2
votes
1answer
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 ...
1
vote
1answer
39 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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 $\tilde{A}$?
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 ...
2
votes
0answers
22 views

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 ...
1
vote
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 ...
1
vote
1answer
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 ...
2
votes
2answers
37 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
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 ...
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
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 ...
0
votes
1answer
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 ...
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
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 ...
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
20 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
38 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
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 ...
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
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, ...
1
vote
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 ...
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
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 ...
0
votes
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 ...
0
votes
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 $A\otimes_\alpha B$ ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
1answer
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. ...
3
votes
2answers
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 ...
0
votes
1answer
20 views

The commutative tensor product norm

Definition 3.3.3 (Maximal norm) Given $A$ and $B$, we define the maximal C*-norm on $A\odot B$ to be $$||x||_{max}=sup\{||\pi(x)||:\pi:A\odot B\rightarrow B(H) a *-homomorphism\}.$$ for $x\in A\odot ...
1
vote
1answer
55 views

The state on C*-algebra

Let $C$ be a C*-algebra, $A\subset C$ be a C*-subalgebra of $C$ and $B=A'\cap C$ (here, $A'$ denotes the commutant of $A$). If $\xi$ is a state on $C$ and we take an positive element $b\in B$, then ...
1
vote
1answer
32 views

An application of Hahn-Banach (separation) theorem

Here is a quotation of a book: Let $S(A)$ denote the state space of a C*-algebra $A$ and $M\subset S(A)$ denote a weak-$*$ closed convex set. Assume there is a state $\psi$ which does not belong ...
3
votes
0answers
33 views

The norm of operators in $B(H\otimes K)$

Let $H, K$ be separable Hilbert spaces and $P_{1} \leq P_{2} \leq ...$ be finite rank projections in $B(H)$ such that $P_{n}$ has rank $n$ and $||P_{n}h-h||\rightarrow 0$ for all $h\in H$. Then, how ...
2
votes
0answers
31 views

If there is already enough room to add all projections, does passing to matrices change anything?

Throughout, $A$ denotes a $*$-algebra. We always assume $A$ is representable in the sense that $A$ can be embedded into $B(H)$ for some Hilbert space $H$. The particular embedding is not important, ...
0
votes
1answer
53 views

Maximal norm of tensor product

Definition 3.3.3. (Maximal Norm) Given $A$ and B (two C*-algebra), we define the maximal C*-norm on $A \odot B$ to be $$||x||_{max}=sup\{||\pi(x)||: \pi: A\odot B\rightarrow B(H) ...
0
votes
1answer
30 views

How to verify $H\otimes K \cong \bigoplus\limits_{i\in I}H$

Let $H,~K$ be the Hilbert space. if $\{v_{j}\}_{j\in J}\subset H$ and $\{w_{i}\}_{i\in I}\subset K$ are the orthonormal bases, then how to construct the isomorphic mapping: $H\otimes K \rightarrow ...
0
votes
1answer
17 views

$*$-homomorphism of the tensor product

Let $A,~B,~C,~D$ be the C*-algebras and the "$\odot$" denotes the algebraic tensor product. Proposition 3.1.16 (Tensor product morphisms). Given $*$-homomorphisms $\phi: A\rightarrow C$ and $\psi: ...
2
votes
1answer
36 views

A linearly independent set about approximate units

Let $B$ be a C*-algebra and $\{b_{i}\}_{i=1}^{n}\subset B$ be linearly independent. If we take $\{f_{k}\}\subset B$ which is approximate units, then can we find a large $k$, such that $\{b_{1}f_{k}, ...
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
37 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
29 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
58 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
0
votes
1answer
39 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 ...
2
votes
1answer
44 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...