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 and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
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Connected components of group of unitaries in Calkin algebra

Let $H$ be a separable infinite dimensional Hilbert space. Denote the Calkin algebra by $Q(H)=B(H)/K(H)$, and $U(Q(H))$ the group of unitaries in $Q(H)$. I'm trying to show that the map $F: U(Q(H))/...
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Unitary element in an AF $C^*$-algebra can be approximated by sequence of unitaries

Let $A$ be a unital AF $C^*$-algebra. Write $A=\overline{\bigcup_{k\in \mathbb{N}}A_k}$ where each $A_k$ is a unital (with the same unit of $A$) finite dimensional $C^*$ subalgebra. Suppose $u\in A$ ...
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22 views

There is no trace on Cuntz algebra

Here is a general explanation why purely infinite $C^*$-algebras admit no tracial states: Non-existence Tracial states. Is my following explanation for non existence of trace on Cuntz algebra $O_n$ (...
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26 views

SOT-isomorphic C*-algebras

Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras. Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that: $p_n \rightarrow 1$ in strong operator ...
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27 views

Projective limit of finite dimensional C* algebras

Let $A$ be a separable unital $C^*$-algebra and $A$ = $I_0 \supset I_1 \supset I_2 \supset \ldots$ Be a sequence of ideals in $A$ such that: $I_k$ is ideal in $I_m$ when $k \geq m$ $\bigcap I_k = \...
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Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
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about representations of a simple $C^*$-algebra

We know that every simple $C^*$-algebra is primitive, say it has a faithful non-zero irreducible representation. The converse is not necessarily true. An counterexample is just the $B(H)$ when $H$ is ...
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38 views

prove that elements in $K_1(A)$ coincide

Let $A$ be a unital $C^*$-algebra $u\in A$ unitary and $s\in A$ isometry. I already proved that $sus^*+(1-ss^*)$ is an unitary. Why is $[u]_1=[sus^*+(1-ss^*)]_1\in K_1(A)$? Basic definitions: ...
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almost unitaries are close to a unitary element

I need help to prove the following exercise: Let $\epsilon >0$. Show that there exists $\delta >0$ with the property: If $A$ is a unital $C^*$-algebra and $x\in A$ such that $\|x^*x-1\|<\...
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Can you give an examples of non commutative non C*algebras?

Are there examples apart from $B(X)$ where $X$ is not a Hilbert space and not finite dimensional. Do they have a characterization or representation?
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Norm of a unital homomorphism

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\...
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30 views

Equivalence of some characterization of pure states

I'm looking for a reference or a proof for these well-known facts in $C^*$-algebras theory for which, however, I havent found any clearly written proof of the same type of ones I will sketch. ...
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41 views

Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
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1answer
19 views

homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
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40 views

Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
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66 views

Frechet derivative of square root on positive elements in some $C^*$-algebra

Let $A$ - is some unital $C^*$ algebra, and $P$ is set of all strictly positive elements in $A$. We can define map $\sqrt{?} : P \to A$ which takes positive element and returns its (unique) strictly ...
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46 views

Differential $\sqrt{1+B(x,x)}$ map in $C^*$-algebra

Let $A$ is $C^*$-algebra and $P \subset A$ is subset of all elements $a \in A$ such that $a > 0$ (nonnegative) and $||a|| < \frac{1}{\sqrt{1-q}}$ (norm bounded) for some $0 < q < 1$. Let $...
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54 views

Semi-direct product of groups

The situation is the following: Let be $G$ a locally compact (Hausdorff) group such that $G = H \rtimes_{\alpha} N$ is the semi direct product of locally compact groups $N$ and $H$. Let $A$ be a C$^*$-...
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$C^*$-algebra desription of the complex Clifford algebra

I read somewhere a discription of the complex Clifford algebra as a $C^*$-algebra, but I don't know where... Is the complex Clifford algebra the universal $C^*$-algebra generated by elements $1$ and $...
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1answer
29 views

equivalent trace-conditions on $C^*$-algebras

Let $A$ be a $C^*$-algebra and $\tau:A\to\mathbb{C}$ linear. Claim: the following conditions are equivalent: $\tau(ab)=\tau(ba)$ for all $a,b\in A$ $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$ $\tau(uau^...
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$A$ has a countable dense subset. How to describe a possible countable dense subset in $M_n(A)$?

Let $A$ be a C$^*$-algebra and $M$ be a countable dense subset in $A$. Let $M_n(A)$ be the $C^*$-algebra of $n\times n$-matrices with entries in $A$, $n\in\mathbb{N}$. Then $M_n(A)$ should have a ...
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Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
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Bott projection as $K_1$ class

Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book $K$-theory and $C^*$-algebras) given by $b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{...
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1answer
36 views

Definition of C$^\ast$-algebra: which conditions can be deduced from the others?

