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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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In a $C^*$ algebra why $\|a\|\not =\rho(a)$ for any $a$?

In a $C^*$ algebra why $\|a\|\not =\rho(a)$ for any $a$? Where $\rho(a)$ is the spectral radius. It can be shown that the equality holds for self-adjoint elements. Then that can be used to show that the Gelfand transform is an isometry. Thus…
Sorfosh
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Equivalent condition for being Positive element of a C*-algebra

This is a problem from the book "A course in functional analysis" by John B. Conway (Chapter 8 Exercise 8.9). If $\mathcal A$ is an unital $\textit C^*$-algebra. For $a\in \mathcal A$, $a\ge 0$ if and only if $f(a)\ge 0$ for every state $f$. One…
Noob mathematician
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Show that $pAp$ is a $C^*$-subalgebra of a $C^*$-algebra $A$.

Let $A$ be a $C^*$-algebra and $p \in A$ a projection, i.e. an element satisfying $p=p^* = p^2$. Is it true that $pAp := \{pup: u \in A\}$ is a $C^*$-subalgebra of $A$? Attempt: Yes, clearly $pAp$ is a complex subalgebra of $A$ so it suffices to…
user745578
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Finding nontivial projections corresponding to a Normal element

In a $C^*$ algebra if we consider a normal element, say $x$, such that spectrum of $x$ is $\left\lbrace-1,1 \right\rbrace$,then can we find two non-trivial projections $p$ and $q$ such that $pq=0$? I'm trying to figure out the answer. My approach…
Leo
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Compact element in a C$^*$-algebra?

Can we define a compact element of an arbitrary C$^*$-algebra $A$? For example, what are the compact elements of C$_0(X)$?
Jo Jomax
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The two projections in an $ C^*$ algebra

Let $A$ be a $C^∗$-algebra, $a \in A$ and let $p,q \in A$ be orthogonal projections (i.e. selfadjoint idempotents with $pq = 0$). Suppose that a is positive and $pap = 0$. Show that $paq = 0$ Could you check my solution of following problem ( i am…
A.Kazakov
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Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces

Let $H_1$, $H_2$ be two separable Hilbert spaces. Let $\mathscr{A}_1$, $\mathscr{A}_2$ be two isomorphic finite-dimensional $C^*$-algebras of operators acting on $H_1$, $H_2$ respectively. Suppose that any eigenvalue of any operator from these…
AAK
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Two isomorphic $C^*$-algebras. What is the isomorphism between corresponding Hilbert spaces?

Let $H$ be a separable Hilbert space. Suppose that $\mathscr{A}$ and $\mathscr{B}$ are some unital $C^*$-algebras of operators acting on $H$, not necessary coinciding with $C^*$-algebra of all the possible operators acting on $H$. Suppose that they…
AAK
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why the irrational rotation algebra $A_{\theta}$ is $C(T^{2})$ when $\theta =0$

since the irrational rotation algebra $A_{\theta}$ is commutative when $\theta =0$, it has the form $C(X)$ for some space $X$ and by universal property of $A_{\theta}$, there is a homomorphism from $A_{\theta}$ to $C(T^{2})$ , so there must be a…
knot
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If $A$ is a unital direct limit of C*- algebras, why can we assume that the connecting maps are unital?

I know that we may assume that each $\phi _{n}$ is injective. Then how to show that we may assume $\phi_n$ is unit preserving when $n \geq N$ for some $N$ ?
user540663
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Do inverse limits always exist in the category of C$^{*}$-algebras

I know that direct limit always exist. But if we follow the same proof, constructing the direct product, how do we know the norm is finite？
user540663
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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 surjective inverse system for which at least one…
J.E.M.S
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positive elements in a $C^*$-algebra

Let $A$ be a $C^*$-Algebra and $a\in A$. I'm stuck in the proof of: $a\ge 0\iff $ it is $\varphi(a)\ge 0$ for all states (=positive linear functionals with norm 1) $\varphi:A\to\mathbb{C}$. Proof: $\Rightarrow$ is no problem, because if $a\ge 0$,…
asaed
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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 elements of a $C^*$-algbera: If $\mathcal{A}$…
JZS
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Existence of a approximate unit $U_{n}^{2}$ for a $ C^{*}$-algebra $

if ${U_{n}}$ is an approximate unit for a $C^{*}$-algebra A. Is ${U_{n}^{2}}$ is an approximate unit for a $C^{*}$-algebra A? Thank in advance.
reza
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