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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Are elements of a $C^*$-Algebra strictly positive iff their spectrum is strictly positive?

Let $A$ be a $C^*$-Algebra. An element $a\in A$ is said to be positive iff $a=a^*$ and the spectrum $\sigma(a)$ is nonnegative, ie. $\sigma(a)\subset[0,\infty)$. This is equivalent to $\varphi(a)\ge ...
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0answers
11 views

Irreducible representation of $C^*(D_\infty)$

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$ Ultimately, I'm interested in finding a ...
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28 views

K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
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13 views

Primitive ideal space of C*(Z2*Z2)

Find the primitive ideal space, the center, a continuous field of $C^*(Z_2*Z_2)$. Here, $C^*(Z_2*Z_2)$ is the full group $C^*$-algebra. I know the definitions of all of them, but I'm having hard ...
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32 views

Universal properties of certain crossed products

So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. ...
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1answer
35 views

A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
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18 views

Determining primitive ideal space of C* algebra

What is the general way of determining the space of primitive ideals of the C* algebra if there is any? Thanks.
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34 views

A Lemma about the operator space

The following lemma comes from the book "C*-algebras Finite-Dimensional Approximations" by N.P. Brown and N. Ozawa P379 Lemma 13.2.3 Let $X_{i}\in B(H_{i})$ (i=1,2) be unital operator subspaces ...
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1answer
48 views

why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$?

Let $A$ be a $c^*$algebra, $x\in A$ and $f:\sigma(x^*x)\to\mathbb{C}$ continuous and $f(0)=0$ ($\sigma(x^*x)$ is the spectrum of $x^*x$). Why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$? It is ...
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1answer
21 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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28 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
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1answer
38 views

A question about $n$-dimensional operator space

Let $F_{n-1}$ be the free group of rank $n-1$ and $C^{*}(F_{n-1})$ be the universal group C*-algebra of $F_{n-1}$. And if $E_{n}$ is the $n$-dimensional operator space in $C^{*}(F_{n-1})$ spanned by ...
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1answer
27 views

Characterization of positive elements of a sub-C*-algebra of $B(H)$

Let $A$ be a non-unital sub-C*-algebra of $B(H)$. I want to show that if $T\in A$ and $\lambda \in \mathbb{C}$ are such that $T+\lambda I_H\geq 0$ then $T$ is self-adjoint and $\lambda \geq 0$. Let ...
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1answer
45 views

Operator matrix is invertible if and only if its determinant is invertible

Let $A,B,C,D$ are pairwise commutative operators on a Hilbert space $H$, then a necessary and sufficient condition that the operator matrix $$\begin{pmatrix} A&B\\C&D\end{pmatrix}$$ be ...
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1answer
32 views

Natural Ordering of the Class of Hermitian Preserving Maps

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices: For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that ...
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2answers
29 views

realizing/ understanding $C^*(\phi_g(C([0,1])))$ and "support projection of an element of a $C^*$-algebra

someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let $C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}$, $g\in C([0,1])$ be a positive ...
2
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1answer
25 views

Does every Hausdorff Compact space admits a (Radon) measure with full support?

The title says it all. I know a bit of C*-algebra theory, so using Gelfand and Riesz theorems, this is equivalent to Does every unital abelian C*-algebra admit a faithful state? I believe this ...
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2answers
30 views

Why the self-adjointness condition for positivity of an element of a C*-algebra?

A positive element x of a C*-algebra A is a self-adjoint element whose spectrum is contained in the non-negative reals. If there's a faithful finite-dimensional representation of A where the ...
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69 views

understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
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1answer
44 views

Questions about the Kapalansky density theorem

I'm studying Takesaki's Theory of operator algebras book by myself. The following is a theorem from that book: I have several questions about this proof: 1- He claims, in the first line of ...
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1answer
53 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
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19 views

Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and ...
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36 views

Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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1answer
31 views

positive part of an element of C* algebra

Consider A = $\bigl( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \bigr)$ B = $\bigl( \begin{smallmatrix} s & 0 \\ 0 & t \end{smallmatrix} \bigr)$ as elements of $M_2(C)$ ...
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0answers
30 views

Finite dimensional C*-algebras and spectrum of each its elements

Let $A$ be a finite dimensional C*-algebra. when I say finite dimensional I mean there is $x_1,...,x_n\in A$ such that $A=span\{x_1,...,x_n\}$(C*- algebra generated by $\{x_1,...,x_n\}$)(is it ...
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21 views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
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15 views

A Vietoris (hyperspace) functor for commutative unital $C^\ast$-algebras

There is a well known hyperspace functor $V \colon \mathbf{KHaus} \to \mathbf{KHaus}$ on the category of compact Hausdorff spaces. This is defined as follows: For objects $V(X) = \{K \subseteq X ...
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1answer
32 views

Gelfand duality restricted to the category of Stone spaces

It is well known that the category of unital commutative $C^\ast$-algebras is dually equivalent to the category $\mathbf{KHaus}$ of compact Hausdorff spaces. I have found that restricting the ...
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1answer
37 views

