# Tagged Questions

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### A simple question about positive element in C*-algebra

I am reading a book about C*-algebra. There is a quotation below. An $operator~system$ $E$ is a closed self-adjoint subspace of a unital C*-algebra $A$ such that $1_{A}\in E$. The $n \times n$ ...
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### A proposition in C*-algebra

Problem: Let $A$ and $B$ be C*-algebra and $\varphi:A \rightarrow B$ be a contractive completely positive map. $A_{\varphi}=\{a\in A: \varphi(a^{\ast}a)=\varphi(a)^{\ast}\varphi(a)$ and ...
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### The excision theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". I can not understand the proof of the excision theorem in the fundamental facts of the book. Theorem 1.4.10(Excision) Let ...
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### An exercise in C*-algebra

Let $A$ be a C*-algebra, $\phi$ be a pure state and $L=\{a\in A:\phi(a^{\ast}a)=0\}$, how to prove that $L+L^*\subseteq ker\phi$. ($L^*=\{a^{\ast}: a\in L\}$) I think it is an easy exercise, ...
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### How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
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### A theorem about approximate units in C*-algebra

I am reading a book about C*-algebra. I encounter a theorem without proof. Could someone help me to complete its proof or give me some hints. Theorem 2.1. Let $I$ be an ideal of C*-algebra $A$. Then ...
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### The unitary implementation of $*$-isomorphism of $B(H)$

Is it possible to construct $*$-isomorphism of (factor von Neumann) algebra $B(H)$ which is not unitary implementable?
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### Automorphism of $W^*$ algebra

Let $\mathfrak{A}$ be von Neumann algebra. It is in particular $C^*$ algebra. Is it true that every $*$-isomorphism of $\mathfrak{A}$ is also $W^*-$isomorphism? (Note that every $*$-isomorphism of ...
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### If $\lambda$ is isolated in $\sigma(u)$, then $E(\left\{\lambda\right\})(H)=\ker(u-\lambda)$.

This is a Question 2.11 from Murphy's book: C$^*$-algebras and Operator Theory: Let $H$ be a Hilbert space. Let $u\in B(H)$ be a normal operator with spectral resolution of the identity $E$. (a) ...
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### Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
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### C* Algebra textbook recommendation

I have read the first two chapters from Analysis Now and the chapter on C* algebras (chptr 8?). I'm taking a course on C* algebras in the spring and am currently overwhelmed with the choices. I'd ...
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### Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
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### equivalent? algebraic definition of a partial isometry in a C*-algebra

An element $a\in\mathfrak{A}$ (unital C*-algebra) is a partial isometry if $a^*\cdot a$ is projection. Can one recover the equivalent caracterizations of a partial isometry in ...
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### Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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### An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
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### A question about tensor product

If $A$ is an algebra, $M_{n}(A)$ denotes the algebra of all $n\times n$ matrices with entries in $A$. The operations are defined just as for scalar matrices. If $A$ is a *-algebra, so is $M_{n}(A)$, ...
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### A question about range projection in von Neumann algebra.

I am reading a book about C*-algebra. And I meet with a problem. Recall the range projection of an operator $a\in B(H)$ is the projection on the closure of $\{a(\eta):\eta\in H\}$(Here, $H$ is a ...
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### why for every $f\in C(\sigma(x))$ we have $\Phi (f(x))= f(\Phi(x))$?

