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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 .

2807 questions
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Explicit description of $C^{\ast}$-algebra generated by a subset

Given a $C^{\ast}$-algebra $\mathcal{A}$ and a subset $S\subseteq\mathcal{A}$, denote by $C^{\ast}(S)\subseteq\mathcal{A}$ the minimal $C^{\ast}$-algebra in $\mathcal{A}$ containing $S$. This is what I mean by the $C^{\ast}$-algebra generated by…
MWL
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Cyclic representations on $C(X)$: if $\pi_\mu\oplus\pi_\nu$ is cyclic, then $\mu\perp\nu$

I'm using the definitions/conventions of $C^*$-algebras and their representations from Conway's Functional Analysis (VIII.5). If I need to supplement any definition, let me know. Let $X$ be a compact Hausdorff space. Let $\mu$ be a positive Borel…
Sha Vuklia
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Contractive unital linear map is positive

Reading Brown Ozawa book on C* algebras they seem to imply any contractive unital linear map between C* algebras is positive. Does this hold true? And if so why? I don't see why something so general would be positive
sirjoe
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Norm on a certain quotient C$^*$-algebra

Let $\{A_i\}_{i\in I}$ be an arbitrary family of C$^*$-algebras, we may define their product as $$\prod_{i\in I}A_i=\{(a_i);~\sup_{i\in I}\|a_i\|<\infty\}.$$ We can also define $$\sum_{i\in I}A_i=\{(a_i);~\text{For any }\epsilon>0\text{ there only…
user505379
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Can distinct elements of a $C^*$-algebra be separated by a maximal left ideal?

Let $A$ be a $C^*$-algebra, and let $f\neq g\in A$. Does there exist a maximal ideal $J\trianglelefteq A$ with $f+J\neq g+J$? I'm particularly interested in the case of $A=B(\mathcal H)$, and why things aren't obvious. In this case, $f\neq g$…
Ashwin Trisal
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How to understand the spectrum of a C*-algebra?

I know that the spectrum of an element $x$ in a unital C*-algebra $A$ is defined as $$\operatorname{Sp}_{A} x=\left\{\lambda\in\mathbb{C}\mid (x-\lambda\cdot1)\ \text{is not invertible}\right\}.$$ Reference materials I am reading all seem to assume…
Frenzy Li
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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 involution is conjugate transposition, I think the second…
Jeffrey
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C*-algebra of polynomials?

Let $A$ be a C*-algebra. Consider its cartesian square $A^2$ and define a multiplication on $A^2$ by the identity $$ (x_0,x_1)\cdot (y_0,y_1)=(x_0y_0,x_0y_1+x_1y_0),\qquad x_0,x_1,y_0,y_1\in A $$ This turns $A^2$ into the algebra of "polynomials of…
Sergei Akbarov
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Why is $1 + i c^{-1/2}dc^{-1/2}$ invertible?

I am reading a proof of this theorem: If $a,b$ are positive elements of a $C^\ast$ algebra and $a \le b$ then $a^{1/2}\le b^{1/2}$. I don't understand one step in the proof. I understand this: Let $t > 0$ and $c,d$ be such that $c + i d = (t + b +…
user167889
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non-faithful KMS states

I am wondering whether a state $\omega$ on a $C^*$-algebra which is KMS (http://en.wikipedia.org/wiki/KMS_state) with respect to the group of automorphisms $\tau^t$, $t\in\mathbb{R}$, and at a given inverse temperature $\beta$ can be non-faithful.…
user162526
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Free product of $C^*$-algebras

i want to figure out what the "free product of two $C^*$-algebras" precisely is. Thus suppose $A$ and $B$ are $C^*$-algebras. How to find or construct $A\ast B$? I readed something about the $C^*$-algebra generated by $A$ and $B$. Is $A\ast B$…
Gerd13
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Why isn't complex conjugation a *-homomorphism?

By Gelfand-duality with $1$, the set $\{pt\}$ is homeomorphic to $\mathrm{Spec}(C(\{pt\})) \cong \mathrm{Spec}(\mathbb{C})$, the set of nonzero *-homomorphisms $\mathbb{C} \rightarrow \mathbb{C}$ with the pointwise convergence topology. In…
user98595
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a condition for a $C^*$ algebra to have a unit?

I am looking for a hint or a methodology to approach this problem, showing in Arveson's A short course on spectral theory: Let $X$ be a compact Hausdorff space and let $F$ be a proper closed subset of $X$. Let $A$ be the ideal of all functions $f\in…
NotaChoice
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A question about a part in a proof regarding approximate identities in $C^\ast$-algebra

Let $A$ be a $C^\ast$-algebra with approximate identity $\{e_\lambda\}_{\lambda\in\Lambda}$. The author states the following comment in the trenches of a proof: Since $t^2\leq t$ on $[0,1]$, it follows $e^2_\lambda\leq e_\lambda$. Intuitively it…
Rough_Manifolds
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Question about paths of unitaries in matrices over a C*-algebra

Let $A$ be a unital C*-algebra and let $U(A)$ be its unitary group. Let $U_2(A)$ be the unitary group of the C*-algebra $M_2(A)$ of 2-by-2 matrices over $A$. Let $u \in U(A)$. Suppose there is a (norm-continuous) path from $\left(…
Mike F
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