this corollary that $B(l^{2})$ is not exact ...
1
vote
1answer
43 views

A simple description of $ {C^{*}}(\Gamma) \otimes_{\sigma} {C^{*}}(\Gamma) $ when $ \Gamma $ is finite.

Problem. Let $ \Gamma $ be a discrete group. Denote its full group $ C^{*} $-algebra by $ {C^{*}}(\Gamma) $. If $ \Gamma $ is a finite group, then is it true that $ {C^{*}}(\Gamma) \odot ...
2
votes
2answers
39 views

A definition of discrete group

Definition: A discrete group $\Gamma$ is called residually finite if there exist subgroups $\Gamma\supset\Gamma_{1}\supset\Gamma_{2}\supset...$ such that each $\Gamma_{i}$ is a finite-index, normal ...
0
votes
1answer
24 views

The commutant of reduced C*-algebra of a discrete group

For a discrete group $\Gamma$ we let $\lambda: \Gamma \rightarrow B(l^{2}(\Gamma))$ denote the left regular representation and $\rho$ denote the right regular representation. The reduced C*-algebra of ...
2
votes
1answer
19 views

An extension of representation

Let $A,~B$ be two C*-algebras, if $A$ is an ideal in $B$, then do we have that any representation of $A$ can extend to a representation of $B$?
1
vote
1answer
18 views

The norm on tensor product

Here is a quotation of a book: Let $B, ~C$ be unital C*-algebras and $A$ be a nonunital C*-algebra, $\|\cdot\|_{\alpha}$ be a C*-norm on $B\odot C$ (the tensor product) and $\|\cdot\|_{\beta}$ be ...
0
votes
1answer
44 views

The quotient embedding of tensor product

Here is a quotation of a book: Let $A, B$ be two $C^*$-algebras and $J\subset A$ be a $C^*$-subalgebra, then there is a dense embedding $$\frac{A\odot B}{J\odot ...
1
vote
1answer
28 views

Exact sequence of tensor product

Here is a quotation of a book: Proposition 3.7.1. If $0 \rightarrow J \rightarrow A \rightarrow (A/J)\rightarrow 0$ is an exact sequence, then for every $B$, the natural sequence $$0 \rightarrow ...
0
votes
0answers
26 views

The canonical quotient map between two tensor product [duplicate]

Let $A, C$ be two C*-algebras. Does there exist a canonical quotient map from $A\otimes_{max} C\rightarrow A\otimes C$? $A\otimes_{max} C$ (resp. $A\otimes C$) denote the completion of $A\odot B$ ...
1
vote
1answer
11 views

A simple question about Lance's weak expectation property.

Here is a quotation of a book: Definition 3.6.7. A C*-algebra $A\subset B(H)$ is said to have Lance's weak expectation property (WEP) if there exists a u.c.p map $\Phi: B(H)\rightarrow A^{**}$ ...
1
vote
1answer
27 views

A equivalent proposition of contractive completely positive map

Proposition 3.6.6. Let $A\subset B$ (C*-algebras) be an inclusion. Then the following are equivalent: (1). there exists a c.c.p.(contractive completely positive) map $\phi: B\rightarrow A^{**}$ such ...
1
vote
1answer
23 views

A proof of a proposition of tensor product

Proposition 3.6.5.(The Trick) Let $A\subset B$ and $C$ be C*-algebras, $||.||_{\alpha}$ be a C*-norm on $B\odot C$ and $||.||_{\beta}$ be the C*-norm on $A\odot C$ obtained by restricting ...
0
votes
1answer
22 views

The hereditary subalgebra

If $B$ is a C*-algebra and $A\subset B$ is a hereditary subalgebra, then , taking $\{e_{n}\}$ be the approximate unit of $A$, can we verify $e_{n}be_{n} \in A$ for every $b\in B$?
0
votes
1answer
40 views

The point-ultraweak convergence of contractive completely positive map

Let $A$ and $C$ be C*-algebras. If $\phi_{n}: A \rightarrow C$ is a c.c.p (contractive completely positive) map, then the point-ultraweak cluster point of the map $\phi_{n}$ is still a c.c.p. map? ...
1
vote
1answer
24 views

Completely bounded map and minimal tensor products

Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p.(completely positive) maps. Then the algebraic tensor product map $$\phi\odot\psi: ...
0
votes
1answer
34 views

The proof of (continuity of tensor product maps) theorem

Here is a proof of (continuity of tensor product maps) theorem: Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p. maps. Then the algebraic ...
0
votes
1answer
26 views

The restriction homomorphism

Let $||.||_{\alpha}$ be a C*-norm on $A\odot B$, $A\otimes_{\alpha} B$ be the completion and $\xi$ be a state on $A\otimes_{\alpha} B$. We definte the restrictions $\xi|_{A}$ and $\xi|_{B}$ as ...
0
votes
1answer
38 views

An exercise about minimal norm

Exercise 3.4.1. Let $\pi: A \otimes B \rightarrow C$ be a $*$-homomorphism which is injective when restricted to $A\odot B$. Show that $\pi$ must be injective on all of $A\otimes B$. Is this still ...
0
votes
1answer
26 views

The unitarily equivalent between two representations

Here is a quotation of a book: Let $\phi$ and $\psi$ be the faithful states on $A$ and $B$ respectively, and let $||.||_{\alpha}$ be any C*-norm on $A\odot B$ (algebraic tensor product). As we know, ...
1
vote
1answer
33 views

The tensor product of $M_{n}(\mathbb{C})$

There is a quotation below: Let $\{e_{i,j}\}_{1\leq i, j\leq n}$ be a system of matrix units fro $M_{n}(\mathbb{C})$ and consider $$\sum\limits_{i,j=1}^{n}e_{j, i}\otimes e_{j, i}.$$ A ...
0
votes
1answer
23 views

The restriction of representation of $A\otimes_{\alpha} B$

Let $||.||_{\alpha}$ be a C*-norm on $A\odot B$, $A\otimes_{\alpha} B$ be the completion and $\xi$ be a state on $A\otimes_{\alpha} B$. Let ($\pi_{\xi}, H_{\xi}, v_{\xi}$) be the GNS triplet and ...
0
votes
1answer
31 views

Several questions about state space

Here are several questions about the state space of a C*-algebra $A$: Let $A$ be a unital and separable C*-algebra, can we find a faithful state $\phi \in S(A)$. ( The $S(A)$ denotes the state space ...
0
votes
1answer
48 views

A lemma about the pure states

There is a quotation of a book: Lemma 3.4.5. Assume that both $A$ and $B$ are unital and abelian C*-algebras. Then for every C*-norm $\|\cdot\|_{\alpha}$ on $A\odot B$ and pair of pure states ...
0
votes
1answer
23 views

The GNS of a pure state on $A\otimes_\alpha B$

Let $A, B$ be the C*-algebras and $\|\cdot\|_\alpha$ be a C*-norm on $A\odot B$, $\xi$ be a state on $A\otimes_\alpha B$. (Here, $\odot$ denotes the algebraic tensor product and $A\otimes_\alpha B$ ...
0
votes
1answer
25 views

How to verify the isomorphism between two C*-algebra

Let $B$ be a C*-subalgebra of a unital C*-algebra $A$, how to verify $C^{*}(B, 1_{A})\cong \tilde{B}$? Here, $C^{*}(B, 1_{A})$ denotes the C*-algebra generated by $B$ and $1_{A}$, meanwhile the ...