In a book about $C^*$-algebra, in the section of continuous functional calculus says that: Suppose $x$ is a normal element of $C^*$-algebra $A$, then the continuous functional calculus ...
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I am reading Lin Hua xin's book "An introduction to the classification of amenable C*-algebras" and i am confused with the lemma 1.7.12 in this book. Lemma 1.7.12 Let $A$ be a C*-algebra and $f\in ... 1answer 41 views ### A question about strongly continuous. I am reading a book about C*-algebra. In the book, Let$\phi$be a linear functional on$B(H)$($H$denotes a Hilbert space), if$\phi$is strongly continuous, therefore, there exist vectors ... 2answers 46 views ### A question about a linear bounded operator (in hilbert space) I am reading a book about C*-algebra. When i study von Neumann algebras in this book, i meet with a problem. In the book, If$H$is a Hilbert space, we write$H^{(n)}$for the orthogonal sum of n ... 1answer 49 views ### Stone's theorem for 1-parameter groups of unitary multipliers? Let$A$be a nonunital C*-algebra and let$M(A)$denote its multiplier algebra. Let$(u_t)_{t \in \mathbb{R}}$be a strictly continuous 1-parameter group of unitary multipliers. That is,$u_t x \to x$... 1answer 118 views ### A simple question about *-homomorphism in C*-algebra Let$A$and$B$be C*-algebra,$h\colon A\rightarrow B$is *-homomorphism. If$a\in A_{\operatorname{sa}}$, then$\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ... 1answer 57 views ### A confusion in C*-algebra I am reading a book about C*-algebra. But I meet with some problems. In the book, the author says: If$I$is an ideal in a C*-algebra$A$, then$B=I\cap I^{\ast}$is a C*-subalgebra. However, I ... 1answer 107 views ### Confusion in Gelfand theorem in C*-algebra. I am reading HX Lin's book, named "An introduction to the classification of amenable C*-algebras", I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a ... 1answer 39 views ### A question about compact Hausdorff space Let$X$be a compact Hausdorff space and$C(X)$be the set of continuous functions on$X$. And$F$is a closed subspace of$X$. If the$f\in C(X)$such that$f|_{F}=0$is only zero function( i.e. ... 1answer 101 views ### If$a$and$b$commute in a$C^*$-algebra and$a$is normal, then$f(a)$and$b$commute for any continuous$f$I'm trying to find a way to demonstrate the following: Let$(A,*,\|\cdot\|)$be a unital$C^*$-algebra. If$a,b\in A$commute and$a\in A$is normal (i.e.$a^*a=aa^*$), then for every continuous ... 1answer 97 views ### A question on multiplicative linear functional on Banach algebra. I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume$A$is a non-unital C*-algebra and$\tilde{A}$is its unitization (the elements of the form ... 1answer 44 views ### How to prove, that the ordering on positive bounded operators agrees with ordering of their ranges? Hypothesis: Assume, that$A$and$B$are positive bounded operators (on some Hilbert space$H$) and$A\geq B \geq 0$. Then${\rm range}(A) \supset {\rm range}(B)$. The textbook "$C^*$-algebras by ... 1answer 141 views ### The set of all continuous functions on a locally compact Hausdorff space. I am reading a book about C*-algebra. There is a example that i could not understand. Let$X$be a locally compact Hausdorff space and$C_{0}(X)$be the set of all continuous functions vanishing at ... 1answer 71 views ### The operator norm of complex matrices Let$M_{n}$be the algebra of$n\times n$complex matrices. By identifying$M_{n}$with$B(\mathbb{C^{n}})$, the set of all bounded linear maps from the n-dimensional Hilbert space$\mathbb{C^{n}}$to ... 0answers 41 views ### Representations of a C*-algebra of bounded Borel functions Let$X$be a compact Hausdorff space. Let$B(X)$be the C*-algebra of bounded Borel measureable functions on$X$(under the supremum norm). I am curious whether the (say unital)$*$-representations of ... 