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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Involution on the set of all multipliers of $A$ ($A$ is a $C^*$-algebra)

Let $A$ be a $C^*$-algebra. $M(A)$ denotes the set of all multipliers of $A$, i.e. $m\in M(A)$ means that there is a map $m^*:A\to A$ such that $m(a)^*b=a^*m^*(b)$ for all $a,b\in A$. I want to know ...
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Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad Q^2=Q=Q^*$$ Order them by: $$A=A^*:\quad A\geq0:\iff\sigma(A)\geq0$$ Then one has: $$P\leq Q:\quad P=PQ=QP$$ How ...
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Sot convergence of a sequence of operators implies uniform convergence

Let $H$ be a Hilbert Space. Let $\{A_n\}$ be a sequence of bounded operators in $H$, and $A\in B(H)$. If $\|A_nf - Af\|\to 0$ uniformly for $f\in H_{\|.\|=1}\ $, prove that $\|A_n - A\|\to 0$. ...
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If a C*-algebra $A=\overline{\bigcup S}$, where $S$ is a class of prime C*-subalgebras, then $A$ is prime.

This is question 5.6 of Murphy's C$^*$-Algebras and Operator Theory: Let $S$ be a set of C*-subalgebras of a C*-algebra $A$ that is upwards-directed, that is, if $B,C\in S$, then there exists ...
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Are there infinite-dimensional, artinian C*-algebras?

A ring is artinian if it has no infinite descending chains of ideals. Of course finite-dimensional algebras are artinian. I'm wondering if it's possible to have an artinian C*-algebra (or Banach ...
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1answer
20 views

In what topology is the unitary group of a unital C*-algebra locally compact?

If a unital $C^*$-algebra $\mathcal{A}$ is finite-dimensional then the unitary group $\mathcal{U}(\mathcal{A})$ is compact with respect to the norm topology. My question is: what if ...
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Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
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1answer
56 views

Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
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Strict positiveness on a C*-algebra given by generators and relations.

Let $A$ be a C*-algebra with generators $a_1,a_2,\ldots,a_n$ and some (non-important) relations (the relations imply that $\|a_i\|\leq 1$, so that $A$ exists). Among the given relations we have that ...
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41 views

If a normal element of a C* algebra has real spectrum, then it is self-adjoint

Let $A$ be a $C^*\!$-algebra. Suppose $x$ is a normal element of $A$ and $\operatorname{spect}(x)$ lies in $\mathbb{R}$. Prove that $x$ is self-adjoint. I tried the following: using ...
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58 views

Is the supremum norm the only $ C^{*} $-norm on $ {C_{c}}(X) $, equipped with the usual pointwise operations?

Let $ X $ be a locally compact Hausdorff space. Then $ {C_{c}}(X) $ is a commutative $ * $-algebra with respect to addition, multiplication, scalar multiplication and conjugation (all pointwise ...
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1answer
18 views

does this theorem imply that every $C^*$-algebra has an approximate unit?

I read that every $C^*$-algebra has a approximate unit. But we proved only the following theorem in lecture: Let $A$ be a $C^*$-algebra, $I\subseteq A$ an ideal which is dense in $A$. Then there is ...
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product $ab$ of postive elements $a,b$ is again positive, if $ab=ba$.

Let $A$ be a $C^*$-algebra, $a,b\in A$ positive elements (this means self-adjoint and the spectrum lies in $[0,\infty)$). In general, $ab$ isn't positive, for example consider the matrices ...
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Questions about stable rank of inductive limit of $C^\ast$-algebras

Let $A$ be an inductive limit of $\{A_n\}$ which are stable rank one. In Huaxin Lin's book An introduction to the classification of amenable $C^\ast$-algebra. The author assume that $\{A_n\}$ and $A$ ...
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unitization of a unital $C^*$-algebra

I have a little question about unitization of a $C^*$-algebra. If $A$ is a non-unital $C^*$-algebra, set $A_1=A\oplus\mathbb{C}$ as vector spaces and define a multiplikation, involution and a norm in ...
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Centralizer of $C^*$ algebra

Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra. If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for ...
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68 views

Commutativity criteria for $C^*$-algebras

If for any $x, y\in A$ a $C^*$-algebra, $0\leq x\leq y\implies x^2\leq y^2$, then it is true that A is commutative ? It is easy to show that the implication $0\leq x\leq y\implies x^2\leq y^2$ is not ...
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19 views

