A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

learn more… | top users | synonyms

0
votes
0answers
12 views

Embedding in Generated C*-Algebra

Given C*-algebras $\mathcal{A}$ and $\mathcal{A}'$. Suppose they have common elements: $$\mathcal{A}\cap\mathcal{A}'\neq\varnothing$$ Then is there a generated C*-algebra containing both and is it ...
0
votes
0answers
32 views

Intersection of C*-Algebras again C*-Algebra

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{A}'$. Is their intersection necessarily a C*-algebra again? So I started like this: $$A,B\in\mathcal{A}\cap\mathcal{A}'\implies ...
1
vote
1answer
22 views

Nonunital C*-Algebras: Morphism contractive?

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Suppose it misses a unit $1\notin\mathcal{A}$. Consider a *-morphism $\pi:\mathcal{A}\to\mathcal{B}$. Then it is contractive: ...
3
votes
1answer
42 views

C*-algebra of polynomials?

Let $A$ be a C*-algebra. Consider its cartesian square $A^2$ and define a multiplication on $A^2$ by the identity $$ (x_0,x_1)\cdot (y_0,y_1)=(x_0y_0,x_0y_1+x_1y_0),\qquad x_0,x_1,y_0,y_1\in A $$ This ...
2
votes
0answers
15 views

Extened of a representation

The following is a part of a theorem of Folland's book: Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$. ...
3
votes
1answer
39 views

Show that an operator is well-defined

Let $v\in B(H)$, Define $u:|v|H\to H$ such that $u(|v|\xi) = v\xi$ . To show the map $u$ is well-defined, the author writes $$\||v|\xi\|^2=\langle v^*v\xi,\xi\rangle = \|v\xi\|^2$$ But I do not know ...
2
votes
1answer
19 views

Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in ...
3
votes
1answer
22 views

A question about essential ideal

Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define $$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I, ~for ~all~ x\in I\}.$$ Then, how to prove $I$ ...
2
votes
2answers
46 views

Equality of two operators

The following is a fact in Murphy's C*-algebras and operator theory page 49: Suppose $u,v \in B(H)$, where $H$ is a Hilbert space, then $u=v$ if and only if $\langle u\xi,\xi\rangle = \langle ...
3
votes
1answer
15 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
0
votes
0answers
22 views

Do infinitesimals split from dimension groups?

Let $G$ be a countable dimension group (i.e. a partially ordered abelian group that is directed, unperforated, and satisfies the Riesz interpolation property) with order unit $u\in G^{+}$. Let ...
1
vote
1answer
72 views

If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
-4
votes
0answers
179 views

$s \in L^{1}(H)$ $\iff$ $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $

Let $H$ be a separable Hilbert space, and let $L^1(H)$ be the space of trace-class operators on $H$. I'd like to prove that $s\in L^{1}(H)$ if and only if there exists $\{ x_{i} \} , \{ y_{i} \} ...
3
votes
1answer
27 views

If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
1
vote
1answer
17 views

$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
2
votes
0answers
19 views

Equivalence between the GNS representation of two different positive linear functionals

Let $\varphi $ be a positive linear functional on $C^*$-algebra $A$ and let $(\pi _{\varphi},H_\varphi ,\xi)$ be the associated GNS representation. Let $\psi \in A_+^*$. Show that the two next ...
1
vote
1answer
17 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element ...
2
votes
0answers
57 views

$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
2
votes
1answer
27 views

A question on the minimal tensor norm

Given two C*-algebras $A$ and $B$ and let $A_1$ and $B_1$ be their C*-subalgebras. Can we conclude that $A_1 \otimes_\min B_1$ is a subalgebra of $A \otimes_\min B$? I think that this is not true, ...
3
votes
0answers
41 views

Creating Bratteli diagrams for Riesz groups

The Effros-Handelman-Shen-theorem tells you that Riesz groups are the same as dimension groups -- i.e. any ordered, unperforated abelian group with the Riesz interpolation property can be realised as ...
3
votes
0answers
59 views

Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
0
votes
1answer
40 views

The cone over separable C*-algebra is also separable?

For a C*-algebra $A$, the cone over $A$ is $CA=C_{0}(0,1]\otimes A$ , My question: If $A$ is separable, $CA$ is also separable?
0
votes
1answer
40 views

Two questions about orthogonal projections on Hilbert space

Let $l_{k}^{2}$ denote the k-dimensional Hilbert space and $\oplus_{1}^{\infty} l_{k}^{2}$ be the infinite direct sum of $l_{k}^{2}$. Let $P_{M}\in ...
1
vote
1answer
20 views

Joint spectrum of $\{a_1,…,a_n\}$

Let $\{a_1,...,a_n\}$ be commuting normal operators on a Hilbert space. Put $A:= C^*(1,a_1,...,a_n)$. By Gelfand theorem ,abelian C*-algebra $A$ is identified with the algebra $C(\Omega)$ of all ...
4
votes
1answer
29 views

The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
1
vote
2answers
36 views

Existence of a approximate unit $U_{n}^{2}$ for a $ C^{*}$-algebra $

if ${U_{n}}$ is an approximate unit for a $C^{*}$-algebra A. Is ${U_{n}^{2}}$ is an approximate unit for a $C^{*}$-algebra A? Thank in advance.
0
votes
1answer
53 views

Soft questions: $C^\ast$-dynamical systems

I have read some papers about $C^\ast$-dynamical systems. But there are still some questions in my mind which I can not answer. When is the $C^\ast$-dynamical system introduced? Why is the ...
2
votes
1answer
23 views

Algebra of matrices — equivalence of norms

Let $A = M_n(\mathbb C)$. Then it is possible to endow this $\ast$-algebra with several different norms (see here): $$ \|a\|_1 = \max_j \sum_i |a_{ij}|$$ $$ \|a\|_\infty = \max_i \sum_j |a_{ij}|$$ ...
3
votes
1answer
20 views

Why is $1 + i c^{-1/2}dc^{-1/2}$ invertible?