0answers 160 views ### Positive elements in a C*-algebra [closed] Prove that if$a$is an element in a$C^*$-algebra$A$, then$a$is positive if and only if$f(a) \geq 0$for every state$f$on$A$. 2answers 65 views ###$a^*a$has a non-negative spectrum I am learning$C^*$-algebra, especially I work on the proof of the Gelfand-Naimark theorem. In many books such as the one of Arveson, it looks that the following lemma is the key stone of the proof: ... 0answers 76 views ### Projections in group$C^*$-algebras Let$G$be an amenable, discrete and infinite group. Cosinder its group C*-algebra$C^*(G)$canonically represented on$B(\ell_2(G))$by the left-regular representation$x\mapsto \delta_x$. Take the ... 0answers 47 views ### Masas in quotients Let$A$be a von Neumann algebra and let$B$be a norm-closed ideal of$A$(but not necessarily WOT-closed). What one has to assume about$A$and$B$to ensure that if$M\subset A$is a maximal ... 0answers 47 views ### Is an exact operator, unitary equivalent to a banded operator? Let$H$be an infinite dimensional separable Hilbert space and$B(H)$the algebra of bounded operators. Definition : Let$(e_{n})_{n \in \mathbb{N}}$be an orthonormal basis.$T \in B(H)$is ... 1answer 127 views ### Does an irreducible operator generate an exact$C^{*}$-algebra? Let$H$be an infinite dimensional separable Hilbert space and$B(H)$the algebra of bounded operators. Definition : An operator$T \in B(H)$is irreducible if$W^{*}(T)=B(H)$. Definition : A ... 1answer 101 views ### Does an irreducible operator generate a nuclear$C^{*}$-algebra? Let$H$be an infinite dimensional separable Hilbert space and$B(H)$the algebra of bounded operators. Definition : An operator$T \in B(H)$is irreducible (Halmos 1968) if its commutant$\{ T\}'$... 1answer 37 views ### Analytic continuation of one parameter subgroup: group property preserved? Let$(\mathcal{A},\alpha)$be a C* dynamical system, i.e.$\mathcal{A}$is a unital C*-algebra and$\{\alpha_t\}_{t\in \mathbb{R}}$a strongly continuous one-parameter group of *-automorphisms. For ... 1answer 46 views ### Analyticity of C*-algebra valued functions Let$\mathcal{A}$be a unital C*-algebra and consider a function$f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that$f$is analytic, i.e. can be locally ... 1answer 59 views ### Why does$L^1(\mathbb{R})$not have the C* property? Consider the space$L^1(\mathbb{R})$which is a Banach-algebra when taking the convolution as the algebra product and even posses the B*-property if one takes$f^*(t)=\overline{f(-t)}$as the ... 2answers 124 views ### Hahn-Banach Theorem in the C*-algebra What is the Hahn-Banach Theorem in the C*-algebra(or W*-algebra maybe)? If B is an nondense subalgebra of C*-algebra(or W*-algebra maybe), can we get an state f of A which is always zero at the ... 1answer 120 views ### Uniqueness of the involution on a$C^*$-algebra indication please Let$A$be a C*-algebra. Suppose that there exists on$A$another involution$x\rightarrow x^{\#}$such that$||xx^{\#}||=||x||^2$, for all$x\in A$. Prove that$x^{\ast}=x^{\#}$, ... 4answers 175 views ### What makes irreducible representations nice? Let$\mathcal{A}$be a C*-algebra and$(H,\pi,\Omega)$a cyclic representation. What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations ... 1answer 56 views ### Why is closeness of an ideal useful? In the GNS-construction for an$C^*$-algebra$\mathcal A$(see this script on page 30) one starts with a state$\phi:\mathcal A\rightarrow \mathbb C$(positive linear functional with$\|\phi\|=1$). ... 3answers 164 views ### Sufficient condition for a *-homomorphism between C*-algebras being isometric Let$\mathcal{A},\mathcal{B}$be two unital C*-algebras and consider a *-homomorphism$\pi: \mathcal{A} \rightarrow \mathcal{B}$. I know that in general$\pi$is contractive, i.e.$\vert\vert \pi(A) ...
Let $\mathcal{A}$ be a C*-algebra and $(H,\pi)$ an irreducible representation of $\mathcal{A}$. I want to prove the statement: all $\xi \in H$ are cyclic or $\pi(\mathcal{A})=\{0\}$ and ...