Orthogonal elements in a unitization of a $C^*$-algebra

Let $A$ ne a $C^*$-algebra and $a,b\in A$ self-adjoint. a and b are orthogonal, iff $ab=0$. Let A be nonunital and denote $A_1$ it's unitization, i.e., $A_1\cong A\oplus\mathbb{C}$ as vector spaces. ...
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Describe the GNS construction [closed]

Question: Describe the GNS construction for the C$^*$-algebra $ C[0, 1]$ and for the positive linear functional $\phi $ given by $\phi(f) = f (0)$. What should i do? Should I describe Hilbert space ...
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3answers
55 views

Two normal operator that commutes

Suppose $N\in B(H)$ is normal, and $T\in B(H)$ is invertible. Prove that if $TNT^{-1}$ is normal then $N$ commutes with $T^*T$. I can not any idea to prove it, just I know ...
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40 views

Projective (or inverse) limit of C*-algebras

(I think that the term "inverse limit" is used when the index set is directed) To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and ...
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positive elements in a $C^*$-algebra

Let $A$ be a $C^*$-Algebra and $a\in A$. I'm stuck in the proof of: $a\ge 0\iff $ it is $\varphi(a)\ge 0$ for all states (=positive linear functionals with norm 1) $\varphi:A\to\mathbb{C}$. Proof: ...
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1answer
15 views

What is a support projection in a $C^*$-algebra?

Let $A$ a $C^*$-algebra and consider $a\in A$ self adjoint and $ax=xa$ for all $x\in A$. I want to know: -what is the support projection of $a$? -what is the definition of the support projection of ...
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1answer
33 views

Two-sided closed ideals of $C(X,M_2(\mathbb C))$

Let $X$ be compact and Hausdorff space. I know all closed ideals of $C(X)$. I want to substitute $\mathbb C$ by $M_2(\mathbb C)$. What can we say about two-sided closed ideals of $C(X,M_2(\mathbb ...
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Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
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14 views

C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
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1answer
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intersection of a multiplier algebra with a commutant of a $C^*$-algebra

I have a question about multiplier algebras and commutants of $C^*$-algebras in general. First of all, the question is related to this structure theorem about completely positive order zero maps (you ...
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example of the arens multiplication; I want to unterstand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the arens multiplication on the double dual $A^{**}$ (considered as Banach ...
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1answer
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*-algebra representation, explicit calculation

suppose we are given a vector space basis of a unital *-algebra $\mathcal{A} \subset \mathcal{M}_d(\mathbb{C})$. I found a proof showing that one can find a unitary $U$ such that $$U\mathcal{A}U^* = ...
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1answer
16 views

positive elements in the unitization of a $C^*$-algebra

Let $A$ be a $C^*$-algebra and consider it's unitization $A_1$ whose underlying vector space is the direct sum $A\oplus \mathbb{C}$. I want to know how does positive elements in $A_1$ look like. I ...
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31 views

Completely positive, orthogonaly preserving maps

First of all, here is the setting for my problem: Definition: Let $A$ be a $C^*$-Algebra, $a,b\in A$. We say, a and b are orthogonal, if $ab=ba=a^*b=ab^*=0$. Proposition: Let $A$ be a $C^*$-Algebra, ...
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1answer
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Prove that the map $f: \alpha I + \beta A^*A \mapsto \alpha + \beta ||A||^2 $ is unital

Fix $A \in {\mathcal{A}}$, where $({\mathcal{A}},||\cdot||,*)$ is a $C^*$-algebra with unit $I$. Prove that the linear form $$ f(D_{\alpha,\beta}) = \alpha + \beta ||A||^2,~~D_{\alpha,\beta} \in ...
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equivalence of properties. Is the restriction in (ii) redundant?