I am reading a proof of this theorem: If $a,b$ are positive elements of a $C^\ast$ algebra and $a \le b$ then $a^{1/2}\le b^{1/2}$. I don't understand one step in the proof. I understand this: Let ...
2
votes
2answers
25 views

Positive elements in star algebras

Let $A$ be a $C^\ast$-algebra. Is it possible to prove that if $a \ge 0$ then $ab, ba \ge 0$ if and only if $b \ge 0$?
1
vote
1answer
22 views

Irreducible representation

I know that correspondence every pure state on a C*-algebra $A$, there is an irreducible representation of $A$. Also we have the following theorem: Let $A$ be a C*-algebras and $(\pi,H)$ be an ...
2
votes
1answer
16 views

Restriction of an irreducible representation

Let $A$ be a C*-algebra and $\pi:A \to B(H)$ be a irreducible representation. Could we claim $\pi_{|B}$ is an irreducible representation if $B$ is a C*-subalgebra of $A$ ?
0
votes
2answers
40 views

construction of an injective representation of $C_0(X)$

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of ...
0
votes
1answer
22 views

$b \le \|b\|$ even when $b$ is not normal or self-adjoint?

It is a theorem in $C^\ast$ algebras that if $0\le a \le b$ then $\|a\|\le \|b\|$. The proof given in this book (page 47) starts by asserting that $b \le \|b\|$ because we can use the Gelfand ...
0
votes
1answer
34 views

Unital maps taking values in abelian C*-algebras

It is known that a bounded linear functional $f$ on a unital C*-algebra $A$ is positive if and only if $f(I)\geqslant 0$. Is the same true for bounded linear operators $T\colon A\to C(X)$ with $T(I) = ...
1
vote
1answer
15 views

Positive invertable element of a C*- algebra

The following is Theorem 2.2.5 of Murphy's C*-algebras and operator theory: Let $A$ be an unital C*-algebra and $a,b$ are positive invertable elements, if $a\leq b$, then $0\leq b^{-1}\leq a^{-1}$. ...
0
votes
1answer
75 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
0
votes
1answer
11 views

Why is the restriction of a character here non zero?

Let $A$ be a unital $C^\ast$-algebra, let $a$ be normal, $B$ the $\ast$-subalgebra generated by $1$ and $a$ and $f\in C(\sigma (a))$. Let $C$ be the $\ast$-algebra generated by $1$ and $f(a)$. If ...
0
votes
1answer
41 views

Important and simple example of application for functional calculus?

I reently proved the theorem for unital $C^\ast$-algebras that for $a\in A$ normal there exists a unique unital isometric $\ast$-homomorphism $\varphi : C(\sigma(a))\to A$ with $\varphi(i) = a$ where ...
0
votes
1answer
36 views

positive elements in c*algebras and states

I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$. The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a ...
2
votes
0answers
27 views

Can a “tangent vector of a discrete group” be extended to a tangent vector of its $C^*$-algebra?

This is related to my recent question in MO. I am sure this is trivial, but I have no intuition here, so my apologies from the very beginning. Let $G$ be a discrete group, $A$ a $C^*$-algebra, and ...
0
votes
1answer
24 views

limit of state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
0
votes
1answer
54 views

Positive Elements in a C*algebra

Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with ...
0
votes
2answers
16 views

Unitaries $u$ span $A$ linearly?

I can't understand this paragraph in my book: If $a$ is a self-adjoint element of the closed unit ball of a unital $C^\ast$-algebra $A$ then $1-a^2$ is positive and $u=a + i\sqrt{1-a^2}$ and $v = a - ...
2
votes
1answer
36 views

Typo in Murphy's book: $ \sigma_A(b)= \sigma_B(b) \cup \{0\}$ or $ \sigma_A(b) \cup \{0\}= \sigma_B(b) \cup \{0\}$

On page 45 the book states that for any $\lambda \in \mathbb C \setminus \{0\}$ and any star subalgebra $B$ of a $C^\ast$ algebra $A$ with $1_B \neq 1_A$, $b -\lambda 1_B$ is invertible in $B$ if and ...
0
votes
1answer
15 views

Spectrum of $C^\ast$ subalgebra

Let $A$ be a unital $C^\ast$ algebra. It is stated in this book that for any $C^\ast$ subalgebra we have $\sigma_B(b)\cup\{0\} = \sigma_A(b)\cup\{0\}$. The reasoning why this should be true is this: ...
3
votes
1answer
35 views

Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
-1
votes
2answers
75 views

C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
1
vote
1answer
29 views

Is the image of a von Neumann algebra under a C*-homomorphism a von Neumann algebra as well?

If $\varphi: A\to B$ is a (norm-continuous, unital, involutive) homomorphism of $C^*$-algebras, then the image $\varphi(A)$ is closed in $B$ and therefore is a $C^*$-algebra with the $C^*$-norm ...
1
vote
1answer
37 views

If $A$ is a $*-$ Banach algebra then $\bar A^{wot} = \bar A^{weak^*}$?

If $A$ is a $*-$ subalgebra of $B(H)$, then clearly $\bar A^{weak^*}\subset \bar A^{wot}$ (wot means weak operator topology). Also on every bounded subset of $A$, two topologies equal. Now my question ...