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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How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
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22 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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Why is $ab=ba=a^\ast b=ab^\ast=0$ (orthogonal elements in a $C^\ast$-algebra)?

Let $a,b$ be elements in a $C^\ast$-algebra $A$, such that $$a^\ast ab^\ast b=b^\ast ba^\ast a=0$$ $$a^\ast abb^\ast=bb^\ast a^\ast a=0$$ $$aa^\ast b^\ast b=b^\ast b aa^\ast =0$$ $$aa^\ast ...
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1answer
23 views

composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
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22 views

the $C^\ast$-algebra $M_n(A)$, understanding the $C^\ast$-norm on $M_n(A)$

Let $A$ be a $C^\ast$-algebra. I want to understand $M_n(A)$, the vector space of $n\times n$-matrices with entries in $A$, as a $C^\ast$-algebra. On $M_n(A)$ you can define an involution ...
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1answer
30 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...
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29 views

C*-Algebra: Cyclic Elements

Given a locally compact Hausdorff space $\Omega$. Consider the C*-algebra: ...
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20 views

Is the identity in unital, simple, purely infinite $C^*$-algebra always infinite?

I'd like to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true: If $p$ in $\mathfrak{A}$ ...
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1answer
35 views

Square root of a compact normal operator

Halmos expresses below problem in his book; Problem: If $A$ is a normal operator and if $A^n$ is compact for some positive integer $n$, then $A$ is compact. I have an example in my mind which I ...
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1answer
37 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
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47 views

Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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28 views

Uniqueness of c.p.c. order zero extensions

I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is ...
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38 views

In a C*-algebra $A$, $x$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$.

The question I am having trouble with is the following: Let $A$ be a C$^*$-algebra. Show that an element $x$ of $A$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$. (Hint: If $h\in ...
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2answers
39 views

Question about $C_0(X)$ is unital iff X is compact

I am sorry if this question is trivial I don't know much about $C^*-algebra$ and learning introduction about them from the following link notes about C* algebras. According to the notes given above ...
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49 views

Understanding the bidual of a $C^*$-algebra as a $C^*$-algebra

I have a lot of problems trying to understand the double dual of a $C^*$-algebra. Let $A$ be a $C^*$-algebra, I read that if you endow the bidual Banach space $A^{**}$ of $A$ with the weak-*topology, ...
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1answer
37 views

Closed convex hull of unitaries

If a C*-algebra ${\cal U}$ contains a non-unitary isometry $S$, show that $$\|S-A\|>\frac{1}{2n}$$ for every $A=\sum_{i=1}^n \lambda_iU_i$ which is the convex combination of $n$ unitaries. Thanks ...
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0answers
29 views

Weakly compact left multipliers

This is Exercise 3(a) on p. 157 in Takesaki's Operator algebras. Let $A$ be a C*-algebra. Then each opeator $T_a\colon A\to A$ given by $T_ax = ax$ ($a\in A$) is weakly compact if and only if $A$ is ...
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1answer
52 views

Some questions about Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras

I have some questions about Joachim Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras, which is found in this paper. For this post, let us adopt the ...
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1answer
57 views

Projections: Inequality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P\in\mathcal{A}:\quad P^2=P=P^*$$ Then one has: $$P\leq A\leq 1\implies P=PA=AP$$ And equivalently: $$0\leq A\leq P\implies ...
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24 views

Stone-Cech compactification of a compact space and the multiplier algebra of $C_0(X)$

What is the Stone-Cech compactification $\beta X$ of a compact space $X$, is is $X$ itself? Or does it depend on the definition of compactification, whether the embedding $i:X\to \beta X$ is assumed ...
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1answer
23 views

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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2answers
76 views

Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\leq P':\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P'-P)$$ Then equivalently: ...
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0answers
42 views

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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2answers
77 views

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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38 views

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
25 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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41 views

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
58 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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39 views

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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2answers
62 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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1answer
64 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
22 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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2answers
52 views

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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0answers
26 views

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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2answers
54 views

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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0answers
30 views

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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1answer
73 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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1answer
24 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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3answers
56 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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2answers
56 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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1answer
22 views

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
19 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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48 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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2answers
53 views

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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0answers
15 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
22 views

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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0answers
18 views

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
35 views

*-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
17 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 ...