I have a question about the claim, which I found in a paper: Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. The following are equivalent: ...
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1answer
31 views

Questions about Stinespring's theorem for completely positive maps

I have a question about Stinespring's Theorem: Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space and $L(H)$ the set of bounded linear operators on H. Let $\Phi: A\to L(H)$ be a completely ...
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1answer
17 views

C*-algebra generated by a non-invertible element

Let $x$ be a non-invertible element, and put $A:=C^*(x)$. Let $I$ be a closed ideal of $A$, and $\pi: A\to A/I$ be a natural quotient map. Is it possible that there is an invertible element $y\in A$ ...
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1answer
19 views

Jordan-homomorphism; equivalent properties

Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. I want to know, why the following properties are equivalent: $(i) \phi(ab+ba)=\phi(a)\phi(b)+\phi(b)\phi(a)$ and $(ii) ...
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1answer
26 views

Commutative multiplier algebra

In my course of spectral theory and operator algebras I came across the following exercise: Let $\mathcal{A}=C_0(X)$ where $X$ is a locally compact Hausdorff space. Describe the multiplier algebra ...
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1answer
26 views

Show that $\|(pqp)^{-\frac{1}{2}}pq-pq\|<1$ for projections $p,q$ with $\|p-q\|<1$ in a $C^*$-Algebra

Let $p,q$ be projections in a (unital) $C^*$-Algebra with $\|p-q\|<1$. Definition of $(pqp)^{-\frac{1}{2}}$: Let $pAp$ be the $C^*$-Algebra of elements $pap$, $a\in A$ with unit $p$. Then $pqp$ is ...
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von Neumann algebra associated to the full group C*-algebra

Lance's theorem asserts a discrete group $G$ is amenable if and only if the reduced and full groups C*-algebras coincide. The group von Neumann algebra is the weak closure of the reduced group ...
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Every conditional expectation is normal?

Let $M$ be a von Neumann algebra and let $N$ be a von Neumann subalgebra and let $E$ be a conditional expectation $M\to N$. Let $i$ be the canonical inclusion of $M$ into $M^{**}$. Claim: $E$ is ...
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(Question about the proof of $M(C_0(X))=C_b(X)$ )- Urysohn's lemma for locally compact Hausdorff spaces

I'm stuck on understanding the proof of why the multiplier algebra of $C_0(X)$ can be identified with $C_b(X)$, $X$ is a locally compact Hausdorff space. The proof uses that $C_0(X)$ is an essentiell ...
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1answer
25 views

On existence of square root of positive elements of a unital $C^*$-algebra

Given a unital $\mathcal{C}^*$-algebra $A$ and a positive element $a \in A$, I am trying to prove the existence of a square root $a^{\frac{1}{2}}$ i.e. a positive element $b \in A$ such that $b^2 = ...
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1answer
23 views

Connection between Stinespring's factorization theorem and the spectral theorem for bounded operators

I know at least 2 versions of a Spectral theorem for operators, one of them is the following Theorem: Let H be a separable complex Hilbert space, $A\in L(H)$ self-adjoint ($L(H)$ are the bounded ...
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State of a $ C^{*} $-algebra.

Let $ (\pi,\mathcal{H}) $ be a non-degenerate $ * $-representation of a $ C^{*} $-algebra $ A $, and let $ h \in \mathcal{H} $ with $ \| h \| = 1 $. Define $ f_{h}: A \to \Bbb{C} $ by $ {f_{h}}(a) ...
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1answer
35 views

Completely positive maps

Let $B$ be a commutative C$^*$-algebra and let $M_n$ denote the algebra of $n\times n$ complex matrices. Let $f$ be a state on the tensor product of $B$ and $M_n$, $B\otimes M_n$. How can I show that ...
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Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
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What algebraic structure do self-adjoint operators form?

Consider the $\mathbb{C}$-algebra of all matrices of dimension $n$ over the complex numbers, $Mat_n\mathbb{C}$; We have here a notion of adjointness which is an involution; and thus of ...
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A necessary and sufficient criterion for an element of a multiplier $ C^{*} $-algebra to be positive.

I am trying to find a reference for the following assertion: Let $ A $ be a $ C^{*} $-algebra, and let $ M(A) $ denote its multiplier algebra. Then $ m $ is a positive element of $ M(A) $ if and ...
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1answer
41 views

$||a||\leq \sup_{||b||\leq 1} ||ab||$ in a C*-algebra

I would like to prove that, if $a$ is an element in a C*-algebra then $$\|a\|\leq \sup_{\|b\|\leq 1} \|ab\|$$ It is obvious if the algebra is unital. What if it is not?
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Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...