# Tagged Questions

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

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### A remark on the polar decomposition

Let us focus on $L^1(H)$, the space of trace class operators on a Hilbert space $H$. Assume $\{x_n\}$ converges to $x$ (in the trace class norm). Let $x_n=u_n|x_n|$ and $x=u|x|$ be the polar ...
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### Essential ideals

I am trying to get my head around essential ideals. In literature I found 2 definitions: An ideal $I$ in a C*-algebra $A$ is essential in $A$ (i) if $aI = 0$ implies $a=0$, $a\in A$; or (ii) if ...
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### postliminal $C^*$-algebra

A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ‎...
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### Non spatial isomorphisms

Let $H$ be a Hilbert space. Any unitary operator $u\in B(H)$ induces an spatial isomorphism $\phi_u(x)=uxu^*$ on $B(H)$. Question: Let $\phi:B(H)\to B(H)$ be a surjective *-ismorphism. Is $\phi$ ...
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### Equivalent non-degenerate representations of C*-algebras

For two non-degenerate representations $\pi_j:A\to B(H_{\pi_j})$ ($j=1,2$), we write $\pi_1\sim\pi_2$ if there exists a $w^*$-continuous isometrically isomorphism from $\pi_1(A)''$ onto $\pi_2(A)''$...
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### Non-degenerate representations and central projections

Let $A$ be a C*-algebra and $\pi:A\to B(H)$ a non-degenerate *-representation. We also denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$ in $B(H)$. The predual of the von Neumann ...
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### ‎strictly ‎positive elements

Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎. ‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎ *I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in Inv(A)$...
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### Example of an operator that is not subnormal

In some recent questions the term subnormal operator has appeared. A bounded operator $A$ acting on a Hilbert space $H$ is called subnormal if there exists a Hilbert space $K$ containing $H$ as a ...
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### subnormal operator

I know that ‎$‎‎u\in B(H)$ ‎is a‎ ‎normal ‎operator if ‎‎$‎‎uu^*=u^*u$‎. I ‎know ‎that ‎if ‎‎$‎u‎$‎‎ ‎is ‎subnormal ‎‎‎‎then ‎‎‎ ‎‎$‎‎uu^*‎\neq ‎u^*u$ ‎(like unilateral shift operator). ‎‎ My ...
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### Classification of representations of compact $C^*$ algebras for single operators.

I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed ...
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### Maximal ideals of unital commutative $C^*$-algebra?

Define an ideal of a unital commutative $C^*$-algebra $A$ to be a proper subspace $I$ of $A$ such that $xy,yx\in I$ for all $x\in I$ and $y\in A$. Show that if $\lambda\in \widehat{A}$ (the space ...
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### If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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### Construct a projection satisfying a certain property

Let $\cal G$ be a group of finite order $n$. For every prime divisor $p$ of $n$, construct a projection $P\in \cal N(G)$ such that $\operatorname{tr}_{\cal N(G)}(P)=1/p$. Here $\cal N(G)$ denotes ...
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### semifinite projections

Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$. ( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
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### Functoriality in $K$-theory for $C^*$-algebras or Banach algebras

I'm trying to clear up some confusion I'm having over how one establishes functoriality in $K$-theory for $C^*$-algebras or Banach algebras. Let me stick to $K_0$. Given a *-homomorphism (or bounded ...