Finite dimensional C*-algebra

Let $A$ be a simple and finite dimensional C*-algebra. We first note that $aAb\neq 0$ for every nonzero $a,b\in A$. Let $B$ be a maximal abelian self-adjoint subalgebra of $A$. Being finite ...
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36 views

Positive element of a C*-algebra

Let $A$ be an abelian C*-algebra and $p$ be a projection in $A$. To show $p$ is an extreme point of $A^+_{\|.\|\leq 1}$ suppose there is $b,c\in (A^+)_{\|.\|\leq 1}$ such that $p= \frac{1}{2}(b+c)$ ...
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1answer
23 views

If $A$ is an abelian C*-algebra, and $\tau$ is pure then it is a character on $A$

If $A$ is an abelian C*-algebra,and positive linear functional $\tau$ is pure then it is a character on $A$. Murphy in his book(C*-algebras and operator theory) has below proof: While I think we can ...
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1answer
67 views

Existence of idempotents versus existence of projections in a C*-algebra

Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections? Clearly, when $x$ is ...
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1answer
20 views

Irreducible representations of $C(T,B(X))$

Let $T$ be a compact topological space, $X$ a finite-dimensional Hilbert space, $B(X)$ the algebra of operators in $X$, and $C(T,B(X))$ the $C^*$-algebra of continuous maps from $T$ into $B(X)$ (with ...
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0answers
26 views

Matrix representation of an operator

Murphy says : The pure states of $A=K(H)$ are precisely the states $\omega_x : A\to \Bbb C ~~;~~\omega_x(u) = \langle ux,x\rangle $ where $x$ is a unit vector of Hilbert space $H$ . Then he gives ...
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3answers
48 views

Non-closed ideals in $C^*$-algebras

What is an example of an ideal in a commutative $C^*$-algebra that is not closed? If by chance every ideal in a commutative $C^*$-algebra is closed, how about in non-commutative $C^*$-algebras? ...
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37 views

null power element in a C*-algebra

Let $A$ be a C*-algebra. Show that there is $x\in A$ such that $x^2=0$. I think in abelian C*-algebra $x^2=0$ if and only if $x=0$(because these elements are continuous functions) Also in certain ...
2
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1answer
34 views

positive element in a Banach $*$- algebra

By definition, $a$ is positive in C*-algebra $A$ if $\sigma(a) \subset \Bbb R^+$. I would like to know the definition of a positive element in a Banach $*$-algebra. I think it's the same as the ...
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1answer
28 views

Affiliated algebras

I am searching for articles on affiliated $C^*$-algebras and affiliated von Neumann algebras. I know that Wonorowicz wrote articles about this topic and also Pedersen wrote a section in his book ...
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32 views

Positive linear functional on a C*-algebra is bounded

The following is a theorem of Murphy's C*-algebras and operator theory: My question: I think in the proof of theorem, Murphy uses the assumption $|\tau(a)|<M$ for positive elements $a\in ...
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0answers
84 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
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0answers
19 views

Positive elements in C* algebra

For $x$ in $C^*$ algebra $A$, $|x|=(x^*x)^{1/2}$ and there holds $x \leq |x|$. If $x$ is a positive element in $A$, does it hold $x = |x|$?
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An invertible hermitian element of a C*-algebra has a logarithm

Suppose $ A$ is a C*-algebra. Show that an invertible hermitian element of $A$ has a logarithm. ($a$ has a logarithm if there is an element $b\in A$ such that $e^b=a$) If $a\in A_+$ then it's easy ...
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1answer
18 views

Show that the orthogonal decomposition for a hermitian element of a $C^*$-algebra is unique

I am reading about $C^*$-algebras from Chapter VIII in Conway's A Course in Functional Analysis. I've come across the following proposition which describes the "orthogonal decomposition" for hermitian ...
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1answer
53 views

Maximal ideal space

Let $X$ be a compact space, $x_0\in X$, and define $$A=\{\{f_n\} ; f_n\in C(X), \sup_n\|f_n\|<\infty, and \{f_n(x_0)\} \text{ is a convergent sequence} \} $$ If $\|\{f_n\}\|$ is defined as ...
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1answer
47 views

A question about induced $C^\ast$-algebra

Recently, I read the book Crossed Products of C*-algebras, and meat a question. The question is how to prove $\mathrm{Ind}_c(A,\alpha)$ is dence in $\mathrm{Ind}(A,\alpha)$. On the page 102, the ...
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1answer
31 views

Compact space X is totally disconnected if and only if C(X) is generated by its projections

If $X$ is compact, show that $X$ is totally disconnected if and only if $C(X)$ as a C*-algebra is generated by its projections. My attempt: Suppose $X$ is totally disconnected, then $X=\{x_i\}_{i\in ...
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1answer
55 views

Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
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1answer
33 views

Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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1answer
63 views

Sums of projections in a C*-algebra

Let $A$ be a $C^*$-algebra, and let $p_1, \ldots, p_n \in A$ be projections, meaning $p_i = p_i^* = p_i^2$. Now assume that the sum $p = p_1 + \ldots + p_n$ is also a projection. How can one show that ...
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1answer
24 views

characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that ...