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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subset S is dense in the set of all pure states of a $C^*$-algebra with respect to the weak-$^*$ topology

Let $A$ be $C^*$-algebra and $P(A)$ the set of all states $f:A\to \mathbb{C}$ such that: for all positive $ g\in A^* $ with $g\le f$ there exists $t\in [0,1]$ such that $f=tg$. I.e. $P(A)$ is the set ...
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Crossed product by locally finite group

If a countable discrete group $G$ is the direct limit of finite subgroups $F_i$, and $G$ acts on a compact Hausdorff space $X$, can the crossed product $C^*$-algebra $C(X)\rtimes_r G$ be described in ...
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$K_0$ group of Direct Sum of C*-Algebras

We know that for C*-algebras, $$ K_0(A\oplus B) \stackrel{(*)}{=} K_0(A)\times K_0(B)$$where the $\oplus$ on the L.H.S. is a direct sum of C*-algebras (derived from the notion of direct sum of ...
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Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset ...
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Direct limit with constant homomorphisms?

I have the following direct system of finite-dimensional C*-algebras: $$\mathbb{C} \to \mathbb{C} \oplus \mathbb{C} \to \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C} \to \mathbb{C} \oplus \mathbb{C} ...
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Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for ...
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25 views

The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all ...
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Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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Form of pure states on $M_n(\mathbb{C})$

Related question: The form of the states on an algebra of $n\times n$ matrices with complex entries I have tried to show that pure states on $M_n(\Bbb{C})$ are of the form $\phi(A)=Tr(\rho A)$, just ...
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Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
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Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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K-theory of infinite direct product of copies of the algebra of compact operators

How does one compute the $K$-theory of an infinite (countable) direct product of copies of the algebra of compact operators on Hilbert space? Will it be an infinite direct sum of copies of ...
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Elements of $C(K)^{**}$, do they have a name?

Let $K$ be a compact (Hausdorff) space, and let $C(K)$ be the Banach algebra of contunous functions on $K$ (with the usual $\sup$-norm). The enveloping von Neumann algebra of $C(K)$ is its second dual ...
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15 views

Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
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29 views

Understanding Operator Norm of Matrices

Let $X$ denote the vector space of $n\times n$ complex matrices. To every matrix $A\in X$ one can associate two operator norms: Thinking of $A$ as a map $A\colon \mathbb{C}^n\to \mathbb{C}^n$ or ...
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25 views

Pure states on $C(X)$ are exactly evaluations

Let $X$ be a compact Hausdorff space. I want to show that pure states are of the form $ \phi (f) =f(x)$. By Reisz Represenation Theorem states on $C(X)$ are of the form $\phi (f)= \int fd\mu$ where ...
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Gelfand transform on functions

The Gelfand transformation identifies function spaces $C_0(X)$ for locally compact Haussdorff $X$ with commutative $C^*$ Algebras. Additionally there is a statement that if $f: X \to Y$ is a proper ...
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$L_p$ version of Toeplitz extension

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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51 views

About a relation between isometries

If we have $(T_i)_{i=1}^N$, operators on a Hilbert space, that are also isometries and satisfy the following relation: $$\sum_{i=1}^NT_iT_i^*=Id\quad (1)$$ How can you prove that they must also ...
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Projections and projective modules of $C(X)$

This is a followup to this question I made yesterday (disclaimer: I'm new here and I'm not sure if asking a new but related question is the correct procedure). If $X$ is a connected, compact, ...
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What are the projections of a commutative C* algebra?

I am aware that the commutative C* algebra is $C_0(X)$ for some nice space $X$ but I cannot figure out what the projections should be. The natural candidates (indicator functions on nice subsets of ...
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30 views

Index of an element in C*-algebra

Suppose that $x$ is an element of abstract $C^*$-algebra $A$. For example if $x$ is normal, i.e. $x^*x=xx^*$ then if we use any representation $\pi$ of $A$ on some Hilbert space $H$ then $\pi(x)$ will ...
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To prove that $C_0(X)$ is a $C^*$-algebra, is it necessary to assume that $X$ is a locally compact Hausdorff space?

