A C*-algebra is a Banach algebra together with an isometric involution satisfying (ab)* = b*a* and the C*-identity |a*a| = |a|^2. This characterizes the closed subalgebras of the bounded operators on Hilbert space, closed under taking the adjoint operator. They are at the heart of (spectral-theory) ...

learn more… | top users | synonyms

5
votes
2answers
155 views

Existence of norm for C*-Algebra

I was wondering wether one can always find/construct a norm which turns an algebra into a C*-algebra. For sure, if it exists it is unique, but does it always exist. If not can you provide a simple ...
3
votes
2answers
149 views

How to show weak*-closed in the unit ball?

I have a unital C* algebra $\mathcal A$ and a subset $K \subseteq B_1^* \subseteq \mathcal A^*$ of the unit ball in the dual space. I want to show that K is compact in order to apply the Krein-Milman ...
8
votes
1answer
85 views

Physical interpretation of C* property

From a mathematical point of view, quantum mechanics can be formulated in the language of a non-commutative, unital C*-algebra $\mathcal A$ (of observables). In this context, what does the ...
2
votes
0answers
67 views

Cap product between K-Theory and K-Homology

In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative. Is the commutativity ...
1
vote
0answers
51 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
2
votes
1answer
51 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
3
votes
1answer
48 views

When is the quotient algebra of a unital C* algebra helpful?

Let $\mathcal A$ be a unital C* algebra. Which properties does $\mathcal B \subset \mathcal A$ has to have for it to make sense to form the quotient algebra $\mathcal A / \mathcal B$? In cases ...
1
vote
0answers
75 views

Why is a *-homomorphism isometric, if it maps strictly positive elements to strictly positive elements?

I have the following exercise: Let $\pi:\mathcal A \rightarrow \mathcal B$ be a *-homomorphism between two unital $C^*$ algebras $\mathcal A$ and $\mathcal B$ which maps the unit to the unit. Assume ...
1
vote
1answer
104 views

In what sense are self adjoint elements of C*-algebra “closed”?

Let $\mathcal A$ be a unital C*-algebra. I read the fact, that the self-adjoint elements $\mathcal S:=\{A\in \mathcal A \vert A^*=A\}$ are closed in $\mathcal A$. In what sense is "closed" meant ...
2
votes
1answer
84 views

Norm inequality for positive element of C* algebra

Let $\mathcal A$ be a unital C* algebra and $A\in \mathcal A$ a positive element $A\geq \mathbb 0$, i.e. A is self-adjoint and the spectrum satisfies $\sigma(A)\subset [0,\infty)$. Let $\alpha \in ...
2
votes
0answers
51 views

When does the Gelfand transform map into positive functionals?

Let $\mathcal A$ be a unital C* algebra. Consider the Gelfand transform on the unital, commutative C* algebra $\mathcal A_A$ generated by $A$, i.e. $\Gamma: \mathcal A_A \rightarrow C(\mathcal ...
5
votes
1answer
73 views

characters of a $C^*$-algebra

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
3
votes
1answer
58 views

Does *-operator be automatically continous

In the C*-algebras, does the * -operator be automatically continous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continous. Am I right? ...
1
vote
0answers
74 views

Unitary equivalent

In general, if two irreducible representations of a $C^*$-algebra have the same kernel we can say this two representations are approximately unitarily equivalent. When our $C^*$-algebra is GCR, how to ...
4
votes
1answer
60 views

When is the image of a GNS representation WOT-dense?

Given a $C^*$-algebra $A$ and a state $\rho$ on $A$, let $\pi_\rho$ be the corresponding GNS representation on the Hilbert space $H_\rho$. I would like know when the image of $\pi_\rho$ is WOT-dense ...
4
votes
1answer
155 views

Quotients of C*-algebras

It is known that every unital separable C*-algebra is a quotient of the full group C*-algebra $C^*(F_I)$, where $F_I$ is the free group generated by some index set $I$. Can we drop the ...
2
votes
1answer
105 views

Quotients of the maximal tensor product

Let $A$ and $B$ be C*-algebras and let $\gamma$ be any C*-norm on the algebraic tensor product $A\odot B$. Why is $A\otimes_\gamma B$ a quotient of $A\otimes_{{\rm max}}B$, where $\otimes_{{\rm max}}$ ...
4
votes
0answers
143 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
3
votes
0answers
116 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
1answer
54 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
5
votes
1answer
68 views

States on a C*algebra

A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$. I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
1
vote
1answer
85 views

On the Spectral Theorem

Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure. a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
3
votes
1answer
139 views

States and positive elements in $C^*$-algebras

Let $A$ be a unital $C^*$-algebra and $w$ be a state (i.e a positive linear functional such that $\|w\|=w(1_A)=1$. I'm trying to prove the following:a) if $a$ is selfadjoint and $w(a^2)=w(a)^2$ then ...
2
votes
1answer
72 views

A question concering nuclearity

B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product ...
1
vote
1answer
44 views

Are self-adjoints elements norming?

