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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Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that if $ X $ and $ Y $ are compact Hausdorff spaces,…
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Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras associated with $X$? What is the algebraic analogue of…
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How to understand what a 'noncommutative space' is

Whilst acquainting myself with the fundamentals of $C^*$-algebras and their $K$-theory, I read about how Gelfand Duality allows various $C^*$-algebraic concepts to be directly translated into a geometrical/topological context and vice versa. I…
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$C^*$-algebra which is also a Hilbert space?

Does there exist a nontrivial (i.e. other than $\mathbb{C}$) example of a $C^*$-algebra which is also a Hilbert space (in the same norm, of course)? For $\mathbb{C}^n$ with $n > 1$ the answer is no by uniqueness of norm in $C^*$-algebras, since…
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Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor find a counter example. I wonder whether anyone…
Hui Yu
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What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber bundle $E'$ over $B$ with fiber $C(F,\mathbb C)$,…
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A question about pure state

For every unit vector $x$ in a Hilbert space $H$,let $F_x$ be the linear functional on $\mathcal B(H)$ (bounded linear operators) defined by $F_x(T)=(Tx,x)$. Prove that each $F_x$ is pure state and $\mathcal B(H)$ has pure states which are not like…
Strongart
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Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" imply that $\parallel ab \parallel$ is uniformly…
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When two projections in a C*-algebra are "almost" Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ with $vv^*=p$ and $v^*v=q$. This is considered obvious…
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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 positive square root. How to evaluate its Frechet…
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Why is every positive linear map between $C^*$-algebras bounded?

We know that every positive linear functional on a $C^*$-algebra is bounded. How can we prove every positive linear map between $C^*$-algebras is bounded?
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Square root is operator monotone

This is a fact I've used a lot, but how would one actually prove this statement? Paraphrased: given two positive operators $X, Y \geq 0$, how can you show that $X^2 \leq Y^2 \Rightarrow X \leq Y$ (or that $X \leq Y \Rightarrow \sqrt X \leq \sqrt Y$,…
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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 wonder what is the motivation of this definition.…
No One
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Ideal in $C(X)$

Let $X$ be a compact Hausdorff space, and $C(X)$ the space of complex-valued continuous functions with maximum-norm. The following problem is driving me little nuts. I want to show that every closed Ideal $J\subset C(X)$ is of the form $J_Y = \{f\in…
DinkyDoe
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Isomorphic matrix algebras with non-isomorphic C*-algebras

Let $A$ and $B$ be two $C^{\ast}$-algebras such that their matrix algebras, $M_2(A)$ and $M_2(B)$, are $\ast$-isomorphic $C^\ast$-algebras. Question 1: Are $A$ and $B$ isomorphic $C^\ast$-algebras? In a related case, let $X$ and $Y$ be locally…
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