Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [operator-algebras]

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects of , , , and .

3533 questions
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simple connectedness and abelian C* algebras

From Gelfand-Neumark Theorem, we know that topological properties of a compact Hausdorff space $X$ are encoded in the abelian $C^*$-algebra of continuous complex-valued functions on $X$ (with $||f||= \sup |f|$). For example, the existence of…
nsoum
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Show this representation is irreducible and faithful

Let A be a prime $C^{*}$ algebra and $e= \alpha^{-1} c^{*}c$ for some $c \in A$ which satisfies $(c^{*}c)^{2}=\alpha c^{*}c$. Then $e^{2}=e=e^{*}$ and $eAe=\mathbb Ce$ so $eAe$ may be identified with $\mathbb C$. Endow $Ae$ with an inner product…
Arundhathi
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$\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n)) = \mathcal{K}(L^2(\mathbb{R}^m)) \otimes \mathcal{K}(L^2(\mathbb{R}^n))$?

QUESTION: Is it true that for the algebra of compact operators: $\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n))$ is as a $C^{\ast}$-algebra isomorphic to $\mathcal{K}(L^2(\mathbb{R}^m)) \otimes \mathcal{K}(L^2(\mathbb{R}^n))$? The latter tensor…
user44568
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universal C* algebras

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would like to read the initial papers or some standard…
nsoum
  • 894
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1 answer

Integral Domain and $C^*$-algebra

If $A$ is a commutative $C^*$-algebra with identity, then can it be an integral domain? I think the answer is no, because one can use the Gelfand Transform to get an isomorphism of $A$ with $C(\Omega (A))$ where $\Omega(A)$ is the space of…
Ester
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Extension of elements of Spectrum

Let $A$ be a commutative $C^*$ algebra with unity, i.e a Banach algebra with an involution operation $^*$ satisfying the identity $\|x^{*}x\|=\|x\|^2$ for all $x\in A$ and $B$ be a $C^*$-subalgebra of $A$ having the same unity as that of $A$. If $f:…
Ester
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Extreme points and Matrix Extreme Points

With reference to this paper. Let $V$ be a locally convex space, and $K=(K_n)$ be a compact matrix convex set in $V$. Then as proved in Cor 3.6 in the above paper, we see that if $v\in K_n$ is a matrix extreme point, then it is also extreme in the…
Nirakar Neo
  • 889
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How to prove compactness of matrix convex sets?

I am reading a paper - "the Krein Milman theorem in Operator Convexity"; and the third section there deals with compact matrix convex sets. The first example there states that the matrix interval $[a\mathbb{I},b\mathbb{I}]$ is a compact matrix…
Nirakar Neo
  • 889
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Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here $\mathcal{L}(\mathcal{H})$ denotes the bounded operators on $\mathcal{H}$…
user44568
  • 103
3
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1 answer

C* identity origins

In the context of $C^*$-algebras , why is the $C^*$-identity a "natural" one to choose ? ($||a^* a||=||a||^2$). Some books try to motivate this by noting that bounded operators on a Hilbert space have the above property. Is this an optimal choice in…
nsoum
  • 894
3
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2 answers

Strictly positive element in unital $C^*$ algebra is invertible

Let A be a unital $C^∗$ algebra. $a\in A_+$ is called strictly positive if for all positive linear functional $\phi \in A_+^*$ and $\phi \neq 0$, we have $\phi(a)>0$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertible. My attempt: if…
John
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Subprojection of a finite projection

The question is entirely explained here, in that I wonder why every source seems to regard as obvious the claim that subprojections of finite projections are finite. Here is the link. To me, playing around with the tools and relations at hand…
Jeff
  • 7,139
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Polar decomposition in a finite von Neumann algebra

Suppose $\mathcal{M}$ is a finite von Neumann Algebra and $a \in \mathcal{M}$. Does it follow that we can write $a = |a|u$ where $u$ is a unitary operator. I know we can write $a = |a|u$ where $u$ is a partial isometry.
Mustafa Said
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2
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Exponential map in $C^{*}$-algebra and unitary invariance

Let $A$ be a unital $C^{*}$-algebra. Let $X$ be a closed vector subspace of $A$ which is unitarily invariant in the sense that $uXu^{*}\subseteq X$ for all unitaries $u$ of $A$. I want to show that $ba-ab\in X$ whenever $a\in X$ and $b\in A$. A…
cyc
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Question considering masas

Suppose $M_1, M_2$ are type II_1 factors, A is a masa in $M_1\otimes M_2$, can we find a *-automorphism $\phi$ on $M_1\otimes M_2$ such that there are two masas $A_1\subset M_1,A_2\subset M_2$ and $\phi(A)$ is isomorphism to $ A_1\otimes A_2$? Here,…
ougao
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