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

1
vote
1answer
27 views

Why is the natural map from maximal to reduced C star algebra surjective?

In the book "Kazhdan' property (T)" the third book in this link, page 438, One sentence is "the regular representation defines a surjective *-homomorphism $\lambda_G: C^{*}(G)\to ...
4
votes
1answer
56 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
1
vote
1answer
20 views

Expressing a hereditary subalgebra in terms of a state

Let $\mathcal{A}$ be a C*-algebra and $\phi$ a state on $\mathcal{A}$. Then, it's not hard to see that $\mathcal{L} = \left\{ x : \phi(x^*x)=0 \right\}$ is a closed left ideal of $A$ and so ...
2
votes
1answer
123 views

Direct limits and pullbacks

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow ...
2
votes
0answers
49 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
0answers
25 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 ...
0
votes
1answer
17 views

Need some help understanding why this function is well defined and why it maps into $S^1$

Let $A$ be a unital $C^\ast$ algebra. Let $\varphi$ be the Gelfand transform. Let $u$ be unitary and $f = \varphi (u)$. Let $\ln: \mathbb C \setminus [0,\infty) \to \mathbb C$ denote the natural ...
0
votes
0answers
21 views

A question on finite-dimensional operator system

Let $A$, $B$ be C*-algebras, $E\subset A$ be an operator system and $J\triangleleft B$ be an ideal. Can we verify the union of the subspaces $\frac{E\otimes B}{E\otimes J}$, where $E\subset A$ is a ...
2
votes
1answer
20 views

If $A$ is a primitive / irreducible C*-algebra, then $M(A)$ has trivial center.

Recall some definitions: a sub-C*-algebra $A$ of $B(H)$, the algebra of bounded operators on a Hilbert space $H$, is called irreducible if the only closed $A$-invariant subspaces of $H$ are $0$ and ...
0
votes
1answer
45 views

Why does a real spectrum not have holes?

Apparently, if $A$ is a $C^\ast$-algebra and the spectrum $\sigma(a) \subseteq \mathbb R$ then $\sigma (a)$ does not have holes. I read this in Murphy and then tried to prove it since I didn't ...
2
votes
0answers
26 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
2
votes
1answer
24 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
51 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
61 views

Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
0
votes
1answer
44 views

Faithful Representations of C*-algebras

Can anyone give me an example of a represetation of the algebra $M_n(\mathbb{C})$ that is not faithul? If it's not possible, could you explain me why it is not?
2
votes
1answer
41 views

C*-algebra representations

Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ ...
2
votes
1answer
25 views

non-faithful KMS states

I am wondering whether a state $\omega$ on a $C^*$-algebra which is KMS (http://en.wikipedia.org/wiki/KMS_state) with respect to the group of automorphisms $\tau^t$, $t\in\mathbb{R}$, and at a given ...
2
votes
1answer
24 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
35 views

Where is commutativity of $b$ needed?

I have a question about the following proof: If $e^{ia}-e^{i\lambda}=(a-\lambda)be^{i\lambda}$ and $(a-\lambda)$ is not invertible then $(a-\lambda)x$ is not invertible for all $x$. Why "since $b$ ...
1
vote
1answer
46 views

A question on tensor product of $C^{*}$ algebras

Let $A$ and $B$ be two $C^{*}$ algebras. Assume that every element of the minimal tensor product $A\otimes_{min} B$ is a finite linear combination of simple tensors $a\otimes b$. Can we say that ...
0
votes
2answers
57 views

How to prove this map is injective

Let $A$ be a non-unital $C^\ast$ algebra and let $M(A)$ denote the multiplier algebra and let $\widetilde{A}$ denote the unitisation of $A$. Consider the map $\varphi : \widetilde{A}\to M(A)$ ...
2
votes
1answer
105 views

Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
1
vote
1answer
37 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 ...
1
vote
1answer
30 views

Minimal and maximal unitization of $C^{*}$ algebras

Is there a non unital $C^{*}$ algebra $A$ for which the multiplier algebra $M(A)$ is isomorphic to the minimal unitization $\tilde{A}$?
1
vote
1answer
60 views

Isn't the center of a von Neumann algebra on a separable Hilbert space a hyperfinite von Neumann subalgebra?

this is a very quick, probably dumb, question, I was reading this chapter from "Hochschild cohomology of von Neumann algebras" by Allan Sinclair and Roger M. Smith and I came across this theorem on ...
0
votes
1answer
31 views

Coproduct of $C^*$-algebras

I want to prove that the free product $A*B$ of two unital $C^*$-algebras $A$ and $B$ is a coproduct in the sense of category theory. Remember the construction of $A*B$: Take generators $\{a:a\in ...
3
votes
1answer
136 views

Pure states and irreducible *-representations

If $A$ is a $C^*$-algebra, then a $*$-representations of $A$ is irreducible if and only if the corresponding state is a pure state. What happens if don't insist on $A$ being a $C^*$-algebra? Is one ...
2
votes
1answer
18 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
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
34 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 ...
2
votes
1answer
32 views

Applications Gelfand-Naimark-Segal Theorem.

I'm reading the book An Introduction to Operator Algebras (By Kehe Zhu). but do not know how to do the following exercise: Let $A$ be the commutative C*-algebra $C(\partial D)$. For any $z\in ...
0
votes
1answer
29 views

Free product of $C^*$-algebras

i want to figure out what the "free product of two $C^*$-algebras" precisely is. Thus suppose $A$ and $B$ are $C^*$-algebras. How to find or construct $A\ast B$? I readed something about the ...
0
votes
1answer
21 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
42 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
25 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 ...
3
votes
1answer
57 views

Module homomorphism

Let $A$ be a Banach algebra with norm $\|.\|_A$ and $X$ be a Banach space with norm $\|.\|_X$. If there exists a operation $.:A\times X\to X$ such that for any $a,b\in A$ and $x,y\in X$ we have ...
3
votes
1answer
37 views

Non-existence Tracial states

We know that every non-zero finite dimensional C*-algebra has a tracial state. I am searching for an example of a simple C* algebra without tracial state with explaination. I think you have to look to ...
2
votes
2answers
39 views

About a spectrum of a C*-algebra

Let $A$ be an unital commutative C*-algebra. Show that the spectrum of $A$ is disconnected iff there is a projection $p \in A$ not trivial.
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
39 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
23 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
31 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
23 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 ...