I know this seems like a dumb question. But I can't see why we need the condition that $X$ is a locally compact Hausdorff space when we define $C_0(X)$. I think many properties still hold without ...
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square root of positive elements preserve order

Let $A$ be a $C^*$ algebra. Show that if $0 \le a \le b$ then $\sqrt a \le \sqrt b$. I've shown that this is true in case $b$ is invertible, here is my proof: $$\|a^{1/2}b^{-1/2}\|^2 = ...
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40 views

What is the motivation for Murray-von Neumann equivalence

Definition: If $p,q$ are projections in a $C^*$-algebra $A$, we say that they are Murray-von Neumann equivalent, and we write $p\sim q$, if there exist $u\in A$ such that $p=u^*u$ and $q=uu^*$. I ...
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Second dual of Calkin algebra

1- I am looking for any information concerning the second dual of Calkin algebra $\frac{B(H)}{K(H)}$. 2- What about the non-degenerate representations of the Calkin algebra?
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Is there an example of a non von Neumann algebra with this property?

What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties: 1) For every $T\in A$, The orthogonal projection ...
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34 views

Is the hermitian condition a must?

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$ In this post, I saw a comment stating that "More generally, if $a$ is normal then $∥a^n∥=∥a∥^n$ for each positive ...
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An strongly open set which is not measurable in the weak operator topology.

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: j\in J\}$$ ...
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52 views

Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, ...
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32 views

Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: ...
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24 views

When set of projections is closed under multiplication

Let $\mathcal A\subset B(H)$ be an unital $C^*$ algebra of operators on a Hilbert space $H$. Let's denote by $\mathcal P$ the set of projections in $\mathcal A$, that is $\mathcal P:=\{a\in \mathcal A ...
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53 views

Projective sequence of C*Algebras by factors of embedded ideals isomorphic to algebra

Let $A$ be a $C^*$-algebra and $$A = I_1 \supset I_2 \supset I_3 \supset\ldots$$ be a sequence of embedded ideals in $A$ such that $\bigcap_{i=1}^\infty I_i = 0$. Is it true, that the projective limit ...
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Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
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37 views

Sequence of ideals in C*-algebra such that its intersection are not zero but intersection of dense sets are zero

Let $$A_0 \supset A_1 \supset A_2 \supset A_3 \supset\ldots$$ be a sequence of $C^*$-algebras, such that $A_n$ is an ideal in $A_m$ (while $n>m$), and $K_i \subset A_i$ be a dense ${}^*$-subalgebra ...
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21 views

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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Left adjoint functor to forgetful functor from C*-algebras to *-algebras category [closed]

Does exist left adjoint functor of forgetful functor from category of C*-algebras to category *-algebras?
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Does universal enveloped C*-algebra is continuous functor?

Let $K$ is category of *-algebras that have next property: for each $x \in B$ (where $B$ is *-algebra) $\sup_{\pi - bounded}||\pi(x)|| < \infty$ where $\pi : B \to B(H)$ - is some bounded ...
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When is the set $\{ x^* x : x \in A\} $ a cone in a *-algebra?

So in a $C^*$-algebra, every positive element can be written as $x^*x$, and the set of positive elements form a cone. What if we remove all information about the norm? Say $A$ is an algebra, with ...
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35 views

Uniqueness of positive square root of postive element in C* algebra

If a is a positive element then it has a unique positive square root, i.e. a unique b positive such that b^2=a. I understand the existence part of the proof. If ...
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Is the algebra of universally integrable functions a von Neumann algebra?

I would like to continue this discussion. Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel ...
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С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
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Meaning of non-degenerate representation in $C^*$-algebras

A representation of a $C^*$-algebra, $A$, is a pair $(H,\pi)$ where $H$ is a Hilbert space and $\pi$ is a *-homomorphism from $A$ to $B(H)$. A representation is non-degenerate if $\{\pi(a)h:a\in A, ...
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Why is the canonical approximate identity positive?

Let $\{u_n\}$ be an approximative identity for C*-algebra A. Why the element $1-u_n$ is also positive? I'm asking about it because in some theorem we use something like this: $\mid \mid a-u_n a \mid ...
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Why is $(\sqrt{P})^2=P$ where $P$ is a positive operator on a Hilbert space?

The following is a proposition regarding positive operators on a Hilbert space in Douglas's Banach Algebra Techniques in Operator Theory: Corollary 4.32 is as the following: I understand that the ...
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$C^\ast$ condition implies $B^\ast$ condition

By $C^\ast$ condition I understand $\|A^\ast A\|=\|A^\ast\|\|A\|$ and for $B^\ast$, $\|A^\ast A\|=\|A\|^2$. I know these conditions are equivalent even NOT assuming the involution is isometric, but I ...
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Morphism: Unitization

Given C*-Algebras $\mathcal{A}$ and $\mathcal{B}$. (Possibly unital!) Morphisms are contractive: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\|\varphi\|\leq1$$ (Possibly nonunital!) How to apply ...
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A point-wise separation Hahn-Banach theorem in C*-algebras

Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$. Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...