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) ...

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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 ...
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16 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 ...
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33 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 $b$ ...
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41 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 ...
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56 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 $\varphi : \widetilde{A}\to M(A)$ ...
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92 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 ...
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34 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 ...
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29 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 $\tilde{A}$?
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57 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 ...
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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 ...
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135 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 ...
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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 ...
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The exact sequence of C*-algebra

Let $A, B$ be a C*-algebra and $\bar{A}$ be the unitization of $A$. If $||.||_{\alpha}$ is the C*-norm on $\bar{A}\odot B$ (algebraic tensor product), then can we prove the following sequence is a ...
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1answer
21 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 ...
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32 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 ...
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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 ...
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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 ...
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28 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 ...
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20 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 ...
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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 $B$?
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1answer
17 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 ...
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41 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 ...
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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 ...
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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 $A\odot B$ ...
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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 $\Phi: B(H)\rightarrow A^{**}$ ...
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26 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 ...
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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 ...
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1answer
54 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 ...
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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 ...
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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.
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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 $b\in B$?
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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? ...
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1answer
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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: ...
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1answer
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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 ...
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22 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 ...
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1answer
47 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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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 ...
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1answer
25 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, ...
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1answer
31 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 ...
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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 ...
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1answer
29 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 ...
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1answer
25 views

In a C*-algebra, does $|xyz| \leq |xz|^\frac{1}{2} | y| \ | xz|^\frac{1}{2}$ hold?

Let $x,y,z$ be elements of a C*-algebra $A$. Does the inequality $$|xyz| \leq |xz|^\frac{1}{2} | y| \ | xz|^\frac{1}{2}$$ hold in the positive cone of $A$? Here $|x| := (x^*x)^\frac{1}{2}$. If ...
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1answer
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Normal bounded operator

Let $T$ be a bounded normal operator on a Hilbert space. Now I have to show that $T$ is self-adjoint if and only if $\sigma(T) \subset \mathbb{R}$. I already know that for an Abelian unital ...
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1answer
46 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 ...
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1answer
22 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 $A\otimes_\alpha B$ ...
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1answer
23 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 ...
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1answer
39 views

An interesting phenomenon of $C^*$-tensor product

On the algebraic tensor product space of $C^*$-algebra, I try to find an example whose maximal $C^*$-norm is not the minimal $C^*$-norm, but it seems as it is impossible to do this because the finite ...
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1answer
28 views

Questions about multiplier algebra and corona algebra

When I read N.E. Wegge-Olsen's book K-theory and C-star-algebras_ A friendly approach I meet the following two problems about standard isomophisms: For any $C^\ast$-algebra $\mathcal{A}$, is ...
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98 views

application of c*algebras

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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52 views

an invariant of $C^{*}$ algebras

consider the following property (invariant) for complex $C^{*}$ algebras: "$T(x)=x^{*}$ is the only non zero $\mathbb{R}$-linear map on $A$ which satisfies $T(x)T(y)=T(yx)$." Questions: 1)Some ...