A C$^\ast$-algebra is a Banach algebra $A$ with an involution, i.e. a map $\ast$ such that: $(x^\ast)^\ast=x$ for all $x\in A$; $(x+y)^\ast=x^\ast+y^\ast$ for all $x,y\in A$; $(ax)^\ast=\overline ax^...
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A comparison between absolute values of functionals

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
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1answer
23 views

In a C*-algebra, pure states which share the same kernel are equal

I'm reading C*-Algebras by Jacques Dixmier. And in the proof of 2.9.5, it says Let $A$ be a C*-algebra. If $f$ and $f'$ are two pure states which have the same kernel, then $f=f'$. It should ...
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Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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37 views

Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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1answer
42 views

Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
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$A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
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1answer
51 views

Elementary proof that $a - 1$ is not invertible, for self-adjoint $a$ with $\lVert a \rVert = 1$

Assume $a \in A$ where $A$ is a unital $C^*$-algebra. If $\lVert a \rVert = 1$ and $a^*=a$ we know that $1 \in \sigma(a)$, the spectrum of $a$. This follows from the fact that $\lVert a \rVert = r(a) =...
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Classifying Unitaries on the Circle with K-Theory

Let $S^n$ be the $n$-sphere. I'm trying to understand the "meaning" of the $\mathbb{Z}$ factors in $$ K_0(C(S^{2n+1}))\cong\mathbb{Z}$$ and $$ K_0(C(S^{2n}))\cong\mathbb{Z}\oplus\mathbb{Z}$$ So $S^n$...
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1answer
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Split exact sequence of $C^*$-algebras

Let $\left\{A_i\right\}_{i\in I}$ be a countable set of $C^*$-algebras $A_i$ and $ \bigoplus_{i\in\mathbb{N}}A_i$ the direct sum as a $C^*$-algebra. Let $A_i^1$ the unitization of $A_i$ and $c_0$ the ...
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1answer
33 views

Convergence of specific sequence in specific reduced group algebra equivalent to convergence of norms

Let $A = C_r^*(S_\infty)$ where $S_\infty$ - is permutation group of natural numbers fixing all but a finite number of element. Let $A_n = C_r^*(S_n)$ - subspaces of $A$ and $P_n : A \to A$ is ...
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1answer
24 views

Convergence of specific sequence in reduced group algebra

I have $B = C_r^* (S_\infty)$ - reduced group algebra of permutation group of naturam numbers fixing all but a finite number of element. $B$ have countable family of subalgebras: $B_n = C_r^* (S_n)$ ...
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1answer
38 views

Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
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Waterhouse example in category of $C^*$ algebras

I am trying to produce an example of surjective inverse system of $C^*$ algebras with empty inverse limit in analogy to Waterhouse example. So far I was trying something weaker namely to find ...
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1answer
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When is an isometry a complete isometry?

Let $A$ be a unital subspace of a $C^*$-algebra. Then every contractive linear map $\varphi$ with $\varphi(1) = 1$ from $A$ into a commutative $C^*$-algebra is completely contractive. Can anything ...
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1answer
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the standard even grading on $M_2(A)$ and $A\otimes \mathbb{K}$

I have a question about a passage in Blackadar's book about K-Theory. Let $A$ be a (ungraded) $C^*$-algebra. There is a grading on $M_2(A)$ with $M_2(A)^{(0)}$ the diagonal matrices and $M_2(A)^{(1)}$...
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Direct limit of certain $C^*$ algebras is simple

Let $X$ be a compact Hausdorff space. Let $(x_n)$ be a sequence in $X$.Assume $X$ has no isolated points. Define $A_n = C(X, M_{2^n}(\mathbb{C}) )$ and define $\phi_{n+1,n} : A_n \to A_{n+1}$ by $$\...
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1answer
35 views

On Schauder basic systems in universal enveloped algebra of system of countable family of bounded selfadjoint operators

Let $A = C^*(1,T_1,T_2, ... | T_i^* = T_i, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of selfadjoint operators. I want to know as more as possible about that algebra,...
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1answer
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If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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1answer
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Why does Conway state $\partial\sigma(a)=\sigma(a)$ when $a$ is a Hermitian element of a $\mathcal{C}^\ast$-algebra?

Here is the offending proof: $\bf 1.14.$ Proposition. Let $\scr A$ and $\scr B$ be $C^*$-algebras with a common identity and norm such that $\scr A\subseteq \scr B$. If $a\in\scr A$, then $\sigma_{...
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Formal series which uniformly bounded in each representation of universal $C^*$ algebra converge

Let $A = C^*(T_1,T_2,...|T_i^* = T_i, ||T_i||\leqslant 1)$ - universal $C^*$ algebra of countable family of selfadjoint operators. I have formal series $x = \sum_{i_1,...,i_k} \alpha_{i_1, ..., i_k} ...
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1answer
18 views

Unitization of Suspension

Let $A$ a C*-algebra (unital or not). Its suspension is defined to be: $$ S(A) \equiv A\otimes C_0((0,1);\,\mathbb{C}) $$where $C_0$ denotes all continuous functions which vanish at infinity. We ...
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41 views

Convergence of sequence of vectors in $C^*$-algebra

Let $B$ is $C^*$-algebra and $x_i \in B$ - linear independent vector system, $\alpha_i \in \mathbb{C}$ such that: $$\|x_i\| = 1$$ $$\lim_{N \to \infty} \|\sum_{k=1}^N \alpha_k x_k\| = \lim_{N \to \...
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2answers
68 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
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1answer
29 views

Why is $C(\beta \mathbb{R})/C_0(\mathbb{R})\cong C(\beta \mathbb{R}\setminus \mathbb{R})$ as $C^*$-algebras?

Let $\beta \mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$ (with euclidean topology) and $C_0(\mathbb{R})$ the $C^*$-algebra of continuous complex-valued functions vanishing at infinity....
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definition of Kasparov modules, what is a degree 1 operator?

I read the book "K-theory for operator algebras" and I have a question about the definition of Kasparov modules for graded $C^*$-algebras $A$ and $B$. Definition: Let $A$ and $B$ graded $C^*$-...