Let $\mathsf A$ be a C*-algebra and let $\mathsf{A}^*$ be its dual space. Is it true that for $f\in \mathsf A^*$ we have $$\|f\|=\sup\{|f(x)|\colon\; \|x\|=1\mbox{ and }x^* = x\}?$$
0
votes
0answers
33 views

Finding Strictly Positive Elements [duplicate]

I need to find the set of strictly positive elements in the $C^*$-algebra $C_0(\Omega)$ where $\Omega$ is a locally compact Hausdorff space. Clearly, the set will be contained in $ \{ f \in ...
2
votes
2answers
132 views

Positive elements in $C^*$-algebras

I'm trying to prove the following, and I'm not sure if the proof is correct? If $A,B$ are $C^*$-algerbas, and $f$ is a $*$-homomorphism from $A$ onto $B$ then $f(A_+)=B_+$.Proof: let $a\in A_+$ then ...
2
votes
1answer
77 views

A question about positive elements in $C^*$ algebras

Let $A$ be a $C^*$-algebra If $a\in A$ is positive, is it true that for any $0<\alpha<\frac{1}{2}$ we have $$\left(a+\frac{1}{n}1\right)^{\frac{-1}{2}}a^{\frac{1}{2}-\alpha}$$is self adjoint?A ...
2
votes
1answer
317 views

strictly positive elements in $C^*$-algebra

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to find the following:a)What are the strictly ...
6
votes
1answer
288 views

Cube root in $ C^{*}$-algebra.

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
2
votes
1answer
59 views

Help with proving: If $X$ is a Hilbert $A$-$B$-module, then $ \| _A \langle x,x \rangle \| = \| \langle x,x \rangle _B \| $ for all $x\in X $.

Sorry, I posted a related question last week on here, but I'm still having trouble and this is a little different, I hope it's OK. Thank you! ( proof that this is an isometric map (on a $C^*$-module) ...
3
votes
1answer
203 views

$*$-homomorphism between matrix algebras

Let $\theta:M_3(\mathbb{C}) \to M_7(\mathbb{C})$ be a $*$-homomorphism. Since matrix algebras over a field are simple, we know that $\theta$ must be injective. After reading this I conclude that, up ...
1
vote
0answers
67 views

Does a $C^*$ subalgebra of the centralizer of a unitary representation always contain the unit?

I am studying a theorem in Folland's "Course in Abstract Harmonic Analysis" where the following ingredients/assumptions are needed: $G$ a locally compact group, $\pi$ a unitary representation of ...
0
votes
1answer
52 views

Separable reducing subspace of a representation

I'm looking for a hint or a reference to understand what's going on in the following problem. Suppose that $A$ is a unital $C^*$-algebra, and let $\pi : A \rightarrow B(H)$ be a representation. ...
0
votes
1answer
58 views

proof that this is an isometric map (on a $C^*$-module)

Are my steps right? I'm not sure about the statement in bold below. Let $A$ be a $C^*$-algebra. Let $X$ be an $A$-module. Let $x\in X$, let $a= \langle x,x \rangle $ Define $\lambda _a (z) = az$, ...
3
votes
1answer
191 views

Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
10
votes
1answer
285 views

In a C*-algebra, put $a^*a \sim aa^*$. Transitivity fails?

Idle curiosity drove me to wonder about the following question. Let $A$ be a C*-algebra. Define a binary relation $\sim$ on the cone $A^{\geq 0}$ of positive elements by putting $x \sim y$ whenever ...
1
vote
1answer
105 views

State space is weak* compact

I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* ...
2
votes
1answer
51 views

Set of states not compact

I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course). I think $K(H)$, the compact operators over a Hilbert space ...
3
votes
1answer
54 views

Why should we use inverse homemorphism here?

I am a brand-new comer in dynamical system. I find it interesting that when defining ergodicity of classical dynamical system $(X,\sigma)$, they use $\mu(\sigma^{-1}(E))$ there. Since $\sigma$ is a ...
3
votes
2answers
87 views

*-algebra, isomorphic to a C*-algebra

A C-*-algebra A is *-isomorphic to a *-algebra B. Any examples when in this case B is not A C-*-algebra?
0
votes
1answer
184 views

positive linear functionals are bounded in $C^*$-algebras

Why is every positive linear functional bounded in $C^*$-algebras?
3
votes
1answer
71 views

Double centralizers in the Murphy book

I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake: ...
2
votes
1answer
59 views

Trying to prove: Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle $ is norm dense in $E$.

Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle$ is norm dense in $E$. I am having trouble proving this. I believe $\langle E,E\rangle $ is a $C^*$-algebra. If I can show this, then the ...
2
votes
1answer
77 views

A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
2
votes
2answers
94 views

Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want ...
10
votes
2answers
687 views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
4
votes
2answers
98 views

$(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. ...
4
votes
1answer
111 views

Spectrum of elements in $C^*$-subalgebras

Assume $\mathcal{A}$ is a $C^*$-algebra with unit $1$ and $\mathcal{B}\subset\mathcal{A}$ is a $C^*$-subalgebra (i.e. a closed $*$-subalgebra) such that $1\in\mathcal{B}$. It is said that under these ...
6
votes
0answers
105 views

K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...