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Questions tagged [c-star-algebras]

A C-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^=b^a^$ and the C-identity $\Vert a^a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$.

For bounded operators on a given Hilbert space, C-algebras characterize topologically closed subalgebras of ${\mathcal B}({\mathcal H})$ (in operator norm), also closed under taking the adjoint operator. C-algebras are at the heart of and are extensively used in .

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Explicit isomorpshim between $C^*(\mathbb{Z})$ and $C(S^1)$

I know that for an abelian group $G$, $C^*(G)$ is isomorphic to $C(\hat{G})$. Hence we have $C^*(\mathbb{Z})$ isomorphic to $C(\hat{\mathbb{Z}})$ which is inturn isomorphic to $C(S^1)$. Can we have an explicit isomorphism between $C^*(\mathbb{Z})$…
budi
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$\ast$-preserving homomorphism between $C^{\ast}$-algebras

An elementary question: A $\ast$-preserving homomorphism between $C^{\ast}$-algebras is positive. Is there any condition that makes a positive homomorphism, $\ast$-preserving?
user78800
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Murray V.N equivalence of projections in finite C* algebras.

In general, Murray von Neumann equivalence of projections is not the same as unitary equivalence. In fact in a unital infinite C* algebra 1 is Murray von Neumann equivalent to a proper subprojection p and so these are going to be murray von Neumann…
sirjoe
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Why are these two definitions of amenability equivalent?

Let $A$ be a C*-algebra. 1, $A$ is said to be amenable if every derivation from $A$ to some dual Banach $A$-bimodule is inner. 2, $A$ is said to be amenable if for every finite set $F\subset A$ and $\epsilon>0$ there is some $M_n$ and contractive…
Sui
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Representation of a C* algebra

Let $A$ be a $C^*$- algebra. Let $\pi$ be a representation of $A$ on a Hilbert space $H$. Let $p \in A$ be a projection. Consider the $C^*$- algebra $pAp$. Let $P=\pi(p)$. I want to represent $pAp$ in $PH$. Is it correct if I define $\tilde \pi: pAp…
budi
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C* algebra isomorphism

I have two unital $C^*$ algebras $A$ and $B$. Let $X\subset A$ and $Y\subset B$ be such that span $X$ and span $Y$ are dense * subalgebras of $A$ and $B$ respectively. I have a map $\Psi: Span ~X \to Span ~Y$ which is a *-homomorphism. Also I have a…
budi
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Question on matrix decomposition in a C*-algebra.

I am reading a book entitled an introduction to the classification of amenable C*-algebras, and the notion of matrix decomposition has come up several times(without any definition or proof). Such as on page 146, it reads, Define…
Sui
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How to calculate $\left\|\left(\begin{array}{}0&F\\F^*&0\end{array}\right)\right\|$ in $M_2(A)$?

My approach: let $v$ be a partial isometry, outside $A$, such that $v^*va=av^*v=a$ and $vv^*a=avv^*=0$ and $vv=0$ for every $a\in A$. Then by defining $\varphi\left(\begin{array}{}a&b\\c&d\end{array}\right)=a+bv+v^*c+v^*dv$ we see $M_2(A)$ is…
Sui
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A question on hereditary C*-subalgebra.

From: An introduction to the classification of amenable C*-algebras. Lemma 3.5.8. Let $A$ be a C*-algebra satisfying the condition that every hereditary C*-subalgebra contains at least two mutually orthogonal nonzero positive elements. Then for any…
Sui
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For a C*-algebra $A$ and arbitrary $x\in A$, is $\overline{x^*Ax}$ and $\overline{x^*xA x^*x}$ the same algebra?

It is obvious that $\overline{x^*xx^*Axx^*x}=\overline{x^*xAx^*x}$ but I have no idea whether $\overline{x^*Ax}=\overline{x^*xAx^*x}$ or not. Any hint would be appreciated.
Sui
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Extension of states

Let $X$ be a Locally compact Hausdorff space and $Y$ is a compact subspace of $X$. Let $\phi$ be a state on $C(Y)$. Then can we extend to $\phi$ to $C_0(X)$? Suppose if $T:f \mapsto f|_X$ is the homomorphism from $C_0(X)$ to $C(Y)$, then will the…
budi
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States on the $C^*$ algebra $C_0(X)$

Let $X$ be a locally compact second countable space. Consider the $C^*$- algebra $C_0(X)$. Is it true that there is a one to one correspondence between states on $C_0(X)$ and Borel probability measures on $X$? Or is it true only $X$ is compact? In…
budi
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Characters of a $C^*$- algebra

By a Character of a commutative unital $C^*$-algebra $A$ means a non zero homomorphism from $A$ to $\mathbb{C}$. Is it true that characters preserves positive elements? Or if $\phi$ is a character and $p$ is a projection, is it true that $\phi(p)=0$…
budi
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Countibility of resolvent set in C* algebra

I'm going through the topic $C^*$ algebra and facing few questions. It would be great if you people could help me to clear the doubts. Q3. Does there exists some $X$ belonging to a $C^*$ algebra such that resolvent set is countable. Ans. If suppose…
Leo
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Application of continuous functional calculas

I'm going through the topic C* algebra and facing few questions . It would be great if you people could help me to clear the doubts. Q2. Let $x$ and $y$ be two positive elements in a C* algebra such that ( Then $x$ and $y$ are normal so we can…
Leo
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