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).

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$b \le \|b\|$ 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 ...
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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) = ...
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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 $0\leq b^{-1}\leq a^{-1}$. ...
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74 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 ...
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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 ...
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30 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 ...
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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 ...
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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 ...
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22 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 $\varphi$ ...
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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 ...
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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 - ...
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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 ...
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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: ...
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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 ...
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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 ...
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28 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 ...
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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 ...
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60 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
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C*-algebra generated by the symmetric on 3 elements

I want compute $C^*(S_3)$ where $S_3$ is the symmetric group on $\{1,2,3\}$ and $C^*(S_3)$ is the (full) C*-algebra generated by $S_3$. My attempt: Since $S_3$ is a finite group, $C^*(S_3)=C_c(S_3)$ ...
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30 views

How can I argue that this is an isomorphism?

Let $A$ be a unital $C^\ast$ algebra and let $B$ be a (not necessarily unital) $C^\ast$-subalgebra such that $B \oplus \mathbb C = A$. I want to argue that the map $\varphi : \widetilde{B} \to A$, ...
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54 views

A question about maximal and minimal tensor product

Let $A, B$ be two C*-algebras and $\pi: A\otimes_{\max} B\rightarrow M_{n}(\mathbb{C})$ be a representations, then this $\pi$ can factor through the minimal tensor product $A\otimes_{\min} B$ ? (That ...
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A isomorphism between full group C*-algebras of free group

Fix $n\in \mathbb{N}$ and let $\mathbb{F}_{n}$ be the rank-$n$ free group, can we use the universal property to illustrate the following isomorphism: $$C^{*}(\mathbb{F}_{n}\times \mathbb{F}_{n})\cong ...
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2answers
69 views

Positive elements of a $C^*$-algbera form a poset

My knowledge of $C^*$-algebras is very little. We call an element positive if $a=b^*b$ for some $b$ and make a relation on all positive elements by saying $$ b \geqslant a \iff b-a \text{ is ...
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1answer
40 views

I need help with a proof: invertibility of $b-\lambda$ in $B$ iff $b-\lambda $ invertible in $A$

Let $A$ be a unital $C^\ast$ algebra and let $B$ be a $\ast$ subalgebra such that $B \oplus \mathbb C = A$ and such that the unit in $B$, $1_B$, is not equal to the unit in $A$. I am trying to show: ...
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1answer
56 views

Need some help understanding one step in this proof of homeomorphism $\Omega (C(X)) \cong X$

Let $X$ be a compact Hausdorff space and $\Omega (C(X))$ the space of characters on $C(X)$. I am showing that the map $x \mapsto e_x$ where $e_x$ is evaluation at $x$ is surjective (I already showed ...
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15 views

Choosing a subalgebra that separates points

Recall the (complex) Stone Weierstrass theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the $\ast$-algebra of continuous maps $X \to \mathbb C$. Then any $\ast$-subalgebra of ...
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Is this really the argument used here?

Let $A$ be some normed vector space and let $A^\ast$ denote its dual. Then if for all $\varphi \in A^\ast$ we have $\varphi (a) = 0$ then $a=0$. This is an argument that is frequently used. Consider ...
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Please can you check my proof of the spectral mapping theorem?

I would like someone to help me check if I understand this proof and for this reason I would like to give the proof here in my own words. The statement I am proving is this: Let $A$ be a unital ...
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Norm of an operator

Suppose $\{\xi_i\}_{i\in I}$ is an orthonormal system of Hilbert space $H$ and $T\in B(H)$. For each $i\in I$, let $\alpha_i$ be a scalar of modulus one such that $$|(T\xi_i,\xi_i)| = \alpha_i ...
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38 views

Norm of a sequence

The following is a theorem that I have some difficulty at it. I do not know how the author shows that $\alpha \in \ell^1$. Please help me. Thanks in advance.
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A question about spectral measure

The following is a part of a theorem of Takesaki's Operator theory: Let $T$ be an positive operator. Suppose $T = \int_0^{\|T\|} \lambda \, de(\lambda)$ is the spectral measure of $T$. Also put ...
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46 views

How to prove the following isomorphism?

Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow ...
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A question about finite-rank projection

Let $B, C$ be two C*-algebras and $\sigma_{0}: B\rightarrow C$ be *-homomorphism such that $\sigma_{0}$ is injective. Then, for a finite set $F\subset B$ of the unit ball and $\varepsilon>0$, Can ...
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2answers
55 views

Bounded measurable functions

Suppose $X$ is a compact space and $B(X)$ denotes the bounded Borel measurable function space. Let $f\in B(X)$. There is a sequence of step functions $\{\phi_n\}$ such that $\phi_n\to f$ (point wise). ...
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1answer
27 views

A lemma about quasicentral-approximate-unit

Here is a lemma about quasicentral-approximate-unit: Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that ...
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1answer
21 views

QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital ...
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1answer
36 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
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1answer
43 views

Please help me getting the big picture: what is this theorem for?

There is a theorem in Murphy's book on operator theory and $C^\ast$-algebras: Let $u$ be a unitary element in a unital $C^\ast$-algebra $A$. Then if $\sigma(u) \subsetneq S^1$ then there exists a ...
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1answer
18 views

What is the result that implies $\ell^1$ is isomorphic to $C(K)$

In this answer here t.b. writes that $\ell^1(\mathbb Z)$ would then have to be isomorphic to a space of the form $C(K)$ with $K$ compact (metrizable and infinite). What is the result they are ...
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$\ell^p$ with pointwise multiplication — example of $C^\ast$ algebra?

I was trying to think of some examples of $C^\ast$-algebras and I think $\ell^p$ with pointwise multiplication would be a good example. My reasoning is that if $a_n, b_n$ are in $\ell^p$ then ...
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56 views

A question on non commutative ring or algebra

Assume that $R$ is a ring such that $R=I+J$ where $I$ and $J$ are 2 -sided ideal.(This is not a direct sum) If $I$ and $J$ are commutative does it implies that $R$ is a commutative ring? Please ...
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34 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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Is exponential function in a C*-algebra injective on self-adjoint elements?

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?
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1answer
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Infinite projections in the Cuntz algebra

I am studying the Cuntz algebra $\mathcal{O}_n$, $(n \ge 2)$ with generators $S_1, S_2, \ldots, S_n$ and in my class notes there is a statement about the projections $S_1S_1^*, S_2S_2^*, \ldots, ...
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On closedness of $C^\ast$ subalgebras

By definition of a $C^\ast$ subalgebra it is a closed subalgebra. Why does it need to be closed? This is a restriction that is not required in the case of a Banach subalgebra. (although I can't ...
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Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
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1answer
33 views

Image of $C^\ast$-algebra is closed?

Let $A$ be a non-zero commutative $C^\ast$ algebra and let $\varphi : A \to B$ be a homomorphism of star algebras. Please could someone help me how to show that $\varphi(A)$ is closed in $B$?
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Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
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1answer
53 views

The set of analytic functions on unit circle is not a C*-algebra

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed ...
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1answer
22 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...